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Theorem poimirlem1 33381
Description: Lemma for poimir 33413- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem2.1 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
poimirlem2.2 (𝜑𝑇:(1...𝑁)⟶ℤ)
poimirlem2.3 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem1.4 (𝜑𝑀 ∈ (1...(𝑁 − 1)))
Assertion
Ref Expression
poimirlem1 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
Distinct variable groups:   𝑗,𝑛,𝑦,𝜑   𝑗,𝐹,𝑛,𝑦   𝑗,𝑀,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝑈,𝑗,𝑛,𝑦

Proof of Theorem poimirlem1
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 poimirlem2.3 . . . . 5 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
2 f1of 6124 . . . . 5 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)⟶(1...𝑁))
31, 2syl 17 . . . 4 (𝜑𝑈:(1...𝑁)⟶(1...𝑁))
4 poimir.0 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
54nncnd 11021 . . . . . . . 8 (𝜑𝑁 ∈ ℂ)
6 npcan1 10440 . . . . . . . 8 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
75, 6syl 17 . . . . . . 7 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
84nnzd 11466 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
9 peano2zm 11405 . . . . . . . 8 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
10 uzid 11687 . . . . . . . 8 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
11 peano2uz 11726 . . . . . . . 8 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
128, 9, 10, 114syl 19 . . . . . . 7 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
137, 12eqeltrrd 2700 . . . . . 6 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
14 fzss2 12366 . . . . . 6 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
1513, 14syl 17 . . . . 5 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
16 poimirlem1.4 . . . . 5 (𝜑𝑀 ∈ (1...(𝑁 − 1)))
1715, 16sseldd 3596 . . . 4 (𝜑𝑀 ∈ (1...𝑁))
183, 17ffvelrnd 6346 . . 3 (𝜑 → (𝑈𝑀) ∈ (1...𝑁))
19 fzp1elp1 12379 . . . . . 6 (𝑀 ∈ (1...(𝑁 − 1)) → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1)))
2016, 19syl 17 . . . . 5 (𝜑 → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1)))
217oveq2d 6651 . . . . 5 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
2220, 21eleqtrd 2701 . . . 4 (𝜑 → (𝑀 + 1) ∈ (1...𝑁))
233, 22ffvelrnd 6346 . . 3 (𝜑 → (𝑈‘(𝑀 + 1)) ∈ (1...𝑁))
24 imassrn 5465 . . . . . . . . . 10 (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ ran 𝑈
25 frn 6040 . . . . . . . . . . 11 (𝑈:(1...𝑁)⟶(1...𝑁) → ran 𝑈 ⊆ (1...𝑁))
261, 2, 253syl 18 . . . . . . . . . 10 (𝜑 → ran 𝑈 ⊆ (1...𝑁))
2724, 26syl5ss 3606 . . . . . . . . 9 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (1...𝑁))
2827sselda 3595 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (1...𝑁))
29 poimirlem2.2 . . . . . . . . . . 11 (𝜑𝑇:(1...𝑁)⟶ℤ)
3029ffvelrnda 6345 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ∈ ℤ)
3130zred 11467 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ∈ ℝ)
3231ltp1d 10939 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) < ((𝑇𝑛) + 1))
3331, 32ltned 10158 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ≠ ((𝑇𝑛) + 1))
3428, 33syldan 487 . . . . . . 7 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (𝑇𝑛) ≠ ((𝑇𝑛) + 1))
35 poimirlem2.1 . . . . . . . . . . 11 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
36 breq1 4647 . . . . . . . . . . . . . . 15 (𝑦 = (𝑀 − 1) → (𝑦 < 𝑀 ↔ (𝑀 − 1) < 𝑀))
37 id 22 . . . . . . . . . . . . . . 15 (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1))
3836, 37ifbieq1d 4100 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 1) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)))
39 elfzelz 12327 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ ℤ)
4016, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℤ)
4140zred 11467 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℝ)
4241ltm1d 10941 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀 − 1) < 𝑀)
4342iftrued 4085 . . . . . . . . . . . . . 14 (𝜑 → if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1))
4438, 43sylan9eqr 2676 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 − 1))
4544csbeq1d 3533 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑀 − 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
468, 9syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
47 elfzm1b 12402 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))))
4840, 46, 47syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))))
4916, 48mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1)))
50 oveq2 6643 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
5150imaeq2d 5454 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 − 1))))
5251xpeq1d 5128 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1}))
5352adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1}))
54 oveq1 6642 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
5540zcnd 11468 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℂ)
56 npcan1 10440 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
5755, 56syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
5854, 57sylan9eqr 2676 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
5958oveq1d 6650 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
6059imaeq2d 5454 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑀 − 1)) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (𝑀...𝑁)))
