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Theorem fprodss 14614
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)
fprodss.3 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)
fprodss.4 (𝜑𝐵 ∈ Fin)
Assertion
Ref Expression
fprodss (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem fprodss
Dummy variables 𝑓 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fprodss.1 . . 3 (𝜑𝐴𝐵)
2 sseq2 3611 . . . . 5 (𝐵 = ∅ → (𝐴𝐵𝐴 ⊆ ∅))
3 ss0 3951 . . . . 5 (𝐴 ⊆ ∅ → 𝐴 = ∅)
42, 3syl6bi 243 . . . 4 (𝐵 = ∅ → (𝐴𝐵𝐴 = ∅))
5 prodeq1 14575 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
6 prodeq1 14575 . . . . . . 7 (𝐵 = ∅ → ∏𝑘𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
76eqcomd 2627 . . . . . 6 (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘𝐵 𝐶)
85, 7sylan9eq 2675 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
98expcom 451 . . . 4 (𝐵 = ∅ → (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
104, 9syld 47 . . 3 (𝐵 = ∅ → (𝐴𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
111, 10syl5com 31 . 2 (𝜑 → (𝐵 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
12 cnvimass 5449 . . . . . . . . 9 (𝑓𝐴) ⊆ dom 𝑓
13 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)
14 f1of 6099 . . . . . . . . . . 11 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓:(1...(#‘𝐵))⟶𝐵)
1513, 14syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))⟶𝐵)
16 fdm 6013 . . . . . . . . . 10 (𝑓:(1...(#‘𝐵))⟶𝐵 → dom 𝑓 = (1...(#‘𝐵)))
1715, 16syl 17 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → dom 𝑓 = (1...(#‘𝐵)))
1812, 17syl5sseq 3637 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(#‘𝐵)))
19 f1ofn 6100 . . . . . . . . . . . . 13 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓 Fn (1...(#‘𝐵)))
2013, 19syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓 Fn (1...(#‘𝐵)))
21 elpreima 6298 . . . . . . . . . . . 12 (𝑓 Fn (1...(#‘𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
2220, 21syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
2315ffvelrnda 6320 . . . . . . . . . . . . 13 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓𝑛) ∈ 𝐵)
2423ex 450 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (1...(#‘𝐵)) → (𝑓𝑛) ∈ 𝐵))
2524adantrd 484 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ((𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴) → (𝑓𝑛) ∈ 𝐵))
2622, 25sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) → (𝑓𝑛) ∈ 𝐵))
2726imp 445 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → (𝑓𝑛) ∈ 𝐵)
28 fprodss.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2928ex 450 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴𝐶 ∈ ℂ))
3029adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴𝐶 ∈ ℂ))
31 eldif 3569 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐵𝐴) ↔ (𝑘𝐵 ∧ ¬ 𝑘𝐴))
32 fprodss.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)
33 ax-1cn 9946 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
3432, 33syl6eqel 2706 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ ℂ)
3531, 34sylan2br 493 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝐵 ∧ ¬ 𝑘𝐴)) → 𝐶 ∈ ℂ)
3635expr 642 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (¬ 𝑘𝐴𝐶 ∈ ℂ))
3730, 36pm2.61d 170 . . . . . . . . . . . 12 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
3837adantlr 750 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
39 eqid 2621 . . . . . . . . . . 11 (𝑘𝐵𝐶) = (𝑘𝐵𝐶)
4038, 39fmptd 6346 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
4140ffvelrnda 6320 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ (𝑓𝑛) ∈ 𝐵) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
4227, 41syldan 487 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
43 eqid 2621 . . . . . . . . 9 (ℤ‘1) = (ℤ‘1)
44 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (#‘𝐵) ∈ ℕ)
45 nnuz 11675 . . . . . . . . . 10 ℕ = (ℤ‘1)
4644, 45syl6eleq 2708 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (#‘𝐵) ∈ (ℤ‘1))
47 ssid 3608 . . . . . . . . . 10 (1...(#‘𝐵)) ⊆ (1...(#‘𝐵))
4847a1i 11 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (1...(#‘𝐵)) ⊆ (1...(#‘𝐵)))
4943, 46, 48fprodntriv 14608 . . . . . . . 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))
50 eldifi 3715 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴)) → 𝑛 ∈ (1...(#‘𝐵)))
5150, 23sylan2 491 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ 𝐵)
52 eldifn 3716 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴)) → ¬ 𝑛 ∈ (𝑓𝐴))
5352adantl 482 . . . . . . . . . . . 12 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ¬ 𝑛 ∈ (𝑓𝐴))
5422adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
5550adantl 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → 𝑛 ∈ (1...(#‘𝐵)))
5655biantrurd 529 . . . . . . . . . . . . 13 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑓𝑛) ∈ 𝐴 ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
5754, 56bitr4d 271 . . . . . . . . . . . 12 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑓𝑛) ∈ 𝐴))
5853, 57mtbid 314 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ¬ (𝑓𝑛) ∈ 𝐴)
5951, 58eldifd 3570 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ (𝐵𝐴))
60 difss 3720 . . . . . . . . . . . . 13 (𝐵𝐴) ⊆ 𝐵
61 resmpt 5413 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
6260, 61ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
6362fveq1i 6154 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
64 fvres 6169 . . . . . . . . . . 11 ((𝑓𝑛) ∈ (𝐵𝐴) → (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
