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Theorem fprodssdc 11582
Description: Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)
Hypotheses
Ref Expression
fprodss.1 (𝜑𝐴𝐵)
fprodss.2 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
fprodssdc.a (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)
fprodss.3 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)
fprodss.4 (𝜑𝐵 ∈ Fin)
Assertion
Ref Expression
fprodssdc (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘,𝑗   𝜑,𝑘   𝑗,𝑘
Allowed substitution hints:   𝜑(𝑗)   𝐶(𝑗,𝑘)

Proof of Theorem fprodssdc
Dummy variables 𝑓 𝑝 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fprodss.1 . . 3 (𝜑𝐴𝐵)
2 sseq2 3179 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊆ ∅))
3 ss0 3463 . . . . 5 (𝐴 ⊆ ∅ → 𝐴 = ∅)
42, 3syl6bi 163 . . . 4 (𝐵 = ∅ → (𝐴𝐵𝐴 = ∅))
5 prodeq1 11545 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6 prodeq1 11545 . . . . . . 7 (𝐵 = ∅ → ∏𝑘𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
76eqcomd 2183 . . . . . 6 (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘𝐵 𝐶)
85, 7sylan9eq 2230 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
98expcom 116 . . . 4 (𝐵 = ∅ → (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
104, 9syld 45 . . 3 (𝐵 = ∅ → (𝐴𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
111, 10syl5com 29 . 2 (𝜑 → (𝐵 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
12 cnvimass 4987 . . . . . . . . 9 (𝑓𝐴) ⊆ dom 𝑓
13 simprr 531 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)
14 f1of 5457 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))⟶𝐵)
1513, 14syl 14 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
1612, 15fssdm 5376 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
17 f1ofn 5458 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓 Fn (1...(♯‘𝐵)))
18 elpreima 5631 . . . . . . . . . . . 12 (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
1913, 17, 183syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
2015ffvelcdmda 5647 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) ∈ 𝐵)
2120ex 115 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓𝑛) ∈ 𝐵))
2221adantrd 279 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴) → (𝑓𝑛) ∈ 𝐵))
2319, 22sylbid 150 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) → (𝑓𝑛) ∈ 𝐵))
2423imp 124 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → (𝑓𝑛) ∈ 𝐵)
25 fprodss.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2625ex 115 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴𝐶 ∈ ℂ))
2726adantr 276 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴𝐶 ∈ ℂ))
28 eldif 3138 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐵𝐴) ↔ (𝑘𝐵 ∧ ¬ 𝑘𝐴))
29 fprodss.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)
30 ax-1cn 7895 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
3129, 30eqeltrdi 2268 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ ℂ)
3228, 31sylan2br 288 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝐵 ∧ ¬ 𝑘𝐴)) → 𝐶 ∈ ℂ)
3332expr 375 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (¬ 𝑘𝐴𝐶 ∈ ℂ))
34 eleq1 2240 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
3534dcbid 838 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
36 fprodssdc.a . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)
3736adantr 276 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐵) → ∀𝑗𝐵 DECID 𝑗𝐴)
38 simpr 110 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐵) → 𝑘𝐵)
3935, 37, 38rspcdva 2846 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐵) → DECID 𝑘𝐴)
40 exmiddc 836 . . . . . . . . . . . . . 14 (DECID 𝑘𝐴 → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
4139, 40syl 14 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
4227, 33, 41mpjaod 718 . . . . . . . . . . . 12 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
4342adantlr 477 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
4443fmpttd 5667 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
4544ffvelcdmda 5647 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ (𝑓𝑛) ∈ 𝐵) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
4624, 45syldan 282 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
47 eqid 2177 . . . . . . . . 9 (ℤ‘1) = (ℤ‘1)
48 simprl 529 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℕ)
49 nnuz 9552 . . . . . . . . . 10 ℕ = (ℤ‘1)
5048, 49eleqtrdi 2270 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ (ℤ‘1))
51 ssidd 3176 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ⊆ (1...(♯‘𝐵)))
5247, 50, 51fprodntrivap 11576 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∃𝑚 ∈ (ℤ‘1)∃𝑦(𝑦 # 0 ∧ seq𝑚( · , (𝑛 ∈ (ℤ‘1) ↦ if(𝑛 ∈ (1...(♯‘𝐵)), ((𝑘𝐵𝐶)‘(𝑓𝑛)), 1))) ⇝ 𝑦))
53 eleq1 2240 . . . . . . . . . . . . 13 (𝑗 = (𝑓𝑝) → (𝑗𝐴 ↔ (𝑓𝑝) ∈ 𝐴))
5453dcbid 838 . . . . . . . . . . . 12 (𝑗 = (𝑓𝑝) → (DECID 𝑗𝐴DECID (𝑓𝑝) ∈ 𝐴))
5536ad3antrrr 492 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ∀𝑗𝐵 DECID 𝑗𝐴)
5613ad2antrr 488 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)
