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Theorem isomenndlem 41065
Description: 𝑂 is sub-additive w.r.t. countable indexed union, implies that 𝑂 is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
isomenndlem.o (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
isomenndlem.o0 (𝜑 → (𝑂‘∅) = 0)
isomenndlem.y (𝜑𝑌 ⊆ 𝒫 𝑋)
isomenndlem.subadd ((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))))
isomenndlem.b (𝜑𝐵 ⊆ ℕ)
isomenndlem.f (𝜑𝐹:𝐵1-1-onto𝑌)
isomenndlem.a 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))
Assertion
Ref Expression
isomenndlem (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
Distinct variable groups:   𝐴,𝑎,𝑛   𝐵,𝑛   𝑛,𝐹   𝑂,𝑎,𝑛   𝑋,𝑎   𝑛,𝑌   𝜑,𝑎,𝑛
Allowed substitution hints:   𝐵(𝑎)   𝐹(𝑎)   𝑋(𝑛)   𝑌(𝑎)

Proof of Theorem isomenndlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝜑𝜑)
2 iftrue 4125 . . . . . . . . 9 (𝑛𝐵 → if(𝑛𝐵, (𝐹𝑛), ∅) = (𝐹𝑛))
32adantl 481 . . . . . . . 8 ((𝜑𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) = (𝐹𝑛))
4 isomenndlem.f . . . . . . . . . . 11 (𝜑𝐹:𝐵1-1-onto𝑌)
5 f1of 6175 . . . . . . . . . . 11 (𝐹:𝐵1-1-onto𝑌𝐹:𝐵𝑌)
64, 5syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐵𝑌)
7 ssun1 3809 . . . . . . . . . . 11 𝑌 ⊆ (𝑌 ∪ {∅})
87a1i 11 . . . . . . . . . 10 (𝜑𝑌 ⊆ (𝑌 ∪ {∅}))
96, 8fssd 6095 . . . . . . . . 9 (𝜑𝐹:𝐵⟶(𝑌 ∪ {∅}))
109ffvelrnda 6399 . . . . . . . 8 ((𝜑𝑛𝐵) → (𝐹𝑛) ∈ (𝑌 ∪ {∅}))
113, 10eqeltrd 2730 . . . . . . 7 ((𝜑𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ (𝑌 ∪ {∅}))
1211adantlr 751 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ (𝑌 ∪ {∅}))
13 iffalse 4128 . . . . . . . . 9 𝑛𝐵 → if(𝑛𝐵, (𝐹𝑛), ∅) = ∅)
1413adantl 481 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) = ∅)
15 0ex 4823 . . . . . . . . . . 11 ∅ ∈ V
1615snid 4241 . . . . . . . . . 10 ∅ ∈ {∅}
17 elun2 3814 . . . . . . . . . 10 (∅ ∈ {∅} → ∅ ∈ (𝑌 ∪ {∅}))
1816, 17ax-mp 5 . . . . . . . . 9 ∅ ∈ (𝑌 ∪ {∅})
1918a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑛𝐵) → ∅ ∈ (𝑌 ∪ {∅}))
2014, 19eqeltrd 2730 . . . . . . 7 ((𝜑 ∧ ¬ 𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ (𝑌 ∪ {∅}))
2120adantlr 751 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ ¬ 𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ (𝑌 ∪ {∅}))
2212, 21pm2.61dan 849 . . . . 5 ((𝜑𝑛 ∈ ℕ) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ (𝑌 ∪ {∅}))
23 isomenndlem.a . . . . 5 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))
2422, 23fmptd 6425 . . . 4 (𝜑𝐴:ℕ⟶(𝑌 ∪ {∅}))
25 isomenndlem.y . . . . 5 (𝜑𝑌 ⊆ 𝒫 𝑋)
26 0elpw 4864 . . . . . . 7 ∅ ∈ 𝒫 𝑋
27 snssi 4371 . . . . . . 7 (∅ ∈ 𝒫 𝑋 → {∅} ⊆ 𝒫 𝑋)
2826, 27ax-mp 5 . . . . . 6 {∅} ⊆ 𝒫 𝑋
2928a1i 11 . . . . 5 (𝜑 → {∅} ⊆ 𝒫 𝑋)
3025, 29unssd 3822 . . . 4 (𝜑 → (𝑌 ∪ {∅}) ⊆ 𝒫 𝑋)
3124, 30fssd 6095 . . 3 (𝜑𝐴:ℕ⟶𝒫 𝑋)
32 nnex 11064 . . . . . 6 ℕ ∈ V
3332mptex 6527 . . . . 5 (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅)) ∈ V
3423, 33eqeltri 2726 . . . 4 𝐴 ∈ V
