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Theorem fprodssdc 11469
 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 3152 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊆ ∅))
3 ss0 3434 . . . . 5 (𝐴 ⊆ ∅ → 𝐴 = ∅)
42, 3syl6bi 162 . . . 4 (𝐵 = ∅ → (𝐴𝐵𝐴 = ∅))
5 prodeq1 11432 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6 prodeq1 11432 . . . . . . 7 (𝐵 = ∅ → ∏𝑘𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
76eqcomd 2163 . . . . . 6 (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘𝐵 𝐶)
85, 7sylan9eq 2210 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
98expcom 115 . . . 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 4946 . . . . . . . . 9 (𝑓𝐴) ⊆ dom 𝑓
13 simprr 522 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)
14 f1of 5411 . . . . . . . . . 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 5331 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
17 f1ofn 5412 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓 Fn (1...(♯‘𝐵)))
18 elpreima 5583 . . . . . . . . . . . 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)))
2015ffvelrnda 5599 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) ∈ 𝐵)
2120ex 114 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓𝑛) ∈ 𝐵))
2221adantrd 277 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴) → (𝑓𝑛) ∈ 𝐵))
2319, 22sylbid 149 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) → (𝑓𝑛) ∈ 𝐵))
2423imp 123 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → (𝑓𝑛) ∈ 𝐵)
25 fprodss.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2625ex 114 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴𝐶 ∈ ℂ))
2726adantr 274 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴𝐶 ∈ ℂ))
28 eldif 3111 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐵𝐴) ↔ (𝑘𝐵 ∧ ¬ 𝑘𝐴))
29 fprodss.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)
30 ax-1cn 7808 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
3129, 30eqeltrdi 2248 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ ℂ)
3228, 31sylan2br 286 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝐵 ∧ ¬ 𝑘𝐴)) → 𝐶 ∈ ℂ)
3332expr 373 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (¬ 𝑘𝐴𝐶 ∈ ℂ))
34 eleq1 2220 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
3534dcbid 824 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
36 fprodssdc.a . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)
3736adantr 274 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐵) → ∀𝑗𝐵 DECID 𝑗𝐴)
38 simpr 109 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐵) → 𝑘𝐵)
3935, 37, 38rspcdva 2821 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐵) → DECID 𝑘𝐴)
40 exmiddc 822 . . . . . . . . . . . . . 14 (DECID 𝑘𝐴 → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
4139, 40syl 14 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
4227, 33, 41mpjaod 708 . . . . . . . . . . . 12 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
4342adantlr 469 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
4443fmpttd 5619 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
4544ffvelrnda 5599 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ (𝑓𝑛) ∈ 𝐵) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
4624, 45syldan 280 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
47 eqid 2157 . . . . . . . . 9 (ℤ‘1) = (ℤ‘1)
48 simprl 521 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℕ)
49 nnuz 9457 . . . . . . . . . 10 ℕ = (ℤ‘1)
5048, 49eleqtrdi 2250 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ (ℤ‘1))
51 ssidd 3149 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ⊆ (1...(♯‘𝐵)))
5247, 50, 51fprodntrivap 11463 . . . . . . . 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 2220 . . . . . . . . . . . . 13 (𝑗 = (𝑓𝑝) → (𝑗𝐴 ↔ (𝑓𝑝) ∈ 𝐴))
5453dcbid 824 . . . . . . . . . . . 12 (𝑗 = (𝑓𝑝) → (DECID 𝑗𝐴DECID (𝑓𝑝) ∈ 𝐴))
5536ad3antrrr 484 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ∀𝑗𝐵 DECID 𝑗𝐴)
5613ad2antrr 480 . . . . . . . . . . . . . 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 109 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝑝 ∈ (1...(♯‘𝐵)))
5957, 58ffvelrnd 5600 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑓𝑝) ∈ 𝐵)
6054, 55, 59rspcdva 2821 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → DECID (𝑓𝑝) ∈ 𝐴)
61 f1ocnv 5424 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:𝐵1-1-onto→(1...(♯‘𝐵)))