6160xpeq1d 5128 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (𝑀...𝑁)) × {0}))
6253, 61uneq12d 3760 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = (𝑀 − 1)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))
6362oveq2d 6651 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6449, 63csbied 3553 . . . . . . . . . . . . 13 (𝜑(𝑀 − 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6564adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → (𝑀 − 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6645, 65eqtrd 2654 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6746zcnd 11468 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℂ)
68 npcan1 10440 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℂ → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1))
6967, 68syl 17 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1))
70 peano2zm 11405 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℤ → ((𝑁 − 1) − 1) ∈ ℤ)
71 uzid 11687 . . . . . . . . . . . . . . 15 (((𝑁 − 1) − 1) ∈ ℤ → ((𝑁 − 1) − 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
72 peano2uz 11726 . . . . . . . . . . . . . . 15 (((𝑁 − 1) − 1) ∈ (ℤ‘((𝑁 − 1) − 1)) → (((𝑁 − 1) − 1) + 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
7346, 70, 71, 724syl 19 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) − 1) + 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
7469, 73eqeltrrd 2700 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
75 fzss2 12366 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘((𝑁 − 1) − 1)) → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1)))
7674, 75syl 17 . . . . . . . . . . . 12 (𝜑 → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1)))
7776, 49sseldd 3596 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
78 ovexd 6665 . . . . . . . . . . 11 (𝜑 → (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))) ∈ V)
7935, 66, 77, 78fvmptd 6275 . . . . . . . . . 10 (𝜑 → (𝐹‘(𝑀 − 1)) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
8079fveq1d 6180 . . . . . . . . 9 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛))
8180adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛))
82 ffn 6032 . . . . . . . . . . . 12 (𝑇:(1...𝑁)⟶ℤ → 𝑇 Fn (1...𝑁))
8329, 82syl 17 . . . . . . . . . . 11 (𝜑𝑇 Fn (1...𝑁))
8483adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑇 Fn (1...𝑁))
85 1ex 10020 . . . . . . . . . . . . . . 15 1 ∈ V
86 fnconstg 6080 . . . . . . . . . . . . . . 15 (1 ∈ V → ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))))
8785, 86ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1)))
88 c0ex 10019 . . . . . . . . . . . . . . 15 0 ∈ V
89 fnconstg 6080 . . . . . . . . . . . . . . 15 (0 ∈ V → ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)))
9088, 89ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))
9187, 90pm3.2i 471 . . . . . . . . . . . . 13 (((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)))
92 dff1o3 6130 . . . . . . . . . . . . . . . 16 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun 𝑈))
9392simprbi 480 . . . . . . . . . . . . . . 15 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun 𝑈)
94 imain 5962 . . . . . . . . . . . . . . 15 (Fun 𝑈 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))))
951, 93, 943syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))))
96 fzdisj 12353 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
9742, 96syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
9897imaeq2d 5454 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (𝑈 “ ∅))
99 ima0 5469 . . . . . . . . . . . . . . 15 (𝑈 “ ∅) = ∅
10098, 99syl6eq 2670 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
10195, 100eqtr3d 2656 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅)
102 fnun 5985 . . . . . . . . . . . . 13 (((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))) ∧ ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
10391, 101, 102sylancr 694 . . . . . . . . . . . 12 (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
104 elfzuz 12323 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (ℤ‘1))
10516, 104syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘1))
10657, 105eqeltrd 2699 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘1))
107 peano2zm 11405 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
108 uzid 11687 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
109 peano2uz 11726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
11040, 107, 108, 1094syl 19 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
11157, 110eqeltrrd 2700 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ‘(𝑀 − 1)))
112 peano2uz 11726 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘(𝑀 − 1)) → (𝑀 + 1) ∈ (ℤ‘(𝑀 − 1)))
113 uzss 11693 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 1) ∈ (ℤ‘(𝑀 − 1)) → (ℤ‘(𝑀 + 1)) ⊆ (ℤ‘(𝑀 − 1)))
114111, 112, 1133syl 18 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ (ℤ‘(𝑀 − 1)))
115 elfzuz3 12324 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (1...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑀))
116 eluzp1p1 11698 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ (ℤ𝑀) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
11716, 115, 1163syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
1187, 117eqeltrrd 2700 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