35 feq1 6064 . . . . . 6 (𝑎 = 𝐴 → (𝑎:ℕ⟶𝒫 𝑋𝐴:ℕ⟶𝒫 𝑋))
3635anbi2d 740 . . . . 5 (𝑎 = 𝐴 → ((𝜑𝑎:ℕ⟶𝒫 𝑋) ↔ (𝜑𝐴:ℕ⟶𝒫 𝑋)))
37 fveq1 6228 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎𝑛) = (𝐴𝑛))
3837iuneq2d 4579 . . . . . . 7 (𝑎 = 𝐴 𝑛 ∈ ℕ (𝑎𝑛) = 𝑛 ∈ ℕ (𝐴𝑛))
3938fveq2d 6233 . . . . . 6 (𝑎 = 𝐴 → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) = (𝑂 𝑛 ∈ ℕ (𝐴𝑛)))
40 simpl 472 . . . . . . . . . 10 ((𝑎 = 𝐴𝑛 ∈ ℕ) → 𝑎 = 𝐴)
4140fveq1d 6231 . . . . . . . . 9 ((𝑎 = 𝐴𝑛 ∈ ℕ) → (𝑎𝑛) = (𝐴𝑛))
4241fveq2d 6233 . . . . . . . 8 ((𝑎 = 𝐴𝑛 ∈ ℕ) → (𝑂‘(𝑎𝑛)) = (𝑂‘(𝐴𝑛)))
4342mpteq2dva 4777 . . . . . . 7 (𝑎 = 𝐴 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛))))
4443fveq2d 6233 . . . . . 6 (𝑎 = 𝐴 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛)))))
4539, 44breq12d 4698 . . . . 5 (𝑎 = 𝐴 → ((𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))) ↔ (𝑂 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛))))))
4636, 45imbi12d 333 . . . 4 (𝑎 = 𝐴 → (((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛))))) ↔ ((𝜑𝐴:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛)))))))
47 isomenndlem.subadd . . . 4 ((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))))
4834, 46, 47vtocl 3290 . . 3 ((𝜑𝐴:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛)))))
491, 31, 48syl2anc 694 . 2 (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛)))))
506ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝐵 = ℕ) ∧ 𝑛 ∈ ℕ) → 𝐹:𝐵𝑌)
51 simpr 476 . . . . . . . . . . . . 13 ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
52 id 22 . . . . . . . . . . . . . . 15 (𝐵 = ℕ → 𝐵 = ℕ)
5352eqcomd 2657 . . . . . . . . . . . . . 14 (𝐵 = ℕ → ℕ = 𝐵)
5453adantr 480 . . . . . . . . . . . . 13 ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → ℕ = 𝐵)
5551, 54eleqtrd 2732 . . . . . . . . . . . 12 ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → 𝑛𝐵)
5655adantll 750 . . . . . . . . . . 11 (((𝜑𝐵 = ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛𝐵)
5750, 56ffvelrnd 6400 . . . . . . . . . 10 (((𝜑𝐵 = ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ 𝑌)
58 eqid 2651 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (𝐹𝑛)) = (𝑛 ∈ ℕ ↦ (𝐹𝑛))
5957, 58fmptd 6425 . . . . . . . . 9 ((𝜑𝐵 = ℕ) → (𝑛 ∈ ℕ ↦ (𝐹𝑛)):ℕ⟶𝑌)
6023a1i 11 . . . . . . . . . . . 12 (𝐵 = ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅)))
6155iftrued 4127 . . . . . . . . . . . . 13 ((𝐵 = ℕ ∧ 𝑛 ∈ ℕ) → if(𝑛𝐵, (𝐹𝑛), ∅) = (𝐹𝑛))
6261mpteq2dva 4777 . . . . . . . . . . . 12 (𝐵 = ℕ → (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅)) = (𝑛 ∈ ℕ ↦ (𝐹𝑛)))
6360, 62eqtrd 2685 . . . . . . . . . . 11 (𝐵 = ℕ → 𝐴 = (𝑛 ∈ ℕ ↦ (𝐹𝑛)))
6463feq1d 6068 . . . . . . . . . 10 (𝐵 = ℕ → (𝐴:ℕ⟶𝑌 ↔ (𝑛 ∈ ℕ ↦ (𝐹𝑛)):ℕ⟶𝑌))
6564adantl 481 . . . . . . . . 9 ((𝜑𝐵 = ℕ) → (𝐴:ℕ⟶𝑌 ↔ (𝑛 ∈ ℕ ↦ (𝐹𝑛)):ℕ⟶𝑌))
6659, 65mpbird 247 . . . . . . . 8 ((𝜑𝐵 = ℕ) → 𝐴:ℕ⟶𝑌)
67 f1ofo 6182 . . . . . . . . . . . . . . . 16 (𝐹:𝐵1-1-onto𝑌𝐹:𝐵onto𝑌)
684, 67syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝐵onto𝑌)
69 dffo3 6414 . . . . . . . . . . . . . . 15 (𝐹:𝐵onto𝑌 ↔ (𝐹:𝐵𝑌 ∧ ∀𝑦𝑌𝑛𝐵 𝑦 = (𝐹𝑛)))
7068, 69sylib 208 . . . . . . . . . . . . . 14 (𝜑 → (𝐹:𝐵𝑌 ∧ ∀𝑦𝑌𝑛𝐵 𝑦 = (𝐹𝑛)))
7170simprd 478 . . . . . . . . . . . . 13 (𝜑 → ∀𝑦𝑌𝑛𝐵 𝑦 = (𝐹𝑛))
7271adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦𝑌) → ∀𝑦𝑌𝑛𝐵 𝑦 = (𝐹𝑛))
73 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑦𝑌) → 𝑦𝑌)
74 rspa 2959 . . . . . . . . . . . 12 ((∀𝑦𝑌𝑛𝐵 𝑦 = (𝐹𝑛) ∧ 𝑦𝑌) → ∃𝑛𝐵 𝑦 = (𝐹𝑛))
7572, 73, 74syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑦𝑌) → ∃𝑛𝐵 𝑦 = (𝐹𝑛))
7675adantlr 751 . . . . . . . . . 10 (((𝜑𝐵 = ℕ) ∧ 𝑦𝑌) → ∃𝑛𝐵 𝑦 = (𝐹𝑛))
77 nfv 1883 . . . . . . . . . . . 12 𝑛(𝜑𝐵 = ℕ)
78 nfre1 3034 . . . . . . . . . . . 12 𝑛𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)
79 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝐵 = ℕ ∧ 𝑛𝐵) → 𝑛𝐵)
80 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝐵 = ℕ ∧ 𝑛𝐵) → 𝐵 = ℕ)
8179, 80eleqtrd 2732 . . . . . . . . . . . . . . . 16 ((𝐵 = ℕ ∧ 𝑛𝐵) → 𝑛 ∈ ℕ)
8281adantll 750 . . . . . . . . . . . . . . 15 (((𝜑𝐵 = ℕ) ∧ 𝑛𝐵) → 𝑛 ∈ ℕ)
83823adant3 1101 . . . . . . . . . . . . . 14 (((𝜑𝐵 = ℕ) ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → 𝑛 ∈ ℕ)
8460fveq1d 6231 . . . . . . . . . . . . . . . . 17 (𝐵 = ℕ → (𝐴𝑛) = ((𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))‘𝑛))
85843ad2ant1 1102 . . . . . . . . . . . . . . . 16 ((𝐵 = ℕ ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → (𝐴𝑛) = ((𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))‘𝑛))
86 fvex 6239 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑛) ∈ V
8786, 15ifex 4189 . . . . . . . . . . . . . . . . . . . 20 if(𝑛𝐵, (𝐹𝑛), ∅) ∈ V
8887a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = ℕ ∧ 𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ V)
89 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅)) = (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))
9089fvmpt2 6330 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ if(𝑛𝐵, (𝐹𝑛), ∅) ∈ V) → ((𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))‘𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
9181, 88, 90syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝐵 = ℕ ∧ 𝑛𝐵) → ((𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))‘𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
922adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝐵 = ℕ ∧ 𝑛𝐵) → if(𝑛𝐵, (𝐹𝑛), ∅) = (𝐹𝑛))
9391, 92eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((𝐵 = ℕ ∧ 𝑛𝐵) → ((𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))‘𝑛) = (𝐹𝑛))
94933adant3 1101 . . . . . . . . . . . . . . . 16 ((𝐵 = ℕ ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → ((𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))‘𝑛) = (𝐹𝑛))
95 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝐹𝑛) → 𝑦 = (𝐹𝑛))
9695eqcomd 2657 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝐹𝑛) → (𝐹𝑛) = 𝑦)