62 f1of1 5410 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → 𝐴𝐵)
65 f1elima 5718 . . . . . . . . . . . . . 14 ((𝑓:𝐵1-1→(1...(♯‘𝐵)) ∧ (𝑓𝑝) ∈ 𝐵𝐴𝐵) → ((𝑓‘(𝑓𝑝)) ∈ (𝑓𝐴) ↔ (𝑓𝑝) ∈ 𝐴))
6663, 59, 64, 65syl3anc 1220 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ((𝑓‘(𝑓𝑝)) ∈ (𝑓𝐴) ↔ (𝑓𝑝) ∈ 𝐴))
67 f1ocnvfv1 5722 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑝 ∈ (1...(♯‘𝐵))) → (𝑓‘(𝑓𝑝)) = 𝑝)
6856, 58, 67syl2anc 409 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑓‘(𝑓𝑝)) = 𝑝)
6968eleq1d 2226 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ((𝑓‘(𝑓𝑝)) ∈ (𝑓𝐴) ↔ 𝑝 ∈ (𝑓𝐴)))
7066, 69bitr3d 189 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → ((𝑓𝑝) ∈ 𝐴𝑝 ∈ (𝑓𝐴)))
7170dcbid 824 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → (DECID (𝑓𝑝) ∈ 𝐴DECID 𝑝 ∈ (𝑓𝐴)))
7260, 71mpbid 146 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ∈ (1...(♯‘𝐵))) → DECID 𝑝 ∈ (𝑓𝐴))
7316ad2antrr 480 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
74 simpr 109 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → ¬ 𝑝 ∈ (1...(♯‘𝐵)))
7573, 74ssneldd 3131 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → ¬ 𝑝 ∈ (𝑓𝐴))
7675olcd 724 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → (𝑝 ∈ (𝑓𝐴) ∨ ¬ 𝑝 ∈ (𝑓𝐴)))
77 df-dc 821 . . . . . . . . . . 11 (DECID 𝑝 ∈ (𝑓𝐴) ↔ (𝑝 ∈ (𝑓𝐴) ∨ ¬ 𝑝 ∈ (𝑓𝐴)))
7876, 77sylibr 133 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ∈ (1...(♯‘𝐵))) → DECID 𝑝 ∈ (𝑓𝐴))
79 eluzelz 9431 . . . . . . . . . . . . 13 (𝑝 ∈ (ℤ‘1) → 𝑝 ∈ ℤ)
8079adantl 275 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → 𝑝 ∈ ℤ)
81 1zzd 9177 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → 1 ∈ ℤ)
8248adantr 274 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℕ)
8382nnzd 9268 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℤ)
84 fzdcel 9924 . . . . . . . . . . . 12 ((𝑝 ∈ ℤ ∧ 1 ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → DECID 𝑝 ∈ (1...(♯‘𝐵)))
8580, 81, 83, 84syl3anc 1220 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → DECID 𝑝 ∈ (1...(♯‘𝐵)))
86 exmiddc 822 . . . . . . . . . . 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 788 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑝 ∈ (ℤ‘1)) → DECID 𝑝 ∈ (𝑓𝐴))
8988ralrimiva 2530 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑝 ∈ (ℤ‘1)DECID 𝑝 ∈ (𝑓𝐴))
90 1zzd 9177 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 1 ∈ ℤ)
91 eldifi 3229 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
9291, 20sylan2 284 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ 𝐵)
93 eldifn 3230 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → ¬ 𝑛 ∈ (𝑓𝐴))
9493adantl 275 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ 𝑛 ∈ (𝑓𝐴))
9591adantl 275 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
9619adantr 274 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
9795, 96mpbirand 438 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑓𝑛) ∈ 𝐴))
9894, 97mtbid 662 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ (𝑓𝑛) ∈ 𝐴)
9992, 98eldifd 3112 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ (𝐵𝐴))
100 difss 3233 . . . . . . . . . . . . 13 (𝐵𝐴) ⊆ 𝐵
101 resmpt 4911 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
102100, 101ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
103102fveq1i 5466 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
104 fvres 5489 . . . . . . . . . . 11 ((𝑓𝑛) ∈ (𝐵𝐴) → (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
105103, 104syl5eqr 2204 . . . . . . . . . 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 7856 . . . . . . . . . . . . . . 15 1 ∈ V
108107elsn2 3594 . . . . . . . . . . . . . 14 (𝐶 ∈ {1} ↔ 𝐶 = 1)
10929, 108sylibr 133 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {1})
110109fmpttd 5619 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{1})
111110ad2antrr 480 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{1})
112111, 99ffvelrnd 5600 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {1})
113 elsni 3578 . . . . . . . . . 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 2192 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) = 1)
116 fzssuz 9949 . . . . . . . . 9 (1...(♯‘𝐵)) ⊆ (ℤ‘1)
117116a1i 9 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ⊆ (ℤ‘1))
11885ralrimiva 2530 . . . . . . . 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 11468 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)) = ∏𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
