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Theorem poimirlem32 33571
 Description: Lemma for poimir 33572, combining poimirlem28 33567, poimirlem30 33569, and poimirlem31 33570 to get Equation (1) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimir.i 𝐼 = ((0[,]1) ↑𝑚 (1...𝑁))
poimir.r 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
poimir.1 (𝜑𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅))
poimir.2 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
poimir.3 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 1)) → 0 ≤ ((𝐹𝑧)‘𝑛))
Assertion
Ref Expression
poimirlem32 (𝜑 → ∃𝑐𝐼𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅t 𝐼)(𝑐𝑣 → ∀𝑟 ∈ { ≤ , ≤ }∃𝑧𝑣 0𝑟((𝐹𝑧)‘𝑛)))
Distinct variable groups:   𝑧,𝑛,𝜑   𝑛,𝐹   𝑛,𝑁   𝜑,𝑧   𝑧,𝐹   𝑧,𝑁   𝑛,𝑐,𝑟,𝑣,𝑧,𝜑   𝐹,𝑐,𝑟,𝑣   𝐼,𝑐,𝑛,𝑟,𝑣,𝑧   𝑁,𝑐,𝑟,𝑣   𝑅,𝑐,𝑛,𝑟,𝑣,𝑧

Proof of Theorem poimirlem32
Dummy variables 𝑓 𝑖 𝑗 𝑘 𝑚 𝑝 𝑞 𝑠 𝑔 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . 7 (𝜑𝑁 ∈ ℕ)
21adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑁 ∈ ℕ)
3 oveq1 6697 . . . . . . . . . . . . . 14 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝𝑓 / ((1...𝑁) × {𝑘})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))
43fveq2d 6233 . . . . . . . . . . . . 13 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘}))))
54fveq1d 6231 . . . . . . . . . . . 12 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))
65breq2d 4697 . . . . . . . . . . 11 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏)))
7 fveq1 6228 . . . . . . . . . . . 12 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝𝑏) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏))
87neeq1d 2882 . . . . . . . . . . 11 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝𝑏) ≠ 0 ↔ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
96, 8anbi12d 747 . . . . . . . . . 10 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ((0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
109ralbidv 3015 . . . . . . . . 9 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
1110rabbidv 3220 . . . . . . . 8 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
1211uneq2d 3800 . . . . . . 7 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
1312supeq1d 8393 . . . . . 6 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
141nnnn0d 11389 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
15 0elfz 12475 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
1614, 15syl 17 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑁))
1716snssd 4372 . . . . . . . . 9 (𝜑 → {0} ⊆ (0...𝑁))
18 ssrab2 3720 . . . . . . . . . . 11 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ⊆ (1...𝑁)
19 fz1ssfz0 12474 . . . . . . . . . . 11 (1...𝑁) ⊆ (0...𝑁)
2018, 19sstri 3645 . . . . . . . . . 10 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ⊆ (0...𝑁)
2120a1i 11 . . . . . . . . 9 (𝜑 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ⊆ (0...𝑁))
2217, 21unssd 3822 . . . . . . . 8 (𝜑 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ (0...𝑁))
23 ltso 10156 . . . . . . . . 9 < Or ℝ
24 snfi 8079 . . . . . . . . . . 11 {0} ∈ Fin
25 fzfi 12811 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
26 rabfi 8226 . . . . . . . . . . . 12 ((1...𝑁) ∈ Fin → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ∈ Fin)
2725, 26ax-mp 5 . . . . . . . . . . 11 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ∈ Fin
28 unfi 8268 . . . . . . . . . . 11 (({0} ∈ Fin ∧ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ∈ Fin) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ∈ Fin)
2924, 27, 28mp2an 708 . . . . . . . . . 10 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ∈ Fin
30 c0ex 10072 . . . . . . . . . . . 12 0 ∈ V
3130snid 4241 . . . . . . . . . . 11 0 ∈ {0}
32 elun1 3813 . . . . . . . . . . 11 (0 ∈ {0} → 0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}))
33 ne0i 3954 . . . . . . . . . . 11 (0 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ≠ ∅)
3431, 32, 33mp2b 10 . . . . . . . . . 10 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ≠ ∅
35 0red 10079 . . . . . . . . . . . . 13 ((𝜑𝑁 ∈ ℕ) → 0 ∈ ℝ)
3635snssd 4372 . . . . . . . . . . . 12 ((𝜑𝑁 ∈ ℕ) → {0} ⊆ ℝ)
371, 36ax-mp 5 . . . . . . . . . . 11 {0} ⊆ ℝ
38 elfzelz 12380 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
3938ssriv 3640 . . . . . . . . . . . . 13 (1...𝑁) ⊆ ℤ
40 zssre 11422 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
4139, 40sstri 3645 . . . . . . . . . . . 12 (1...𝑁) ⊆ ℝ
4218, 41sstri 3645 . . . . . . . . . . 11 {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ⊆ ℝ
4337, 42unssi 3821 . . . . . . . . . 10 ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ ℝ
4429, 34, 433pm3.2i 1259 . . . . . . . . 9 (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ ℝ)
45 fisupcl 8416 . . . . . . . . 9 (( < Or ℝ ∧ (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ∈ Fin ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ≠ ∅ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ ℝ)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}))
4623, 44, 45mp2an 708 . . . . . . . 8 sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})
47 ssel 3630 . . . . . . . 8 (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ (0...𝑁) → (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ (0...𝑁)))
4822, 46, 47mpisyl 21 . . . . . . 7 (𝜑 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ (0...𝑁))
4948ad2antrr 762 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑝:(1...𝑁)⟶(0...𝑘)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ (0...𝑁))
50 elfznn 12408 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
51 nngt0 11087 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
5251adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) → 0 < 𝑛)
53 simpr 476 . . . . . . . . . . . . . 14 ((0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → (𝑝𝑏) ≠ 0)
5453ralimi 2981 . . . . . . . . . . . . 13 (∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑠)(𝑝𝑏) ≠ 0)
55 elfznn 12408 . . . . . . . . . . . . . 14 (𝑠 ∈ (1...𝑁) → 𝑠 ∈ ℕ)