97963ad2ant3 1104 . . . . . . . . . . . . . . . 16 ((𝐵 = ℕ ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → (𝐹𝑛) = 𝑦)
9885, 94, 973eqtrrd 2690 . . . . . . . . . . . . . . 15 ((𝐵 = ℕ ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → 𝑦 = (𝐴𝑛))
99983adant1l 1358 . . . . . . . . . . . . . 14 (((𝜑𝐵 = ℕ) ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → 𝑦 = (𝐴𝑛))
100 rspe 3032 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ 𝑦 = (𝐴𝑛)) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
10183, 99, 100syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝐵 = ℕ) ∧ 𝑛𝐵𝑦 = (𝐹𝑛)) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
1021013exp 1283 . . . . . . . . . . . 12 ((𝜑𝐵 = ℕ) → (𝑛𝐵 → (𝑦 = (𝐹𝑛) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))))
10377, 78, 102rexlimd 3055 . . . . . . . . . . 11 ((𝜑𝐵 = ℕ) → (∃𝑛𝐵 𝑦 = (𝐹𝑛) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
104103adantr 480 . . . . . . . . . 10 (((𝜑𝐵 = ℕ) ∧ 𝑦𝑌) → (∃𝑛𝐵 𝑦 = (𝐹𝑛) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
10576, 104mpd 15 . . . . . . . . 9 (((𝜑𝐵 = ℕ) ∧ 𝑦𝑌) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
106105ralrimiva 2995 . . . . . . . 8 ((𝜑𝐵 = ℕ) → ∀𝑦𝑌𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
10766, 106jca 553 . . . . . . 7 ((𝜑𝐵 = ℕ) → (𝐴:ℕ⟶𝑌 ∧ ∀𝑦𝑌𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
108 dffo3 6414 . . . . . . 7 (𝐴:ℕ–onto𝑌 ↔ (𝐴:ℕ⟶𝑌 ∧ ∀𝑦𝑌𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
109107, 108sylibr 224 . . . . . 6 ((𝜑𝐵 = ℕ) → 𝐴:ℕ–onto𝑌)
110 founiiun 39674 . . . . . 6 (𝐴:ℕ–onto𝑌 𝑌 = 𝑛 ∈ ℕ (𝐴𝑛))
111109, 110syl 17 . . . . 5 ((𝜑𝐵 = ℕ) → 𝑌 = 𝑛 ∈ ℕ (𝐴𝑛))
112 uniun 4488 . . . . . . . 8 (𝑌 ∪ {∅}) = ( 𝑌 {∅})
11315unisn 4483 . . . . . . . . 9 {∅} = ∅
114113uneq2i 3797 . . . . . . . 8 ( 𝑌 {∅}) = ( 𝑌 ∪ ∅)
115 un0 4000 . . . . . . . 8 ( 𝑌 ∪ ∅) = 𝑌
116112, 114, 1153eqtrri 2678 . . . . . . 7 𝑌 = (𝑌 ∪ {∅})
117116a1i 11 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝑌 = (𝑌 ∪ {∅}))
11824adantr 480 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐴:ℕ⟶(𝑌 ∪ {∅}))
119 isomenndlem.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ⊆ ℕ)
120119adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐵 ⊆ ℕ)
12152necon3bi 2849 . . . . . . . . . . . . . . . . . 18 𝐵 = ℕ → 𝐵 ≠ ℕ)
122121adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐵 ≠ ℕ)
123120, 122jca 553 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 𝐵 = ℕ) → (𝐵 ⊆ ℕ ∧ 𝐵 ≠ ℕ))
124 df-pss 3623 . . . . . . . . . . . . . . . 16 (𝐵 ⊊ ℕ ↔ (𝐵 ⊆ ℕ ∧ 𝐵 ≠ ℕ))
125123, 124sylibr 224 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐵 ⊊ ℕ)
126 pssnel 4072 . . . . . . . . . . . . . . 15 (𝐵 ⊊ ℕ → ∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵))
127125, 126syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵))
128127adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → ∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵))