1201adantr 274 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴𝐵)
121120resmptd 4914 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
122121fveq1d 5467 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
123 fvres 5489 . . . . . . . . . 10 (𝑚𝐴 → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
124122, 123sylan9req 2211 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
125124prodeq2dv 11445 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
126 fveq2 5465 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐵𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
127 fprodss.4 . . . . . . . . . . . 12 (𝜑𝐵 ∈ Fin)
128127adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐵 ∈ Fin)
12936adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑗𝐵 DECID 𝑗𝐴)
130 ssfidc 6872 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵 ∧ ∀𝑗𝐵 DECID 𝑗𝐴) → 𝐴 ∈ Fin)
131128, 120, 129, 130syl3anc 1220 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴 ∈ Fin)
132120, 13, 131preimaf1ofi 6888 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
133 f1of1 5410 . . . . . . . . . . . 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 5426 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐵))–1-1𝐵 ∧ (𝑓𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
136134, 16, 135syl2anc 409 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
137 f1ofo 5418 . . . . . . . . . . . . 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 5429 . . . . . . . . . . . 12 ((𝑓:(1...(♯‘𝐵))–onto𝐵𝐴𝐵) → (𝑓 “ (𝑓𝐴)) = 𝐴)
140138, 120, 139syl2anc 409 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 “ (𝑓𝐴)) = 𝐴)
141140f1oeq3d 5408 . . . . . . . . . 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 146 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
143 fvres 5489 . . . . . . . . . 10 (𝑛 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
144143adantl 275 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
145120sselda 3128 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝑚𝐵)
14644ffvelrnda 5599 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐵) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
147145, 146syldan 280 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
148126, 132, 142, 144, 147fprodf1o 11467 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
149125, 148eqtrd 2190 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
15048nnzd 9268 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℤ)
15190, 150fzfigd 10312 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
152 eqidd 2158 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) = (𝑓𝑛))
153126, 151, 13, 152, 146fprodf1o 11467 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
154119, 149, 1533eqtr4d 2200 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚))
15525ralrimiva 2530 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)
156155adantr 274 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑘𝐴 𝐶 ∈ ℂ)
157 prodfct 11466 . . . . . . 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 2530 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑘𝐵 𝐶 ∈ ℂ)
160 prodfct 11466 . . . . . . 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 2198 . . . . 5 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
163162expr 373 . . . 4 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
164163exlimdv 1799 . . 3 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
165164expimpd 361 . 2 (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
166 fz1f1o 11254 . . 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 708 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435   ∖ cdif 3099   ⊆ wss 3102  ∅c0 3394  {csn 3560   ↦ cmpt 4025  ◡ccnv 4582   ↾ cres 4585   “ cima 4586   Fn wfn 5162  ⟶wf 5163  –1-1→wf1 5164  –onto→wfo 5165  –1-1-onto→wf1o 5166  ‘cfv 5167  (class class class)co 5818  Fincfn 6678  ℂcc 7713  1c1 7716  ℕcn 8816  ℤcz 9150  ℤ≥cuz 9422  ...cfz 9894  ♯chash 10631  ∏cprod 11429 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-en 6679  df-dom 6680  df-fin 6681  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-ihash 10632  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-proddc 11430 This theorem is referenced by:  fprodsplitdc  11475
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