56 nnre 11065 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
57 nnre 11065 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
58 lenlt 10154 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (𝑛𝑠 ↔ ¬ 𝑠 < 𝑛))
5956, 57, 58syl2an 493 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛𝑠 ↔ ¬ 𝑠 < 𝑛))
60 elfz1b 12447 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑠) ↔ (𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛𝑠))
6160biimpri 218 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ ∧ 𝑛𝑠) → 𝑛 ∈ (1...𝑠))
62613expia 1286 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑛𝑠𝑛 ∈ (1...𝑠)))
6359, 62sylbird 250 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛𝑛 ∈ (1...𝑠)))
64 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = 𝑛 → (𝑝𝑏) = (𝑝𝑛))
6564eqeq1d 2653 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑛 → ((𝑝𝑏) = 0 ↔ (𝑝𝑛) = 0))
6665rspcev 3340 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...𝑠) ∧ (𝑝𝑛) = 0) → ∃𝑏 ∈ (1...𝑠)(𝑝𝑏) = 0)
6766expcom 450 . . . . . . . . . . . . . . . . . 18 ((𝑝𝑛) = 0 → (𝑛 ∈ (1...𝑠) → ∃𝑏 ∈ (1...𝑠)(𝑝𝑏) = 0))
6863, 67sylan9 690 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑠 ∈ ℕ) ∧ (𝑝𝑛) = 0) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝𝑏) = 0))
6968an32s 863 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ∃𝑏 ∈ (1...𝑠)(𝑝𝑏) = 0))
70 nne 2827 . . . . . . . . . . . . . . . . . 18 (¬ (𝑝𝑏) ≠ 0 ↔ (𝑝𝑏) = 0)
7170rexbii 3070 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ (1...𝑠) ¬ (𝑝𝑏) ≠ 0 ↔ ∃𝑏 ∈ (1...𝑠)(𝑝𝑏) = 0)
72 rexnal 3024 . . . . . . . . . . . . . . . . 17 (∃𝑏 ∈ (1...𝑠) ¬ (𝑝𝑏) ≠ 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝𝑏) ≠ 0)
7371, 72bitr3i 266 . . . . . . . . . . . . . . . 16 (∃𝑏 ∈ (1...𝑠)(𝑝𝑏) = 0 ↔ ¬ ∀𝑏 ∈ (1...𝑠)(𝑝𝑏) ≠ 0)
7469, 73syl6ib 241 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (¬ 𝑠 < 𝑛 → ¬ ∀𝑏 ∈ (1...𝑠)(𝑝𝑏) ≠ 0))
7574con4d 114 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) ∧ 𝑠 ∈ ℕ) → (∀𝑏 ∈ (1...𝑠)(𝑝𝑏) ≠ 0 → 𝑠 < 𝑛))
7655, 75sylan2 490 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) ∧ 𝑠 ∈ (1...𝑁)) → (∀𝑏 ∈ (1...𝑠)(𝑝𝑏) ≠ 0 → 𝑠 < 𝑛))
7754, 76syl5 34 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) ∧ 𝑠 ∈ (1...𝑁)) → (∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → 𝑠 < 𝑛))
7877ralrimiva 2995 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) → ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → 𝑠 < 𝑛))
79 ralunb 3827 . . . . . . . . . . . 12 (∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑠 < 𝑛 ↔ (∀𝑠 ∈ {0}𝑠 < 𝑛 ∧ ∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}𝑠 < 𝑛))
80 breq1 4688 . . . . . . . . . . . . . 14 (𝑠 = 0 → (𝑠 < 𝑛 ↔ 0 < 𝑛))
8130, 80ralsn 4254 . . . . . . . . . . . . 13 (∀𝑠 ∈ {0}𝑠 < 𝑛 ↔ 0 < 𝑛)
82 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑎 = 𝑠 → (1...𝑎) = (1...𝑠))
8382raleqdv 3174 . . . . . . . . . . . . . 14 (𝑎 = 𝑠 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
8483ralrab 3401 . . . . . . . . . . . . 13 (∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}𝑠 < 𝑛 ↔ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → 𝑠 < 𝑛))
8581, 84anbi12i 733 . . . . . . . . . . . 12 ((∀𝑠 ∈ {0}𝑠 < 𝑛 ∧ ∀𝑠 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}𝑠 < 𝑛) ↔ (0 < 𝑛 ∧ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → 𝑠 < 𝑛)))
8679, 85bitri 264 . . . . . . . . . . 11 (∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑠 < 𝑛 ↔ (0 < 𝑛 ∧ ∀𝑠 ∈ (1...𝑁)(∀𝑏 ∈ (1...𝑠)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → 𝑠 < 𝑛)))
8752, 78, 86sylanbrc 699 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) → ∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑠 < 𝑛)
88 breq1 4688 . . . . . . . . . . 11 (𝑠 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) → (𝑠 < 𝑛 ↔ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < 𝑛))
8988rspcva 3338 . . . . . . . . . 10 ((sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ∧ ∀𝑠 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑠 < 𝑛) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
9046, 87, 89sylancr 696 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (𝑝𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
9150, 90sylan 487 . . . . . . . 8 ((𝑛 ∈ (1...𝑁) ∧ (𝑝𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
92913adant2 1100 . . . . . . 7 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 0) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
9392adantl 481 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 0)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < 𝑛)
9438zred 11520 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
95943ad2ant1 1102 . . . . . . . . . 10 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) → 𝑛 ∈ ℝ)
9695adantl 481 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → 𝑛 ∈ ℝ)
97 simpr1 1087 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → 𝑛 ∈ (1...𝑁))
98 simpll 805 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → 𝜑)
99 simplr 807 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑘 ∈ ℕ)
100 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℤ)
101100zred 11520 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0...𝑘) → 𝑖 ∈ ℝ)
102 nndivre 11094 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
103101, 102sylan 487 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
104 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0...𝑘) → 0 ≤ 𝑖)
105101, 104jca 553 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0...𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
106 nnrp 11880 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
107106rpregt0d 11916 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
108 divge0 10930 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
109105, 107, 108syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
110 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0...𝑘) → 𝑖𝑘)
111110adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖𝑘)
112101adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ)
113 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
114106adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