129 nfv 1883 . . . . . . . . . . . . . 14 𝑛((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅)
130 simprl 809 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → 𝑛 ∈ ℕ)
131 simprl 809 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → 𝑛 ∈ ℕ)
13287a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ V)
13323fvmpt2 6330 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ∧ if(𝑛𝐵, (𝐹𝑛), ∅) ∈ V) → (𝐴𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
134131, 132, 133syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → (𝐴𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
135134adantlr 751 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → (𝐴𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
13613ad2antll 765 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → if(𝑛𝐵, (𝐹𝑛), ∅) = ∅)
137 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → 𝑦 = ∅)
138137eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → ∅ = 𝑦)
139138ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → ∅ = 𝑦)
140135, 136, 1393eqtrrd 2690 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → 𝑦 = (𝐴𝑛))
141130, 140, 100syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 = ∅) ∧ (𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵)) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
142141ex 449 . . . . . . . . . . . . . . 15 ((𝜑𝑦 = ∅) → ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
143142adantlr 751 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → ((𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
144129, 78, 143exlimd 2125 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → (∃𝑛(𝑛 ∈ ℕ ∧ ¬ 𝑛𝐵) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
145128, 144mpd 15 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 = ∅) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
146145adantlr 751 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ 𝑦 = ∅) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
147 simplll 813 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ ¬ 𝑦 = ∅) → 𝜑)
148 simpl 472 . . . . . . . . . . . . . 14 ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ (𝑌 ∪ {∅}))
149 elsni 4227 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {∅} → 𝑦 = ∅)
150149con3i 150 . . . . . . . . . . . . . . 15 𝑦 = ∅ → ¬ 𝑦 ∈ {∅})
151150adantl 481 . . . . . . . . . . . . . 14 ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 = ∅) → ¬ 𝑦 ∈ {∅})
152 elunnel2 39512 . . . . . . . . . . . . . 14 ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 ∈ {∅}) → 𝑦𝑌)
153148, 151, 152syl2anc 694 . . . . . . . . . . . . 13 ((𝑦 ∈ (𝑌 ∪ {∅}) ∧ ¬ 𝑦 = ∅) → 𝑦𝑌)
154153adantll 750 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ ¬ 𝑦 = ∅) → 𝑦𝑌)
15568adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑌) → 𝐹:𝐵onto𝑌)
156 foelrni 6283 . . . . . . . . . . . . . 14 ((𝐹:𝐵onto𝑌𝑦𝑌) → ∃𝑛𝐵 (𝐹𝑛) = 𝑦)
157155, 73, 156syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑦𝑌) → ∃𝑛𝐵 (𝐹𝑛) = 𝑦)
