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Theorem fproddiv 15994
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 11956 . . . . 5 (1 / 1) = 1
21eqcomi 2744 . . . 4 1 = (1 / 1)
3 prodeq1 15940 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = ∏𝑘 ∈ ∅ (𝐵 / 𝐶))
4 prod0 15976 . . . . 5 𝑘 ∈ ∅ (𝐵 / 𝐶) = 1
53, 4eqtrdi 2791 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = 1)
6 prodeq1 15940 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
7 prod0 15976 . . . . . 6 𝑘 ∈ ∅ 𝐵 = 1
86, 7eqtrdi 2791 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = 1)
9 prodeq1 15940 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10 prod0 15976 . . . . . 6 𝑘 ∈ ∅ 𝐶 = 1
119, 10eqtrdi 2791 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = 1)
128, 11oveq12d 7449 . . . 4 (𝐴 = ∅ → (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶) = (1 / 1))
132, 5, 123eqtr4a 2801 . . 3 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
1413a1i 11 . 2 (𝜑 → (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
15 simprl 771 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
16 nnuz 12919 . . . . . . . . 9 ℕ = (ℤ‘1)
1715, 16eleqtrdi 2849 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
18 fprodmul.2 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1918fmpttd 7135 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
20 f1of 6849 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2120adantl 481 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
22 fco 6761 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2319, 21, 22syl2an 596 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2423ffvelcdmda 7104 . . . . . . . 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 7135 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐶):𝐴⟶ℂ)
27 fco 6761 . . . . . . . . . 10 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2826, 21, 27syl2an 596 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2928ffvelcdmda 7104 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
30 simprr 773 . . . . . . . . . . 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 7008 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
3331, 32sylan 580 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
3431ffvelcdmda 7104 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
35 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
36 eqid 2735 . . . . . . . . . . . . . . 15 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
3736fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
3835, 25, 37syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
39 fproddiv.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → 𝐶 ≠ 0)
4038, 39eqnetrd 3006 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4140ralrimiva 3144 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4241ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
43 nffvmpt1 6918 . . . . . . . . . . . 12 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
44 nfcv 2903 . . . . . . . . . . . 12 𝑘0
4543, 44nfne 3041 . . . . . . . . . . 11 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0
46 fveq2 6907 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
4746neeq1d 2998 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐶)‘𝑘) ≠ 0 ↔ ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
4845, 47rspc 3610 . . . . . . . . . 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 3006 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ≠ 0)
5118, 25, 39divcld 12041 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐵 / 𝐶) ∈ ℂ)
52 eqid 2735 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐵 / 𝐶)) = (𝑘𝐴 ↦ (𝐵 / 𝐶))
5352fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐵 / 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
5435, 51, 53syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
55 eqid 2735 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5655fvmpt2 7027 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5735, 18, 56syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5857, 38oveq12d 7449 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 / 𝐶))
5954, 58eqtr4d 2778 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6059ralrimiva 3144 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6160ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
62 nffvmpt1 6918 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛))
63 nffvmpt1 6918 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
64 nfcv 2903 . . . . . . . . . . . . 13 𝑘 /
6563, 64, 43nfov 7461 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6662, 65nfeq 2917 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
67 fveq2 6907 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
68 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6968, 46oveq12d 7449 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
7067, 69eqeq12d 2751 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7166, 70rspc 3610 . . . . . . . . . 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 7008 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
7431, 73sylan 580 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
75 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7631, 75sylan 580 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7776, 33oveq12d 7449 . . . . . . . . 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 2785 . . . . . . . 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 15929 . . . . . . 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 6907 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
8151fmpttd 7135 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐵 / 𝐶)):𝐴⟶ℂ)
8281adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 / 𝐶)):𝐴⟶ℂ)
8382ffvelcdmda 7104 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) ∈ ℂ)
8480, 15, 30, 83, 74fprod 15974 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (seq1( · , ((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓))‘(♯‘𝐴)))
85 fveq2 6907 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
8619adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
8786ffvelcdmda 7104 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8885, 15, 30, 87, 76fprod 15974 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
89 fveq2 6907 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9026adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
9190ffvelcdmda 7104 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
9289, 15, 30, 91, 33fprod 15974 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴)))
9388, 92oveq12d 7449 . . . . . . 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 2785 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
95 prodfc 15978 . . . . . 6 𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 / 𝐶)
96 prodfc 15978 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
97 prodfc 15978 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶
9896, 97oveq12i 7443 . . . . . 6 (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)
9994, 95, 983eqtr3g 2798 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
10099expr 456 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
101100exlimdv 1931 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
102101expimpd 453 . 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 15743 . . 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 860 1 (𝜑 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  c0 4339  cmpt 5231  ccom 5693  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Fincfn 8984  cc 11151  0cc0 11153  1c1 11154   · cmul 11158   / cdiv 11918  cn 12264  cuz 12876  ...cfz 13544  seqcseq 14039  chash 14366  cprod 15936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-prod 15937
This theorem is referenced by:  fproddivf  16020  bcprod  35718
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