115112, 113, 114ledivmuld 11963 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
116 nncn 11066 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
117116mulid1d 10095 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
118117breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖𝑘))
119118adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖𝑘))
120115, 119bitrd 268 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖𝑘))
121111, 120mpbird 247 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
122 0re 10078 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ ℝ
123 1re 10077 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ
124122, 123elicc2i 12277 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
125103, 109, 121, 124syl3anbrc 1265 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0...𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1))
126125ancoms 468 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
127 elsni 4227 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
128127oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘))
129128eleq1d 2715 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1)))
130126, 129syl5ibrcom 237 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1)))
131130impr 648 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
13299, 131sylan 487 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) ∧ (𝑖 ∈ (0...𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
133 simprr 811 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → 𝑝:(1...𝑁)⟶(0...𝑘))
134 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑘 ∈ V
135134fconst 6129 . . . . . . . . . . . . . . . . 17 ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
136135a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
137 fzfid 12812 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (1...𝑁) ∈ Fin)
138 inidm 3855 . . . . . . . . . . . . . . . 16 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
139132, 133, 136, 137, 137, 138off 6954 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
140 poimir.i . . . . . . . . . . . . . . . . 17 𝐼 = ((0[,]1) ↑𝑚 (1...𝑁))
141140eleq2i 2722 . . . . . . . . . . . . . . . 16 ((𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)))
142 ovex 6718 . . . . . . . . . . . . . . . . 17 (0[,]1) ∈ V
143 ovex 6718 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
144142, 143elmap 7928 . . . . . . . . . . . . . . . 16 ((𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)) ↔ (𝑝𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
145141, 144bitri 264 . . . . . . . . . . . . . . 15 ((𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ (𝑝𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
146139, 145sylibr 224 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘))) → (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
1471463adantr3 1242 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
148 3anass 1059 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)))
149 ancom 465 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) ↔ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁)))
150148, 149bitri 264 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) ↔ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁)))
151 ffn 6083 . . . . . . . . . . . . . . . . . 18 (𝑝:(1...𝑁)⟶(0...𝑘) → 𝑝 Fn (1...𝑁))
152151ad2antrl 764 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → 𝑝 Fn (1...𝑁))
153 fnconstg 6131 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ V → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
154134, 153mp1i 13 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((1...𝑁) × {𝑘}) Fn (1...𝑁))
155 fzfid 12812 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → (1...𝑁) ∈ Fin)
156 simplrr 818 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑝𝑛) = 𝑘)
157134fvconst2 6510 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
158157adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {𝑘})‘𝑛) = 𝑘)
159152, 154, 155, 155, 138, 156, 158ofval 6948 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ ℕ) ∧ (𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘))
160159anasss 680 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ ((𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘))
161150, 160sylan2b 491 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = (𝑘 / 𝑘))
162 nnne0 11091 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
163116, 162dividd 10837 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑘 / 𝑘) = 1)
164163ad2antlr 763 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → (𝑘 / 𝑘) = 1)
165161, 164eqtrd 2685 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)
166 ovex 6718 . . . . . . . . . . . . . 14 (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ V
167 eleq1 2718 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → (𝑧𝐼 ↔ (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼))
168 fveq1 6228 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → (𝑧𝑛) = ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛))
169168eqeq1d 2653 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → ((𝑧𝑛) = 1 ↔ ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1))
170167, 1693anbi23d 1442 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 1) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)))
171170anbi2d 740 . . . . . . . . . . . . . . 15 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 1)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1))))
172 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → (𝐹𝑧) = (𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘}))))
173172fveq1d 6231 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → ((𝐹𝑧)‘𝑛) = ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
174173breq2d 4697 . . . . . . . . . . . . . . 15 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → (0 ≤ ((𝐹𝑧)‘𝑛) ↔ 0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