158 nfv 1883 . . . . . . . . . . . . . 14 𝑛(𝜑𝑦𝑌)
159119sselda 3636 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝐵) → 𝑛 ∈ ℕ)
1601593adant3 1101 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝐵 ∧ (𝐹𝑛) = 𝑦) → 𝑛 ∈ ℕ)
161159, 87, 133sylancl 695 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛𝐵) → (𝐴𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
162161, 3eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝐵) → (𝐴𝑛) = (𝐹𝑛))
1631623adant3 1101 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝐵 ∧ (𝐹𝑛) = 𝑦) → (𝐴𝑛) = (𝐹𝑛))
164 simp3 1083 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝐵 ∧ (𝐹𝑛) = 𝑦) → (𝐹𝑛) = 𝑦)
165163, 164eqtr2d 2686 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝐵 ∧ (𝐹𝑛) = 𝑦) → 𝑦 = (𝐴𝑛))
166160, 165, 100syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝐵 ∧ (𝐹𝑛) = 𝑦) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
1671663exp 1283 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛𝐵 → ((𝐹𝑛) = 𝑦 → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))))
168167adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑌) → (𝑛𝐵 → ((𝐹𝑛) = 𝑦 → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))))
169158, 78, 168rexlimd 3055 . . . . . . . . . . . . 13 ((𝜑𝑦𝑌) → (∃𝑛𝐵 (𝐹𝑛) = 𝑦 → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
170157, 169mpd 15 . . . . . . . . . . . 12 ((𝜑𝑦𝑌) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
171147, 154, 170syl2anc 694 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) ∧ ¬ 𝑦 = ∅) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
172146, 171pm2.61dan 849 . . . . . . . . . 10 (((𝜑 ∧ ¬ 𝐵 = ℕ) ∧ 𝑦 ∈ (𝑌 ∪ {∅})) → ∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
173172ralrimiva 2995 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵 = ℕ) → ∀𝑦 ∈ (𝑌 ∪ {∅})∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛))
174118, 173jca 553 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵 = ℕ) → (𝐴:ℕ⟶(𝑌 ∪ {∅}) ∧ ∀𝑦 ∈ (𝑌 ∪ {∅})∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
175 dffo3 6414 . . . . . . . 8 (𝐴:ℕ–onto→(𝑌 ∪ {∅}) ↔ (𝐴:ℕ⟶(𝑌 ∪ {∅}) ∧ ∀𝑦 ∈ (𝑌 ∪ {∅})∃𝑛 ∈ ℕ 𝑦 = (𝐴𝑛)))
176174, 175sylibr 224 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝐴:ℕ–onto→(𝑌 ∪ {∅}))
177 founiiun 39674 . . . . . . 7 (𝐴:ℕ–onto→(𝑌 ∪ {∅}) → (𝑌 ∪ {∅}) = 𝑛 ∈ ℕ (𝐴𝑛))
178176, 177syl 17 . . . . . 6 ((𝜑 ∧ ¬ 𝐵 = ℕ) → (𝑌 ∪ {∅}) = 𝑛 ∈ ℕ (𝐴𝑛))
179117, 178eqtrd 2685 . . . . 5 ((𝜑 ∧ ¬ 𝐵 = ℕ) → 𝑌 = 𝑛 ∈ ℕ (𝐴𝑛))
180111, 179pm2.61dan 849 . . . 4 (𝜑 𝑌 = 𝑛 ∈ ℕ (𝐴𝑛))
181180fveq2d 6233 . . 3 (𝜑 → (𝑂 𝑌) = (𝑂 𝑛 ∈ ℕ (𝐴𝑛)))
182 uncom 3790 . . . . . . . . 9 ((ℕ ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (ℕ ∖ 𝐵))
183182a1i 11 . . . . . . . 8 (𝜑 → ((ℕ ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (ℕ ∖ 𝐵)))
184 undif 4082 . . . . . . . . 9 (𝐵 ⊆ ℕ ↔ (𝐵 ∪ (ℕ ∖ 𝐵)) = ℕ)
185119, 184sylib 208 . . . . . . . 8 (𝜑 → (𝐵 ∪ (ℕ ∖ 𝐵)) = ℕ)
186183, 185eqtrd 2685 . . . . . . 7 (𝜑 → ((ℕ ∖ 𝐵) ∪ 𝐵) = ℕ)
