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Theorem fprodssdc 11569
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 11532 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6 prodeq1 11532 . . . . . . 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 4986 . . . . . . . . 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 5456 . . . . . . . . . 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 5375 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
17 f1ofn 5457 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓 Fn (1...(♯‘𝐵)))
18 elpreima 5630 . . . . . . . . . . . 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 5646 . . . . . . . . . . . . 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 7882 . . . . . . . . . . . . . . . 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 5666 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
4544ffvelcdmda 5646 . . . . . . . . 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 9539 . . . . . . . . . 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 11563 . . . . . . . 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 5647 . . . . . . . . . . . 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 5469 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:𝐵1-1-onto→(1...(♯‘𝐵)))
62 f1of1 5455 . . . . . . . . . . . . . . 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 5767 . . . . . . . . . . . . . 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 5771 . . . . . . . . . . . . . . 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 9513 . . . . . . . . . . . . 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 9256 . . . . . . . . . . . 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 9350 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℤ)
84 fzdcel 10013 . . . . . . . . . . . 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 9256 . . . . . . . 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 4950 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
102100, 101ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
103102fveq1i 5511 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
104 fvres 5534 . . . . . . . . . . 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 7930 . . . . . . . . . . . . . . 15 1 ∈ V
108107elsn2 3625 . . . . . . . . . . . . . 14 (𝐶 ∈ {1} ↔ 𝐶 = 1)
10929, 108sylibr 134 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {1})
110109fmpttd 5666 . . . . . . . . . . . 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 5647 . . . . . . . . . 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 10038 . . . . . . . . 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 11568 . . . . . . 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 4953 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
122121fveq1d 5512 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
123 fvres 5534 . . . . . . . . . 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 11545 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
126 fveq2 5510 . . . . . . . . 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 6927 . . . . . . . . . . 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 6943 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
133 f1of1 5455 . . . . . . . . . . . 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 5471 . . . . . . . . . . 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 5463 . . . . . . . . . . . . 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 5474 . . . . . . . . . . . 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 5453 . . . . . . . . . 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 5534 . . . . . . . . . 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 5646 . . . . . . . . . 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 11567 . . . . . . . 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 9350 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℤ)
15190, 150fzfigd 10404 . . . . . . . 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 11567 . . . . . . 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 11566 . . . . . . 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 11566 . . . . . . 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 11354 . . 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 4621  cres 4624  cima 4625   Fn wfn 5206  wf 5207  1-1wf1 5208  ontowfo 5209  1-1-ontowf1o 5210  cfv 5211  (class class class)co 5868  Fincfn 6733  cc 7787  1c1 7790  cn 8895  cz 9229  cuz 9504  ...cfz 9982  chash 10726  cprod 11529
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 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
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 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-frec 6385  df-1o 6410  df-oadd 6414  df-er 6528  df-en 6734  df-dom 6735  df-fin 6736  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-fz 9983  df-fzo 10116  df-seqfrec 10419  df-exp 10493  df-ihash 10727  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979  df-clim 11258  df-proddc 11530
This theorem is referenced by:  fprodsplitdc  11575
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