175171, 174imbi12d 333 . . . . . . . . . . . . . 14 (𝑧 = (𝑝𝑓 / ((1...𝑁) × {𝑘})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 1)) → 0 ≤ ((𝐹𝑧)‘𝑛)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) → 0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))))
176 poimir.3 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 1)) → 0 ≤ ((𝐹𝑧)‘𝑛))
177166, 175, 176vtocl 3290 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ∧ ((𝑝𝑓 / ((1...𝑁) × {𝑘}))‘𝑛) = 1)) → 0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
17898, 97, 147, 165, 177syl13anc 1368 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → 0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
179 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
180 simp3 1083 . . . . . . . . . . . . 13 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘) → (𝑝𝑛) = 𝑘)
181 neeq1 2885 . . . . . . . . . . . . . . 15 ((𝑝𝑛) = 𝑘 → ((𝑝𝑛) ≠ 0 ↔ 𝑘 ≠ 0))
182162, 181syl5ibrcom 237 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → ((𝑝𝑛) = 𝑘 → (𝑝𝑛) ≠ 0))
183182imp 444 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ (𝑝𝑛) = 𝑘) → (𝑝𝑛) ≠ 0)
184179, 180, 183syl2an 493 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → (𝑝𝑛) ≠ 0)
185 vex 3234 . . . . . . . . . . . . 13 𝑛 ∈ V
186 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑏 = 𝑛 → ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛))
187186breq2d 4697 . . . . . . . . . . . . . 14 (𝑏 = 𝑛 → (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)))
18864neeq1d 2882 . . . . . . . . . . . . . 14 (𝑏 = 𝑛 → ((𝑝𝑏) ≠ 0 ↔ (𝑝𝑛) ≠ 0))
189187, 188anbi12d 747 . . . . . . . . . . . . 13 (𝑏 = 𝑛 → ((0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑝𝑛) ≠ 0)))
190185, 189ralsn 4254 . . . . . . . . . . . 12 (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑛) ∧ (𝑝𝑛) ≠ 0))
191178, 184, 190sylanbrc 699 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))
19238zcnd 11521 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
193 1cnd 10094 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → 1 ∈ ℂ)
194192, 193subeq0ad 10440 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
195194biimpcd 239 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → 𝑛 = 1))
196 1z 11445 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
197 fzsn 12421 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ ℤ → (1...1) = {1})
198196, 197ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (1...1) = {1}
199 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (1...𝑛) = (1...1))
200 sneq 4220 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → {𝑛} = {1})
201198, 199, 2003eqtr4a 2711 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → (1...𝑛) = {𝑛})
202201raleqdv 3174 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
203202biimprd 238 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
204195, 203syl6 35 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) = 0 → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
205 ralun 3828 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))
206 npcan1 10493 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
207192, 206syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) = 𝑛)
208 elfzuz 12376 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘1))
209207, 208eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
210 peano2zm 11458 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
211 uzid 11740 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
212 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
21338, 210, 211, 2124syl 19 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
214207, 213eqeltrrd 2731 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
215 fzsplit2 12404 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
216209, 214, 215syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
217207oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
218 fzsn 12421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
21938, 218syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) → (𝑛...𝑛) = {𝑛})
220217, 219eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
221220uneq2d 3800 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
222216, 221eqtrd 2685 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑁) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
223222raleqdv 3174 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ ∀𝑏 ∈ ((1...(𝑛 − 1)) ∪ {𝑛})(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
224205, 223syl5ibr 236 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
225224expd 451 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
226225com12 32 . . . . . . . . . . . . . . . . 17 (∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
227226adantl 481 . . . . . . . . . . . . . . . 16 (((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
228204, 227jaoi 393 . . . . . . . . . . . . . . 15 (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))) → (𝑛 ∈ (1...𝑁) → (∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) → ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
229228imdistand 728 . . . . . . . . . . . . . 14 (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))) → ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
230229com12 32 . . . . . . . . . . . . 13 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → (((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))) → (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
231 elun 3786 . . . . . . . . . . . . . 14 ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ↔ ((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}))
232 ovex 6718 . . . . . . . . . . . . . . . 16 (𝑛 − 1) ∈ V
233232elsn 4225 . . . . . . . . . . . . . . 15 ((𝑛 − 1) ∈ {0} ↔ (𝑛 − 1) = 0)
234 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑛 − 1) → (1...𝑎) = (1...(𝑛 − 1)))
235234raleqdv 3174 . . . . . . . . . . . . . . . 16 (𝑎 = (𝑛 − 1) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
236235elrab 3396 . . . . . . . . . . . . . . 15 ((𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ↔ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
237233, 236orbi12i 542 . . . . . . . . . . . . . 14 (((𝑛 − 1) ∈ {0} ∨ (𝑛 − 1) ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