187186eqcomd 2657 . . . . . 6 (𝜑 → ℕ = ((ℕ ∖ 𝐵) ∪ 𝐵))
188187mpteq1d 4771 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛))) = (𝑛 ∈ ((ℕ ∖ 𝐵) ∪ 𝐵) ↦ (𝑂‘(𝐴𝑛))))
189188fveq2d 6233 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛)))) = (Σ^‘(𝑛 ∈ ((ℕ ∖ 𝐵) ∪ 𝐵) ↦ (𝑂‘(𝐴𝑛)))))
190 nfv 1883 . . . . 5 𝑛𝜑
191 difexg 4841 . . . . . . 7 (ℕ ∈ V → (ℕ ∖ 𝐵) ∈ V)
19232, 191ax-mp 5 . . . . . 6 (ℕ ∖ 𝐵) ∈ V
193192a1i 11 . . . . 5 (𝜑 → (ℕ ∖ 𝐵) ∈ V)
19432a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
195194, 119ssexd 4838 . . . . 5 (𝜑𝐵 ∈ V)
196 incom 3838 . . . . . . 7 ((ℕ ∖ 𝐵) ∩ 𝐵) = (𝐵 ∩ (ℕ ∖ 𝐵))
197 disjdif 4073 . . . . . . 7 (𝐵 ∩ (ℕ ∖ 𝐵)) = ∅
198196, 197eqtri 2673 . . . . . 6 ((ℕ ∖ 𝐵) ∩ 𝐵) = ∅
199198a1i 11 . . . . 5 (𝜑 → ((ℕ ∖ 𝐵) ∩ 𝐵) = ∅)
200 simpl 472 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → 𝜑)
201 eldifi 3765 . . . . . . 7 (𝑛 ∈ (ℕ ∖ 𝐵) → 𝑛 ∈ ℕ)
202201adantl 481 . . . . . 6 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → 𝑛 ∈ ℕ)
203 isomenndlem.o . . . . . . . 8 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
204203adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
20531ffvelrnda 6399 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ 𝒫 𝑋)
206204, 205ffvelrnd 6400 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑂‘(𝐴𝑛)) ∈ (0[,]+∞))
207200, 202, 206syl2anc 694 . . . . 5 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘(𝐴𝑛)) ∈ (0[,]+∞))
208159, 206syldan 486 . . . . 5 ((𝜑𝑛𝐵) → (𝑂‘(𝐴𝑛)) ∈ (0[,]+∞))
209190, 193, 195, 199, 207, 208sge0splitmpt 40946 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ((ℕ ∖ 𝐵) ∪ 𝐵) ↦ (𝑂‘(𝐴𝑛)))) = ((Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴𝑛)))) +𝑒^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))))))
210 eqid 2651 . . . . . . . 8 (𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))) = (𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))
211208, 210fmptd 6425 . . . . . . 7 (𝜑 → (𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))):𝐵⟶(0[,]+∞))
212195, 211sge0xrcl 40920 . . . . . 6 (𝜑 → (Σ^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))) ∈ ℝ*)
213212xaddid2d 39848 . . . . 5 (𝜑 → (0 +𝑒^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))))) = (Σ^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))))
21487a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → if(𝑛𝐵, (𝐹𝑛), ∅) ∈ V)
215202, 214, 133syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → (𝐴𝑛) = if(𝑛𝐵, (𝐹𝑛), ∅))
216 eldifn 3766 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℕ ∖ 𝐵) → ¬ 𝑛𝐵)
217216adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → ¬ 𝑛𝐵)
218217iffalsed 4130 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → if(𝑛𝐵, (𝐹𝑛), ∅) = ∅)
219215, 218eqtrd 2685 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → (𝐴𝑛) = ∅)
220219fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘(𝐴𝑛)) = (𝑂‘∅))
221 isomenndlem.o0 . . . . . . . . . . 11 (𝜑 → (𝑂‘∅) = 0)