119114, 118sseldd 3596 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘(𝑀 − 1)))
120 fzsplit2 12351 . . . . . . . . . . . . . . . . . 18 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
121106, 119, 120syl2anc 692 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
12257oveq1d 6650 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
123122uneq2d 3759 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
124121, 123eqtrd 2654 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
125124imaeq2d 5454 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
126 imaundi 5533 . . . . . . . . . . . . . . 15 (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁)))
127125, 126syl6eq 2670 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
128 f1ofo 6131 . . . . . . . . . . . . . . 15 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
129 foima 6107 . . . . . . . . . . . . . . 15 (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
1301, 128, 1293syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
131127, 130eqtr3d 2656 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) = (1...𝑁))
132131fneq2d 5970 . . . . . . . . . . . 12 (𝜑 → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) ↔ (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)))
133103, 132mpbid 222 . . . . . . . . . . 11 (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
134133adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
135 ovexd 6665 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (1...𝑁) ∈ V)
136 inidm 3814 . . . . . . . . . 10 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
137 eqidd 2621 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
138101adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅)
139 fzss2 12366 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁))
140 imass2 5489 . . . . . . . . . . . . . . 15 ((𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁)))
141118, 139, 1403syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁)))
142141sselda 3595 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))
143 fvun2 6257 . . . . . . . . . . . . . 14 ((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)) ∧ (((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
14487, 90, 143mp3an12 1412 . . . . . . . . . . . . 13 ((((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
145138, 142, 144syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
14688fvconst2 6454 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑈 “ (𝑀...𝑁)) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0)
147142, 146syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0)
148145, 147eqtrd 2654 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0)
149148adantr 481 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0)
15084, 134, 135, 135, 136, 137, 149ofval 6891 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
15128, 150mpdan 701 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
15230zcnd 11468 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ∈ ℂ)
153152addid1d 10221 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝑇𝑛) + 0) = (𝑇𝑛))
15428, 153syldan 487 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇𝑛) + 0) = (𝑇𝑛))
15581, 151, 1543eqtrd 2658 . . . . . . 7 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (𝑇𝑛))
156 breq1 4647 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → (𝑦 < 𝑀𝑀 < 𝑀))
157 oveq1 6642 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → (𝑦 + 1) = (𝑀 + 1))
158156, 157ifbieq2d 4102 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)))
15941ltnrd 10156 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝑀 < 𝑀)
160159iffalsed 4088 . . . . . . . . . . . . . 14 (𝜑 → if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)) = (𝑀 + 1))
161158, 160sylan9eqr 2676 . . . . . . . . . . . . 13 ((𝜑𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 + 1))
162161csbeq1d 3533 . . . . . . . . . . . 12 ((𝜑𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑀 + 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
163 ovex 6663 . . . . . . . . . . . . 13 (𝑀 + 1) ∈ V
164 oveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 + 1) → (1...𝑗) = (1...(𝑀 + 1)))
165164imaeq2d 5454 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 + 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 + 1))))
166165xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 + 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 + 1))) × {1}))
167 oveq1 6642 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 + 1) → (𝑗 + 1) = ((𝑀 + 1) + 1))
168167oveq1d 6650 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 + 1) → ((𝑗 + 1)...𝑁) = (((𝑀 + 1) + 1)...𝑁))
169168imaeq2d 5454 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 + 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
170169xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 + 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))
171166, 170uneq12d 3760 . . . . . . . . . . . . . 14 (𝑗 = (𝑀 + 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))
172171oveq2d 6651 . . . . . . . . . . . . 13 (𝑗 = (𝑀 + 1) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
173163, 172csbie 3552 . . . . . . . . . . . 12 (𝑀 + 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))
174162, 173syl6eq 2670 . . . . . . . . . . 11 ((𝜑𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