238231, 237bitri 264 . . . . . . . . . . . . 13 ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ↔ ((𝑛 − 1) = 0 ∨ ((𝑛 − 1) ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...(𝑛 − 1))(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0))))
239 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑎 = 𝑛 → (1...𝑎) = (1...𝑛))
240239raleqdv 3174 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
241240elrab 3396 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} ↔ (𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ (1...𝑛)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)))
242230, 238, 2413imtr4g 285 . . . . . . . . . . . 12 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → 𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}))
243 elun2 3814 . . . . . . . . . . . 12 (𝑛 ∈ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)} → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}))
244242, 243syl6 35 . . . . . . . . . . 11 ((𝑛 ∈ (1...𝑁) ∧ ∀𝑏 ∈ {𝑛} (0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})))
24597, 191, 244syl2anc 694 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → 𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})))
246 fimaxre2 11007 . . . . . . . . . . . . 13 ((({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ∈ Fin) → ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑗𝑖)
24743, 29, 246mp2an 708 . . . . . . . . . . . 12 𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑗𝑖
24843, 34, 2473pm3.2i 1259 . . . . . . . . . . 11 (({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ⊆ ℝ ∧ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ≠ ∅ ∧ ∃𝑖 ∈ ℝ ∀𝑗 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})𝑗𝑖)
249248suprubii 11036 . . . . . . . . . 10 (𝑛 ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ))
250245, 249syl6 35 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → 𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
251 ltm1 10901 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
252 peano2rem 10386 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
25343, 46sselii 3633 . . . . . . . . . . . 12 sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ℝ
254 ltletr 10167 . . . . . . . . . . . 12 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ℝ) → (((𝑛 − 1) < 𝑛𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
255253, 254mp3an3 1453 . . . . . . . . . . 11 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → (((𝑛 − 1) < 𝑛𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
256252, 255mpancom 704 . . . . . . . . . 10 (𝑛 ∈ ℝ → (((𝑛 − 1) < 𝑛𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
257251, 256mpand 711 . . . . . . . . 9 (𝑛 ∈ ℝ → (𝑛 ≤ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
25896, 250, 257sylsyld 61 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
259253ltnri 10184 . . . . . . . . . 10 ¬ sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )
260 breq1 4688 . . . . . . . . . 10 (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ↔ (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < )))
261259, 260mtbii 315 . . . . . . . . 9 (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → ¬ (𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ))
262261necon2ai 2852 . . . . . . . 8 ((𝑛 − 1) < sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1))
263258, 262syl6 35 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → ((𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1)))
264 eleq1 2718 . . . . . . . . 9 (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) ↔ (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)})))
26546, 264mpbii 223 . . . . . . . 8 (sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) = (𝑛 − 1) → (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}))
266265necon3bi 2849 . . . . . . 7 (¬ (𝑛 − 1) ∈ ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1))
267263, 266pm2.61d1 171 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝑘) ∧ (𝑝𝑛) = 𝑘)) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑝𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑝𝑏) ≠ 0)}), ℝ, < ) ≠ (𝑛 − 1))
2682, 13, 49, 93, 267, 179poimirlem28 33567 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ∃𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
269 nn0ex 11336 . . . . . . . . . . . 12 0 ∈ V
270 fzo0ssnn0 12588 . . . . . . . . . . . 12 (0..^𝑘) ⊆ ℕ0
271 mapss 7942 . . . . . . . . . . . 12 ((ℕ0 ∈ V ∧ (0..^𝑘) ⊆ ℕ0) → ((0..^𝑘) ↑𝑚 (1...𝑁)) ⊆ (ℕ0𝑚 (1...𝑁)))
272269, 270, 271mp2an 708 . . . . . . . . . . 11 ((0..^𝑘) ↑𝑚 (1...𝑁)) ⊆ (ℕ0𝑚 (1...𝑁))
273 xpss1 5161 . . . . . . . . . . 11 (((0..^𝑘) ↑𝑚 (1...𝑁)) ⊆ (ℕ0𝑚 (1...𝑁)) → (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
274272, 273ax-mp 5 . . . . . . . . . 10 (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
275274sseli 3632 . . . . . . . . 9 (𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
276 xp1st 7242 . . . . . . . . . 10 (𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑠) ∈ ((0..^𝑘) ↑𝑚 (1...𝑁)))
277 elmapi 7921 . . . . . . . . . 10 ((1st𝑠) ∈ ((0..^𝑘) ↑𝑚 (1...𝑁)) → (1st𝑠):(1...𝑁)⟶(0..^𝑘))
278 frn 6091 . . . . . . . . . 10 ((1st𝑠):(1...𝑁)⟶(0..^𝑘) → ran (1st𝑠) ⊆ (0..^𝑘))
279276, 277, 2783syl 18 . . . . . . . . 9 (𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ran (1st𝑠) ⊆ (0..^𝑘))
280275, 279jca 553 . . . . . . . 8 (𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1st𝑠) ⊆ (0..^𝑘)))
281280anim1i 591 . . . . . . 7 ((𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ((𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1st𝑠) ⊆ (0..^𝑘)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
282 anass 682 . . . . . . 7 (((𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ran (1st𝑠) ⊆ (0..^𝑘)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) ↔ (𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
283281, 282sylib 208 . . . . . 6 ((𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → (𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