5756, 14syl 14 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
58 simpr 110 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝑝 ∈ (1...(♯‘𝐵)))
5957, 58ffvelcdmd 5648 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑓𝑝) ∈ 𝐵)
6054, 55, 59rspcdva 2846 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → DECID (𝑓𝑝) ∈ 𝐴)
61 f1ocnv 5470 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:𝐵1-1-onto→(1...(♯‘𝐵)))
62 f1of1 5456 . . . . . . . . . . . . . . 15 (𝑓:𝐵1-1-onto→(1...(♯‘𝐵)) → 𝑓:𝐵1-1→(1...(♯‘𝐵)))
6356, 61, 623syl 17 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝑓:𝐵1-1→(1...(♯‘𝐵)))
641ad3antrrr 492 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝐴𝐵)
65 f1elima 5768 . . . . . . . . . . . . . 14 ((𝑓:𝐵1-1→(1...(♯‘𝐵)) ∧ (𝑓𝑝) ∈ 𝐵𝐴𝐵) → ((𝑓‘(𝑓𝑝)) ∈ (𝑓𝐴) ↔ (𝑓𝑝) ∈ 𝐴))
6663, 59, 64, 65syl3anc 1238 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ((𝑓‘(𝑓𝑝)) ∈ (𝑓𝐴) ↔ (𝑓𝑝) ∈ 𝐴))
67 f1ocnvfv1 5772 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑝 ∈ (1...(♯‘𝐵))) → (𝑓‘(𝑓𝑝)) = 𝑝)
6856, 58, 67syl2anc 411 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑓‘(𝑓𝑝)) = 𝑝)
6968eleq1d 2246 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ((𝑓‘(𝑓𝑝)) ∈ (𝑓𝐴) ↔ 𝑝 ∈ (𝑓𝐴)))
7066, 69bitr3d 190 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ((𝑓𝑝) ∈ 𝐴𝑝 ∈ (𝑓𝐴)))
7170dcbid 838 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → (DECID (𝑓𝑝) ∈ 𝐴DECID 𝑝 ∈ (𝑓𝐴)))
7260, 71mpbid 147 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → DECID 𝑝 ∈ (𝑓𝐴))
7316ad2antrr 488 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
74 simpr 110 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → ¬ 𝑝 ∈ (1...(♯‘𝐵)))
7573, 74ssneldd 3158 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → ¬ 𝑝 ∈ (𝑓𝐴))
7675olcd 734 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑝 ∈ (𝑓𝐴) ∨ ¬ 𝑝 ∈ (𝑓𝐴)))
77 df-dc 835 . . . . . . . . . . 11 (DECID 𝑝 ∈ (𝑓𝐴) ↔ (𝑝 ∈ (𝑓𝐴) ∨ ¬ 𝑝 ∈ (𝑓𝐴)))
7876, 77sylibr 134 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → DECID 𝑝 ∈ (𝑓𝐴))
79 eluzelz 9526 . . . . . . . . . . . . 13 (𝑝 ∈ (ℤ‘1) → 𝑝 ∈ ℤ)
8079adantl 277 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → 𝑝 ∈ ℤ)
81 1zzd 9269 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → 1 ∈ ℤ)
8248adantr 276 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℕ)
8382nnzd 9363 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℤ)
84 fzdcel 10026 . . . . . . . . . . . 12 ((𝑝 ∈ ℤ ∧ 1 ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → DECID 𝑝 ∈ (1...(♯‘𝐵)))
8580, 81, 83, 84syl3anc 1238 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → DECID 𝑝 ∈ (1...(♯‘𝐵)))
86 exmiddc 836 . . . . . . . . . . 11 (DECID 𝑝 ∈ (1...(♯‘𝐵)) → (𝑝 ∈ (1...(♯‘𝐵)) ∨ ¬ 𝑝 ∈ (1...(♯‘𝐵))))
8785, 86syl 14 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → (𝑝 ∈ (1...(♯‘𝐵)) ∨ ¬ 𝑝 ∈ (1...(♯‘𝐵))))
8872, 78, 87mpjaodan 798 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → DECID 𝑝 ∈ (𝑓𝐴))
8988ralrimiva 2550 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑝 ∈ (ℤ‘1)DECID 𝑝 ∈ (𝑓𝐴))
90 1zzd 9269 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 1 ∈ ℤ)
91 eldifi 3257 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
9291, 20sylan2 286 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ 𝐵)
93 eldifn 3258 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → ¬ 𝑛 ∈ (𝑓𝐴))
9493adantl 277 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ 𝑛 ∈ (𝑓𝐴))
9591adantl 277 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
9619adantr 276 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
9795, 96mpbirand 441 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑓𝑛) ∈ 𝐴))
9894, 97mtbid 672 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ (𝑓𝑛) ∈ 𝐴)
9992, 98eldifd 3139 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ (𝐵𝐴))
100 difss 3261 . . . . . . . . . . . . 13 (𝐵𝐴) ⊆ 𝐵
101 resmpt 4951 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
102100, 101ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
103102fveq1i 5512 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
104 fvres 5535 . . . . . . . . . . 11 ((𝑓𝑛) ∈ (𝐵𝐴) → (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
105103, 104eqtr3id 2224 . . . . . . . . . 10 ((𝑓𝑛) ∈ (𝐵𝐴) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
10699, 105syl 14 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
107 1ex 7943 . . . . . . . . . . . . . . 15 1 ∈ V
108107elsn2 3625 . . . . . . . . . . . . . 14 (𝐶 ∈ {1} ↔ 𝐶 = 1)
10929, 108sylibr 134 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {1})
110109fmpttd 5667 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{1})
111110ad2antrr 488 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{1})
112111, 99ffvelcdmd 5648 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {1})
113 elsni 3609 . . . . . . . . . 10 (((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {1} → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 1)
114112, 113syl 14 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 1)
