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Theorem fproddiv 15943
Description: The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)
Hypotheses
Ref Expression
fprodmul.1 (𝜑𝐴 ∈ Fin)
fprodmul.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fprodmul.3 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
fproddiv.4 ((𝜑𝑘𝐴) → 𝐶 ≠ 0)
Assertion
Ref Expression
fproddiv (𝜑 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
Distinct variable groups:   𝐴,𝑘   𝜑,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem fproddiv
Dummy variables 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1div1e1 11940 . . . . 5 (1 / 1) = 1
21eqcomi 2736 . . . 4 1 = (1 / 1)
3 prodeq1 15891 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = ∏𝑘 ∈ ∅ (𝐵 / 𝐶))
4 prod0 15925 . . . . 5 𝑘 ∈ ∅ (𝐵 / 𝐶) = 1
53, 4eqtrdi 2783 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = 1)
6 prodeq1 15891 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
7 prod0 15925 . . . . . 6 𝑘 ∈ ∅ 𝐵 = 1
86, 7eqtrdi 2783 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = 1)
9 prodeq1 15891 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10 prod0 15925 . . . . . 6 𝑘 ∈ ∅ 𝐶 = 1
119, 10eqtrdi 2783 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = 1)
128, 11oveq12d 7442 . . . 4 (𝐴 = ∅ → (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶) = (1 / 1))
132, 5, 123eqtr4a 2793 . . 3 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
1413a1i 11 . 2 (𝜑 → (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
15 simprl 769 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
16 nnuz 12901 . . . . . . . . 9 ℕ = (ℤ‘1)
1715, 16eleqtrdi 2838 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
18 fprodmul.2 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1918fmpttd 7128 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
20 f1of 6842 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2120adantl 480 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
22 fco 6750 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2319, 21, 22syl2an 594 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2423ffvelcdmda 7097 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
25 fprodmul.3 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2625fmpttd 7128 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐶):𝐴⟶ℂ)
27 fco 6750 . . . . . . . . . 10 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2826, 21, 27syl2an 594 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2928ffvelcdmda 7097 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
30 simprr 771 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
3130, 20syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
32 fvco3 7000 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
3331, 32sylan 578 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
3431ffvelcdmda 7097 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
35 simpr 483 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
36 eqid 2727 . . . . . . . . . . . . . . 15 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
3736fvmpt2 7019 . . . . . . . . . . . . . 14 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
3835, 25, 37syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
39 fproddiv.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → 𝐶 ≠ 0)
4038, 39eqnetrd 3004 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4140ralrimiva 3142 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4241ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
43 nffvmpt1 6911 . . . . . . . . . . . 12 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
44 nfcv 2898 . . . . . . . . . . . 12 𝑘0
4543, 44nfne 3039 . . . . . . . . . . 11 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0
46 fveq2 6900 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
4746neeq1d 2996 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐶)‘𝑘) ≠ 0 ↔ ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
4845, 47rspc 3597 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0 → ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
4934, 42, 48sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0)
5033, 49eqnetrd 3004 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ≠ 0)
5118, 25, 39divcld 12026 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐵 / 𝐶) ∈ ℂ)
52 eqid 2727 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐵 / 𝐶)) = (𝑘𝐴 ↦ (𝐵 / 𝐶))
5352fvmpt2 7019 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐵 / 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
5435, 51, 53syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
55 eqid 2727 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5655fvmpt2 7019 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5735, 18, 56syl2anc 582 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5857, 38oveq12d 7442 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 / 𝐶))
5954, 58eqtr4d 2770 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6059ralrimiva 3142 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6160ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
62 nffvmpt1 6911 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛))
63 nffvmpt1 6911 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
64 nfcv 2898 . . . . . . . . . . . . 13 𝑘 /
6563, 64, 43nfov 7454 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6662, 65nfeq 2912 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
67 fveq2 6900 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
68 fveq2 6900 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6968, 46oveq12d 7442 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
7067, 69eqeq12d 2743 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7166, 70rspc 3597 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7234, 61, 71sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
73 fvco3 7000 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
7431, 73sylan 578 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
75 fvco3 7000 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7631, 75sylan 578 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7776, 33oveq12d 7442 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) / (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
7872, 74, 773eqtr4d 2777 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) / (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
7917, 24, 29, 50, 78prodfdiv 15880 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( · , ((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓))‘(♯‘𝐴)) = ((seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)) / (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴))))
80 fveq2 6900 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
8151fmpttd 7128 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐵 / 𝐶)):𝐴⟶ℂ)
8281adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 / 𝐶)):𝐴⟶ℂ)
8382ffvelcdmda 7097 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) ∈ ℂ)
8480, 15, 30, 83, 74fprod 15923 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (seq1( · , ((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓))‘(♯‘𝐴)))
85 fveq2 6900 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
8619adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
8786ffvelcdmda 7097 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8885, 15, 30, 87, 76fprod 15923 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
89 fveq2 6900 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9026adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
9190ffvelcdmda 7097 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
9289, 15, 30, 91, 33fprod 15923 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴)))
9388, 92oveq12d 7442 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = ((seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)) / (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴))))
9479, 84, 933eqtr4d 2777 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
95 prodfc 15927 . . . . . 6 𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 / 𝐶)
96 prodfc 15927 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
97 prodfc 15927 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶
9896, 97oveq12i 7436 . . . . . 6 (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)
9994, 95, 983eqtr3g 2790 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
10099expr 455 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
101100exlimdv 1928 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
102101expimpd 452 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
103 fprodmul.1 . . 3 (𝜑𝐴 ∈ Fin)
104 fz1f1o 15694 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
105103, 104syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
10614, 102, 105mpjaod 858 1 (𝜑 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1533  wex 1773  wcel 2098  wne 2936  wral 3057  c0 4324  cmpt 5233  ccom 5684  wf 6547  1-1-ontowf1o 6550  cfv 6551  (class class class)co 7424  Fincfn 8968  cc 11142  0cc0 11144  1c1 11145   · cmul 11149   / cdiv 11907  cn 12248  cuz 12858  ...cfz 13522  seqcseq 14004  chash 14327  cprod 15887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-inf2 9670  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221  ax-pre-sup 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-se 5636  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-isom 6560  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-sup 9471  df-oi 9539  df-card 9968  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-div 11908  df-nn 12249  df-2 12311  df-3 12312  df-n0 12509  df-z 12595  df-uz 12859  df-rp 13013  df-fz 13523  df-fzo 13666  df-seq 14005  df-exp 14065  df-hash 14328  df-cj 15084  df-re 15085  df-im 15086  df-sqrt 15220  df-abs 15221  df-clim 15470  df-prod 15888
This theorem is referenced by:  fproddivf  15969  bcprod  35337
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