284283reximi2 3039 . . . . 5 (∃𝑠 ∈ (((0..^𝑘) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
285268, 284syl 17 . . . 4 ((𝜑𝑘 ∈ ℕ) → ∃𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
286285ralrimiva 2995 . . 3 (𝜑 → ∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
287 nnex 11064 . . . 4 ℕ ∈ V
288143, 269ixpconst 7960 . . . . . . 7 X𝑛 ∈ (1...𝑁)ℕ0 = (ℕ0𝑚 (1...𝑁))
289 omelon 8581 . . . . . . . . . 10 ω ∈ On
290 nn0ennn 12818 . . . . . . . . . . 11 0 ≈ ℕ
291 nnenom 12819 . . . . . . . . . . 11 ℕ ≈ ω
292290, 291entr2i 8052 . . . . . . . . . 10 ω ≈ ℕ0
293 isnumi 8810 . . . . . . . . . 10 ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card)
294289, 292, 293mp2an 708 . . . . . . . . 9 0 ∈ dom card
295294rgenw 2953 . . . . . . . 8 𝑛 ∈ (1...𝑁)ℕ0 ∈ dom card
296 finixpnum 33524 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)ℕ0 ∈ dom card) → X𝑛 ∈ (1...𝑁)ℕ0 ∈ dom card)
29725, 295, 296mp2an 708 . . . . . . 7 X𝑛 ∈ (1...𝑁)ℕ0 ∈ dom card
298288, 297eqeltrri 2727 . . . . . 6 (ℕ0𝑚 (1...𝑁)) ∈ dom card
299143, 143mapval 7911 . . . . . . . . 9 ((1...𝑁) ↑𝑚 (1...𝑁)) = {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
300 mapfi 8303 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin)
30125, 25, 300mp2an 708 . . . . . . . . 9 ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin
302299, 301eqeltrri 2727 . . . . . . . 8 {𝑓𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin
303 f1of 6175 . . . . . . . . 9 (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
304303ss2abi 3707 . . . . . . . 8 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
305 ssfi 8221 . . . . . . . 8 (({𝑓𝑓:(1...𝑁)⟶(1...𝑁)} ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}) → {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
306302, 304, 305mp2an 708 . . . . . . 7 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
307 finnum 8812 . . . . . . 7 ({𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin → {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card)
308306, 307ax-mp 5 . . . . . 6 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card
309 xpnum 8815 . . . . . 6 (((ℕ0𝑚 (1...𝑁)) ∈ dom card ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ dom card) → ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card)
310298, 308, 309mp2an 708 . . . . 5 ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card
311 ssrab2 3720 . . . . . . . 8 {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
312311rgenw 2953 . . . . . . 7 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
313 ss2iun 4568 . . . . . . 7 (∀𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ 𝑘 ∈ ℕ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
314312, 313ax-mp 5 . . . . . 6 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ 𝑘 ∈ ℕ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
315 1nn 11069 . . . . . . 7 1 ∈ ℕ
316 ne0i 3954 . . . . . . 7 (1 ∈ ℕ → ℕ ≠ ∅)
317 iunconst 4561 . . . . . . 7 (ℕ ≠ ∅ → 𝑘 ∈ ℕ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
318315, 316, 317mp2b 10 . . . . . 6 𝑘 ∈ ℕ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
319314, 318sseqtri 3670 . . . . 5 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
320 ssnum 8900 . . . . 5 ((((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ dom card ∧ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ⊆ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom card)
321310, 319, 320mp2an 708 . . . 4 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom card
322 fveq2 6229 . . . . . . . 8 (𝑠 = (𝑔𝑘) → (1st𝑠) = (1st ‘(𝑔𝑘)))
323322rneqd 5385 . . . . . . 7 (𝑠 = (𝑔𝑘) → ran (1st𝑠) = ran (1st ‘(𝑔𝑘)))
324323sseq1d 3665 . . . . . 6 (𝑠 = (𝑔𝑘) → (ran (1st𝑠) ⊆ (0..^𝑘) ↔ ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘)))
325 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑔𝑘) → (2nd𝑠) = (2nd ‘(𝑔𝑘)))
326325imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (𝑔𝑘) → ((2nd𝑠) “ (1...𝑗)) = ((2nd ‘(𝑔𝑘)) “ (1...𝑗)))
327326xpeq1d 5172 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = (𝑔𝑘) → (((2nd𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}))
328325imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (𝑔𝑘) → ((2nd𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)))
329328xpeq1d 5172 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = (𝑔𝑘) → (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))
330327, 329uneq12d 3801 . . . . . . . . . . . . . . . . . . 19 (𝑠 = (𝑔𝑘) → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))
331322, 330oveq12d 6708 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑔𝑘) → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))))
332331oveq1d 6705 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑔𝑘) → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})) = (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))
333332fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑔𝑘) → (𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘}))))
334333fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑠 = (𝑔𝑘) → ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏))
335334breq2d 4697 . . . . . . . . . . . . . 14 (𝑠 = (𝑔𝑘) → (0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏)))
336331fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑠 = (𝑔𝑘) → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏))
337336neeq1d 2882 . . . . . . . . . . . . . 14 (𝑠 = (𝑔𝑘) → ((((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
338335, 337anbi12d 747 . . . . . . . . . . . . 13 (𝑠 = (𝑔𝑘) → ((0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
339338ralbidv 3015 . . . . . . . . . . . 12 (𝑠 = (𝑔𝑘) → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
340339rabbidv 3220 . . . . . . . . . . 11 (𝑠 = (𝑔𝑘) → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
341340uneq2d 3800 . . . . . . . . . 10 (𝑠 = (𝑔𝑘) → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
342341supeq1d 8393 . . . . . . . . 9 (𝑠 = (𝑔𝑘) → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
343342eqeq2d 2661 . . . . . . . 8 (𝑠 = (𝑔𝑘) → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