6563, 64syl5eqr 2669 . . . . . . . . . 10 ((𝑓𝑛) ∈ (𝐵𝐴) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
6659, 65syl 17 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
67 1ex 9987 . . . . . . . . . . . . . . 15 1 ∈ V
6867elsn2 4187 . . . . . . . . . . . . . 14 (𝐶 ∈ {1} ↔ 𝐶 = 1)
6932, 68sylibr 224 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {1})
70 eqid 2621 . . . . . . . . . . . . 13 (𝑘 ∈ (𝐵𝐴) ↦ 𝐶) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
7169, 70fmptd 6346 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{1})
7271ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{1})
7372, 59ffvelrnd 6321 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {1})
74 elsni 4170 . . . . . . . . . 10 (((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {1} → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 1)
7573, 74syl 17 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 1)
7666, 75eqtr3d 2657 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) = 1)
77 fzssuz 12332 . . . . . . . . 9 (1...(#‘𝐵)) ⊆ (ℤ‘1)
7877a1i 11 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (1...(#‘𝐵)) ⊆ (ℤ‘1))
7918, 42, 49, 76, 78prodss 14613 . . . . . . 7 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)) = ∏𝑛 ∈ (1...(#‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
801adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝐴𝐵)
8180resmptd 5416 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
8281fveq1d 6155 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
83 fvres 6169 . . . . . . . . . 10 (𝑚𝐴 → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
8482, 83sylan9req 2676 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
8584prodeq2dv 14589 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
86 fveq2 6153 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐵𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
87 fzfid 12720 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (1...(#‘𝐵)) ∈ Fin)
8887, 15fisuppfi 8235 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
89 f1of1 6098 . . . . . . . . . . . 12 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓:(1...(#‘𝐵))–1-1𝐵)
9013, 89syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))–1-1𝐵)
91 f1ores 6113 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐵))–1-1𝐵 ∧ (𝑓𝐴) ⊆ (1...(#‘𝐵))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
9290, 18, 91syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
93 f1ofo 6106 . . . . . . . . . . . . 13 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓:(1...(#‘𝐵))–onto𝐵)
9413, 93syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))–onto𝐵)
95 foimacnv 6116 . . . . . . . . . . . 12 ((𝑓:(1...(#‘𝐵))–onto𝐵𝐴𝐵) → (𝑓 “ (𝑓𝐴)) = 𝐴)
9694, 80, 95syl2anc 692 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓 “ (𝑓𝐴)) = 𝐴)
97 f1oeq3 6091 . . . . . . . . . . 11 ((𝑓 “ (𝑓𝐴)) = 𝐴 → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
9896, 97syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
9992, 98mpbid 222 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
100 fvres 6169 . . . . . . . . . 10 (𝑛 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
101100adantl 482 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
10280sselda 3587 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝑚𝐵)
10340ffvelrnda 6320 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐵) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
104102, 103syldan 487 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
10586, 88, 99, 101, 104fprodf1o 14612 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
10685, 105eqtrd 2655 . . . . . . 7 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
107 eqidd 2622 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓𝑛) = (𝑓𝑛))
10886, 87, 13, 107, 103fprodf1o 14612 . . . . . . 7 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑛 ∈ (1...(#‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
10979, 106, 1083eqtr4d 2665 . . . . . 6 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚))
110 prodfc 14611 . . . . . 6 𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶
111 prodfc 14611 . . . . . 6 𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = ∏𝑘𝐵 𝐶
112109, 110, 1113eqtr3g 2678 . . . . 5 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
113112expr 642 . . . 4 ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (𝑓:(1...(#‘𝐵))–1-1-onto𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
114113exlimdv 1858 . . 3 ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
115114expimpd 628 . 2 (𝜑 → (((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵) → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶))
116 fprodss.4 . . 3 (𝜑𝐵 ∈ Fin)
117 fz1f1o 14382 . . 3 (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)))
118116, 117syl 17 . 2 (𝜑 → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)))
11911, 115, 118mpjaod 396 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  cdif 3556  wss 3559  c0 3896  {csn 4153  cmpt 4678  ccnv 5078  dom cdm 5079  cres 5081  cima 5082   Fn wfn 5847  wf 5848  1-1wf1 5849  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  Fincfn 7907  cc 9886  1c1 9889  cn 10972  cuz 11639  ...cfz 12276  #chash 13065  cprod 14571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161  df-prod 14572
This theorem is referenced by:  fprodsplit  14632
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