115106, 114eqtr3d 2212 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) = 1)
116 fzssuz 10051 . . . . . . . . 9 (1...(♯‘𝐵)) ⊆ (ℤ‘1)
117116a1i 9 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ⊆ (ℤ‘1))
11885ralrimiva 2550 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑝 ∈ (ℤ‘1)DECID 𝑝 ∈ (1...(♯‘𝐵)))
11916, 46, 52, 89, 90, 115, 117, 118prodssdc 11581 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)) = ∏𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
1201adantr 276 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴𝐵)
121120resmptd 4954 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
122121fveq1d 5513 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
123 fvres 5535 . . . . . . . . . 10 (𝑚𝐴 → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
124122, 123sylan9req 2231 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
125124prodeq2dv 11558 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
126 fveq2 5511 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐵𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
127 fprodss.4 . . . . . . . . . . . 12 (𝜑𝐵 ∈ Fin)
128127adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐵 ∈ Fin)
12936adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑗𝐵 DECID 𝑗𝐴)
130 ssfidc 6928 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵 ∧ ∀𝑗𝐵 DECID 𝑗𝐴) → 𝐴 ∈ Fin)
131128, 120, 129, 130syl3anc 1238 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴 ∈ Fin)
132120, 13, 131preimaf1ofi 6944 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
133 f1of1 5456 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))–1-1𝐵)
13413, 133syl 14 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1𝐵)
135 f1ores 5472 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐵))–1-1𝐵 ∧ (𝑓𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
136134, 16, 135syl2anc 411 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
137 f1ofo 5464 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))–onto𝐵)
13813, 137syl 14 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–onto𝐵)
139 foimacnv 5475 . . . . . . . . . . . 12 ((𝑓:(1...(♯‘𝐵))–onto𝐵𝐴𝐵) → (𝑓 “ (𝑓𝐴)) = 𝐴)
140138, 120, 139syl2anc 411 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 “ (𝑓𝐴)) = 𝐴)
141140f1oeq3d 5454 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
142136, 141mpbid 147 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
143 fvres 5535 . . . . . . . . . 10 (𝑛 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
144143adantl 277 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
145120sselda 3155 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝑚𝐵)
14644ffvelcdmda 5647 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐵) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
147145, 146syldan 282 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
148126, 132, 142, 144, 147fprodf1o 11580 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
149125, 148eqtrd 2210 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
15048nnzd 9363 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℤ)
15190, 150fzfigd 10417 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
152 eqidd 2178 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) = (𝑓𝑛))
153126, 151, 13, 152, 146fprodf1o 11580 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
154119, 149, 1533eqtr4d 2220 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚))
15525ralrimiva 2550 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)
156155adantr 276 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑘𝐴 𝐶 ∈ ℂ)
157 prodfct 11579 . . . . . . 7 (∀𝑘𝐴 𝐶 ∈ ℂ → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶)
158156, 157syl 14 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶)
15943ralrimiva 2550 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℂ)
160 prodfct 11579 . . . . . . 7 (∀𝑘𝐵 𝐶 ∈ ℂ → ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑘𝐵 𝐶)
161159, 160syl 14 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑘𝐵 𝐶)
162154, 158, 1613eqtr3d 2218 . . . . 5 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
163162expr 375 . . . 4 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
164163exlimdv 1819 . . 3 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
165164expimpd 363 . 2 (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
166 fz1f1o 11367 . . 3 (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)))
167127, 166syl 14 . 2 (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)))
16811, 165, 167mpjaod 718 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wral 2455  cdif 3126  wss 3129  c0 3422  {csn 3591  cmpt 4061  ccnv 4622  cres 4625  cima 4626   Fn wfn 5207  wf 5208  1-1wf1 5209  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Fincfn 6734  cc 7800  1c1 7803  cn 8908  cz 9242  cuz 9517  ...cfz 9995  chash 10739  cprod 11542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543
This theorem is referenced by:  fprodsplitdc  11588
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