175 1eluzge0 11717 . . . . . . . . . . . . 13 1 ∈ (ℤ‘0)
176 fzss1 12365 . . . . . . . . . . . . 13 (1 ∈ (ℤ‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
177175, 176ax-mp 5 . . . . . . . . . . . 12 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
178177, 16sseldi 3593 . . . . . . . . . . 11 (𝜑𝑀 ∈ (0...(𝑁 − 1)))
179 ovexd 6665 . . . . . . . . . . 11 (𝜑 → (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) ∈ V)
18035, 174, 178, 179fvmptd 6275 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
181180fveq1d 6180 . . . . . . . . 9 (𝜑 → ((𝐹𝑀)‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛))
182181adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹𝑀)‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛))
183 fnconstg 6080 . . . . . . . . . . . . . . 15 (1 ∈ V → ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))))
18485, 183ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1)))
185 fnconstg 6080 . . . . . . . . . . . . . . 15 (0 ∈ V → ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
18688, 185ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))
187184, 186pm3.2i 471 . . . . . . . . . . . . 13 (((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
188 imain 5962 . . . . . . . . . . . . . . . 16 (Fun 𝑈 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
1891, 93, 1883syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
190 peano2re 10194 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
19141, 190syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 + 1) ∈ ℝ)
192191ltp1d 10939 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) < ((𝑀 + 1) + 1))
193 fzdisj 12353 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) < ((𝑀 + 1) + 1) → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅)
194192, 193syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅)
195194imaeq2d 5454 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
196189, 195eqtr3d 2656 . . . . . . . . . . . . . 14 (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
197196, 99syl6eq 2670 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅)
198 fnun 5985 . . . . . . . . . . . . 13 (((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ∧ ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
199187, 197, 198sylancr 694 . . . . . . . . . . . 12 (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
200 fzsplit 12352 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
20122, 200syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
202201imaeq2d 5454 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))))
203 imaundi 5533 . . . . . . . . . . . . . . 15 (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
204202, 203syl6eq 2670 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
205204, 130eqtr3d 2656 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (1...𝑁))
206205fneq2d 5970 . . . . . . . . . . . 12 (𝜑 → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ↔ (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
207199, 206mpbid 222 . . . . . . . . . . 11 (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
208207adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
209197adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅)
210 fzss1 12365 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → (𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
211 imass2 5489 . . . . . . . . . . . . . . 15 ((𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1))))
212105, 210, 2113syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1))))
213212sselda 3595 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))
214 fvun1 6256 . . . . . . . . . . . . . 14 ((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
215184, 186, 214mp3an12 1412 . . . . . . . . . . . . 13 ((((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
216209, 213, 215syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
21785fvconst2 6454 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1)
218213, 217syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1)
219216, 218eqtrd 2654 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
220219adantr 481 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
22184, 208, 135, 135, 136, 137, 220ofval 6891 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
22228, 221mpdan 701 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
223182, 222eqtrd 2654 . . . . . . 7 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹𝑀)‘𝑛) = ((𝑇𝑛) + 1))
22434, 155, 2233netr4d 2868 . . . . . 6 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
225224ralrimiva 2963 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
226 fzpr 12381 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
22716, 39, 2263syl 18 . . . . . . . 8 (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
228227imaeq2d 5454 . . . . . . 7 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = (𝑈 “ {𝑀, (𝑀 + 1)}))
229 f1ofn 6125 . . . . . . . . 9 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
2301, 229syl 17 . . . . . . . 8 (𝜑𝑈 Fn (1...𝑁))
231 fnimapr 6249 . . . . . . . 8 ((𝑈 Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁)) → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈𝑀), (𝑈‘(𝑀 + 1))})
232230, 17, 22, 231syl3anc 1324 . . . . . . 7 (𝜑 → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈𝑀), (𝑈‘(𝑀 + 1))})