344343rexbidv 3081 . . . . . . 7 (𝑠 = (𝑔𝑘) → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
345344ralbidv 3015 . . . . . 6 (𝑠 = (𝑔𝑘) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
346324, 345anbi12d 747 . . . . 5 (𝑠 = (𝑔𝑘) → ((ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) ↔ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
347346ac6num 9339 . . . 4 ((ℕ ∈ V ∧ 𝑘 ∈ ℕ {𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))} ∈ dom card ∧ ∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∃𝑔(𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
348287, 321, 347mp3an12 1454 . . 3 (∀𝑘 ∈ ℕ ∃𝑠 ∈ ((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(ran (1st𝑠) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∃𝑔(𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
349286, 348syl 17 . 2 (𝜑 → ∃𝑔(𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))))
3501ad2antrr 762 . . . 4 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝑁 ∈ ℕ)
351 poimir.r . . . 4 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
352 poimir.1 . . . . 5 (𝜑𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅))
353352ad2antrr 762 . . . 4 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝐹 ∈ ((𝑅t 𝐼) Cn 𝑅))
354 eqid 2651 . . . 4 ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑛) = ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑛)
355 simplr 807 . . . 4 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
356 simpl 472 . . . . . . 7 ((ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘))
357356ralimi 2981 . . . . . 6 (∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∀𝑘 ∈ ℕ ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘))
358357adantl 481 . . . . 5 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∀𝑘 ∈ ℕ ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘))
359 fveq2 6229 . . . . . . . . 9 (𝑘 = 𝑝 → (𝑔𝑘) = (𝑔𝑝))
360359fveq2d 6233 . . . . . . . 8 (𝑘 = 𝑝 → (1st ‘(𝑔𝑘)) = (1st ‘(𝑔𝑝)))
361360rneqd 5385 . . . . . . 7 (𝑘 = 𝑝 → ran (1st ‘(𝑔𝑘)) = ran (1st ‘(𝑔𝑝)))
362 oveq2 6698 . . . . . . 7 (𝑘 = 𝑝 → (0..^𝑘) = (0..^𝑝))
363361, 362sseq12d 3667 . . . . . 6 (𝑘 = 𝑝 → (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ↔ ran (1st ‘(𝑔𝑝)) ⊆ (0..^𝑝)))
364363rspccva 3339 . . . . 5 ((∀𝑘 ∈ ℕ ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ 𝑝 ∈ ℕ) → ran (1st ‘(𝑔𝑝)) ⊆ (0..^𝑝))
365358, 364sylan 487 . . . 4 ((((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ 𝑝 ∈ ℕ) → ran (1st ‘(𝑔𝑝)) ⊆ (0..^𝑝))
366 simpll 805 . . . . . 6 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → 𝜑)
367 poimir.2 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
368366, 367sylan 487 . . . . 5 ((((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧𝐼 ∧ (𝑧𝑛) = 0)) → ((𝐹𝑧)‘𝑛) ≤ 0)
369 eqid 2651 . . . . 5 ((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) = ((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
370 simpr 476 . . . . . . . 8 ((ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
371370ralimi 2981 . . . . . . 7 (∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )) → ∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
372371adantl 481 . . . . . 6 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
373359fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑝 → (2nd ‘(𝑔𝑘)) = (2nd ‘(𝑔𝑝)))
374373imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑝 → ((2nd ‘(𝑔𝑘)) “ (1...𝑗)) = ((2nd ‘(𝑔𝑝)) “ (1...𝑗)))
375374xpeq1d 5172 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑝 → (((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) = (((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}))
376373imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑝 → ((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)))
377376xpeq1d 5172 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑝 → (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))
378375, 377uneq12d 3801 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑝 → ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))
379360, 378oveq12d 6708 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑝 → ((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))))
380 sneq 4220 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑝 → {𝑘} = {𝑝})
381380xpeq2d 5173 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑝 → ((1...𝑁) × {𝑘}) = ((1...𝑁) × {𝑝}))
382379, 381oveq12d 6708 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑝 → (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})) = (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))
383382fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑝 → (𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝}))))
384383fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑘 = 𝑝 → ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) = ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏))
385384breq2d 4697 . . . . . . . . . . . . . 14 (𝑘 = 𝑝 → (0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏)))
386379fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑘 = 𝑝 → (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏))
387386neeq1d 2882 . . . . . . . . . . . . . 14 (𝑘 = 𝑝 → ((((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
388385, 387anbi12d 747 . . . . . . . . . . . . 13 (𝑘 = 𝑝 → ((0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
389388ralbidv 3015 . . . . . . . . . . . 12 (𝑘 = 𝑝 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
390389rabbidv 3220 . . . . . . . . . . 11 (𝑘 = 𝑝 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
391390uneq2d 3800 . . . . . . . . . 10 (𝑘 = 𝑝 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
392391supeq1d 8393 . . . . . . . . 9 (𝑘 = 𝑝 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
393392eqeq2d 2661 . . . . . . . 8 (𝑘 = 𝑝 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
394393rexbidv 3081 . . . . . . 7 (𝑘 = 𝑝 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
395 eqeq1 2655 . . . . . . . . 9 (𝑖 = 𝑞 → (𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