222200, 221syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘∅) = 0)
223220, 222eqtrd 2685 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℕ ∖ 𝐵)) → (𝑂‘(𝐴𝑛)) = 0)
224223mpteq2dva 4777 . . . . . . . 8 (𝜑 → (𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴𝑛))) = (𝑛 ∈ (ℕ ∖ 𝐵) ↦ 0))
225224fveq2d 6233 . . . . . . 7 (𝜑 → (Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴𝑛)))) = (Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ 0)))
226190, 193sge0z 40910 . . . . . . 7 (𝜑 → (Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ 0)) = 0)
227225, 226eqtrd 2685 . . . . . 6 (𝜑 → (Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴𝑛)))) = 0)
228227oveq1d 6705 . . . . 5 (𝜑 → ((Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴𝑛)))) +𝑒^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))))) = (0 +𝑒^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))))))
229203, 25feqresmpt 6289 . . . . . . 7 (𝜑 → (𝑂𝑌) = (𝑦𝑌 ↦ (𝑂𝑦)))
230229fveq2d 6233 . . . . . 6 (𝜑 → (Σ^‘(𝑂𝑌)) = (Σ^‘(𝑦𝑌 ↦ (𝑂𝑦))))
231 nfv 1883 . . . . . . 7 𝑦𝜑
232 fveq2 6229 . . . . . . 7 (𝑦 = (𝐴𝑛) → (𝑂𝑦) = (𝑂‘(𝐴𝑛)))
233162eqcomd 2657 . . . . . . 7 ((𝜑𝑛𝐵) → (𝐹𝑛) = (𝐴𝑛))
234203adantr 480 . . . . . . . 8 ((𝜑𝑦𝑌) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
23525sselda 3636 . . . . . . . 8 ((𝜑𝑦𝑌) → 𝑦 ∈ 𝒫 𝑋)
236234, 235ffvelrnd 6400 . . . . . . 7 ((𝜑𝑦𝑌) → (𝑂𝑦) ∈ (0[,]+∞))
237231, 190, 232, 195, 4, 233, 236sge0f1o 40917 . . . . . 6 (𝜑 → (Σ^‘(𝑦𝑌 ↦ (𝑂𝑦))) = (Σ^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))))
238 eqidd 2652 . . . . . 6 (𝜑 → (Σ^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))) = (Σ^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))))
239230, 237, 2383eqtrd 2689 . . . . 5 (𝜑 → (Σ^‘(𝑂𝑌)) = (Σ^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛)))))
240213, 228, 2393eqtr4d 2695 . . . 4 (𝜑 → ((Σ^‘(𝑛 ∈ (ℕ ∖ 𝐵) ↦ (𝑂‘(𝐴𝑛)))) +𝑒^‘(𝑛𝐵 ↦ (𝑂‘(𝐴𝑛))))) = (Σ^‘(𝑂𝑌)))
241189, 209, 2403eqtrrd 2690 . . 3 (𝜑 → (Σ^‘(𝑂𝑌)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛)))))
242181, 241breq12d 4698 . 2 (𝜑 → ((𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)) ↔ (𝑂 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐴𝑛))))))
24349, 242mpbird 247 1 (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  wpss 3608  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762  cres 5145  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  0cc0 9974  +∞cpnf 10109  cle 10113  cn 11058   +𝑒 cxad 11982  [,]cicc 12216  Σ^csumge0 40897
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-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-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  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-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  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-sumge0 40898
This theorem is referenced by:  isomennd  41066
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