233228, 232eqtrd 2654 . . . . . 6 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = {(𝑈𝑀), (𝑈‘(𝑀 + 1))})
234233raleqdv 3139 . . . . 5 (𝜑 → (∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ∀𝑛 ∈ {(𝑈𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)))
235225, 234mpbid 222 . . . 4 (𝜑 → ∀𝑛 ∈ {(𝑈𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
236 fvex 6188 . . . . 5 (𝑈𝑀) ∈ V
237 fvex 6188 . . . . 5 (𝑈‘(𝑀 + 1)) ∈ V
238 fveq2 6178 . . . . . 6 (𝑛 = (𝑈𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈𝑀)))
239 fveq2 6178 . . . . . 6 (𝑛 = (𝑈𝑀) → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘(𝑈𝑀)))
240238, 239neeq12d 2852 . . . . 5 (𝑛 = (𝑈𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀))))
241 fveq2 6178 . . . . . 6 (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))))
242 fveq2 6178 . . . . . 6 (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘(𝑈‘(𝑀 + 1))))
243241, 242neeq12d 2852 . . . . 5 (𝑛 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
244236, 237, 240, 243ralpr 4229 . . . 4 (∀𝑛 ∈ {(𝑈𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
245235, 244sylib 208 . . 3 (𝜑 → (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
24641ltp1d 10939 . . . . 5 (𝜑𝑀 < (𝑀 + 1))
24741, 246ltned 10158 . . . 4 (𝜑𝑀 ≠ (𝑀 + 1))
248 f1of1 6123 . . . . . . 7 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–1-1→(1...𝑁))
2491, 248syl 17 . . . . . 6 (𝜑𝑈:(1...𝑁)–1-1→(1...𝑁))
250 f1veqaeq 6499 . . . . . 6 ((𝑈:(1...𝑁)–1-1→(1...𝑁) ∧ (𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁))) → ((𝑈𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1)))
251249, 17, 22, 250syl12anc 1322 . . . . 5 (𝜑 → ((𝑈𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1)))
252251necon3d 2812 . . . 4 (𝜑 → (𝑀 ≠ (𝑀 + 1) → (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1))))
253247, 252mpd 15 . . 3 (𝜑 → (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1)))
254240anbi1d 740 . . . . 5 (𝑛 = (𝑈𝑀) → ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚))))
255 neeq1 2853 . . . . 5 (𝑛 = (𝑈𝑀) → (𝑛𝑚 ↔ (𝑈𝑀) ≠ 𝑚))
256254, 255anbi12d 746 . . . 4 (𝑛 = (𝑈𝑀) → (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ (𝑈𝑀) ≠ 𝑚)))
257 fveq2 6178 . . . . . . 7 (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑚) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))))
258 fveq2 6178 . . . . . . 7 (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹𝑀)‘𝑚) = ((𝐹𝑀)‘(𝑈‘(𝑀 + 1))))
259257, 258neeq12d 2852 . . . . . 6 (𝑚 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
260259anbi2d 739 . . . . 5 (𝑚 = (𝑈‘(𝑀 + 1)) → ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1))))))
261 neeq2 2854 . . . . 5 (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝑈𝑀) ≠ 𝑚 ↔ (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1))))
262260, 261anbi12d 746 . . . 4 (𝑚 = (𝑈‘(𝑀 + 1)) → (((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ (𝑈𝑀) ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1)))))
263256, 262rspc2ev 3319 . . 3 (((𝑈𝑀) ∈ (1...𝑁) ∧ (𝑈‘(𝑀 + 1)) ∈ (1...𝑁) ∧ ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1)))) → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
26418, 23, 245, 253, 263syl112anc 1328 . 2 (𝜑 → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
265 dfrex2 2993 . . 3 (∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ¬ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
266 fveq2 6178 . . . . . 6 (𝑛 = 𝑚 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑚))
267 fveq2 6178 . . . . . 6 (𝑛 = 𝑚 → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘𝑚))
268266, 267neeq12d 2852 . . . . 5 (𝑛 = 𝑚 → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)))
269268rmo4 3393 . . . 4 (∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
270 dfral2 2991 . . . . . 6 (∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
271 df-ne 2792 . . . . . . . . 9 (𝑛𝑚 ↔ ¬ 𝑛 = 𝑚)
272271anbi2i 729 . . . . . . . 8 (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚))
273 annim 441 . . . . . . . 8 (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
274272, 273bitri 264 . . . . . . 7 (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
275274rexbii 3037 . . . . . 6 (∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
276270, 275xchbinxr 325 . . . . 5 (∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
277276ralbii 2977 . . . 4 (∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
278269, 277bitri 264 . . 3 (∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
279265, 278xchbinxr 325 . 2 (∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
280264, 279sylib 208 1 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  ∃*wrmo 2912  Vcvv 3195  csb 3526  cun 3565  cin 3566  wss 3567  c0 3907  ifcif 4077  {csn 4168  {cpr 4170   class class class wbr 4644  cmpt 4720   × cxp 5102  ccnv 5103  ran crn 5105  cima 5107  Fun wfun 5870   Fn wfn 5871  wf 5872  1-1wf1 5873  ontowfo 5874  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  𝑓 cof 6880  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059  cmin 10251  cn 11005  cz 11362  cuz 11672  ...cfz 12311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312
This theorem is referenced by:  poimirlem8  33388  poimirlem18  33398  poimirlem21  33401  poimirlem22  33402
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