396395rexbidv 3081 . . . . . . . 8 (𝑖 = 𝑞 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑗 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
397 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑚 → (1...𝑗) = (1...𝑚))
398397imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑚 → ((2nd ‘(𝑔𝑝)) “ (1...𝑗)) = ((2nd ‘(𝑔𝑝)) “ (1...𝑚)))
399398xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑚 → (((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) = (((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}))
400 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑚 → (𝑗 + 1) = (𝑚 + 1))
401400oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑚 → ((𝑗 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
402401imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑚 → ((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)))
403402xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑚 → (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))
404399, 403uneq12d 3801 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑚 → ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))
405404oveq2d 6706 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑚 → ((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))))
406405oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑚 → (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})) = (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))
407406fveq2d 6233 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑚 → (𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝}))) = (𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝}))))
408407fveq1d 6231 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) = ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏))
409408breq2d 4697 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ↔ 0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏)))
410405fveq1d 6231 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) = (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏))
411410neeq1d 2882 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → ((((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0 ↔ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0))
412409, 411anbi12d 747 . . . . . . . . . . . . . 14 (𝑗 = 𝑚 → ((0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ (0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
413412ralbidv 3015 . . . . . . . . . . . . 13 (𝑗 = 𝑚 → (∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0) ↔ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)))
414413rabbidv 3220 . . . . . . . . . . . 12 (𝑗 = 𝑚 → {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)} = {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)})
415414uneq2d 3800 . . . . . . . . . . 11 (𝑗 = 𝑚 → ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}) = ({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}))
416415supeq1d 8393 . . . . . . . . . 10 (𝑗 = 𝑚 → sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
417416eqeq2d 2661 . . . . . . . . 9 (𝑗 = 𝑚 → (𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ 𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
418417cbvrexv 3202 . . . . . . . 8 (∃𝑗 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
419396, 418syl6bb 276 . . . . . . 7 (𝑖 = 𝑞 → (∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) ↔ ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
420394, 419rspc2v 3353 . . . . . 6 ((𝑝 ∈ ℕ ∧ 𝑞 ∈ (0...𝑁)) → (∀𝑘 ∈ ℕ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ) → ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))
421372, 420mpan9 485 . . . . 5 ((((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑝 ∈ ℕ ∧ 𝑞 ∈ (0...𝑁))) → ∃𝑚 ∈ (0...𝑁)𝑞 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑏) ∧ (((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))
422350, 140, 351, 353, 368, 369, 355, 365, 421poimirlem31 33570 . . . 4 ((((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) ∧ (𝑝 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ≤ })) → ∃𝑚 ∈ (0...𝑁)0𝑟((𝐹‘(((1st ‘(𝑔𝑝)) ∘𝑓 + ((((2nd ‘(𝑔𝑝)) “ (1...𝑚)) × {1}) ∪ (((2nd ‘(𝑔𝑝)) “ ((𝑚 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑝})))‘𝑛))
423350, 140, 351, 353, 354, 355, 365, 422poimirlem30 33569 . . 3 (((𝜑𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < ))) → ∃𝑐𝐼𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅t 𝐼)(𝑐𝑣 → ∀𝑟 ∈ { ≤ , ≤ }∃𝑧𝑣 0𝑟((𝐹𝑧)‘𝑛)))
424423anasss 680 . 2 ((𝜑 ∧ (𝑔:ℕ⟶((ℕ0𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑘 ∈ ℕ (ran (1st ‘(𝑔𝑘)) ⊆ (0..^𝑘) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (((1st ‘(𝑔𝑘)) ∘𝑓 + ((((2nd ‘(𝑔𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝑔𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑏) ≠ 0)}), ℝ, < )))) → ∃𝑐𝐼𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅t 𝐼)(𝑐𝑣 → ∀𝑟 ∈ { ≤ , ≤ }∃𝑧𝑣 0𝑟((𝐹𝑧)‘𝑛)))
425349, 424exlimddv 1903 1 (𝜑 → ∃𝑐𝐼𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅t 𝐼)(𝑐𝑣 → ∀𝑟 ∈ { ≤ , ≤ }∃𝑧𝑣 0𝑟((𝐹𝑧)‘𝑛)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  {csn 4210  {cpr 4212  ∪ ciun 4552   class class class wbr 4685   Or wor 5063   × cxp 5141  ◡ccnv 5142  dom cdm 5143  ran crn 5144   “ cima 5146  Oncon0 5761   Fn wfn 5921  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937  ωcom 7107  1st c1st 7208  2nd c2nd 7209   ↑𝑚 cmap 7899  Xcixp 7950   ≈ cen 7994  Fincfn 7997  supcsup 8387  cardccrd 8799  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   ≤ cle 10113   − cmin 10304   / cdiv 10722  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ℤ≥cuz 11725  ℝ+crp 11870  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504   ↾t crest 16128  topGenctg 16145  ∏tcpt 16146   Cn ccn 21076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-dvds 15028  df-rest 16130  df-topgen 16151  df-pt 16152  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-lp 20988  df-cn 21079  df-cnp 21080  df-t1 21166  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-hmph 21607  df-ii 22727 This theorem is referenced by:  poimir  33572
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