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Theorem fproddiv 14530
 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 10596 . . . . 5 (1 / 1) = 1
21eqcomi 2619 . . . 4 1 = (1 / 1)
3 prodeq1 14478 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = ∏𝑘 ∈ ∅ (𝐵 / 𝐶))
4 prod0 14512 . . . . 5 𝑘 ∈ ∅ (𝐵 / 𝐶) = 1
53, 4syl6eq 2660 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = 1)
6 prodeq1 14478 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
7 prod0 14512 . . . . . 6 𝑘 ∈ ∅ 𝐵 = 1
86, 7syl6eq 2660 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = 1)
9 prodeq1 14478 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10 prod0 14512 . . . . . 6 𝑘 ∈ ∅ 𝐶 = 1
119, 10syl6eq 2660 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = 1)
128, 11oveq12d 6567 . . . 4 (𝐴 = ∅ → (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶) = (1 / 1))
132, 5, 123eqtr4a 2670 . . 3 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
1413a1i 11 . 2 (𝜑 → (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
15 simprl 790 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ ℕ)
16 nnuz 11599 . . . . . . . . 9 ℕ = (ℤ‘1)
1715, 16syl6eleq 2698 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
18 fprodmul.2 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
19 eqid 2610 . . . . . . . . . . 11 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
2018, 19fmptd 6292 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
21 f1of 6050 . . . . . . . . . . 11 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))⟶𝐴)
2221adantl 481 . . . . . . . . . 10 (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → 𝑓:(1...(#‘𝐴))⟶𝐴)
23 fco 5971 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(#‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
2420, 22, 23syl2an 493 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
2524ffvelrnda 6267 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
26 fprodmul.3 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
27 eqid 2610 . . . . . . . . . . 11 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
2826, 27fmptd 6292 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐶):𝐴⟶ℂ)
29 fco 5971 . . . . . . . . . 10 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(#‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
3028, 22, 29syl2an 493 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
3130ffvelrnda 6267 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
32 simprr 792 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)
3332, 21syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))⟶𝐴)
34 fvco3 6185 . . . . . . . . . 10 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
3533, 34sylan 487 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
3633ffvelrnda 6267 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
37 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
3827fvmpt2 6200 . . . . . . . . . . . . . 14 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
3937, 26, 38syl2anc 691 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
40 fproddiv.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → 𝐶 ≠ 0)
4139, 40eqnetrd 2849 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4241ralrimiva 2949 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4342ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
44 nffvmpt1 6111 . . . . . . . . . . . 12 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
45 nfcv 2751 . . . . . . . . . . . 12 𝑘0
4644, 45nfne 2882 . . . . . . . . . . 11 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0
47 fveq2 6103 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
4847neeq1d 2841 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐶)‘𝑘) ≠ 0 ↔ ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
4946, 48rspc 3276 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0 → ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
5036, 43, 49sylc 63 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0)
5135, 50eqnetrd 2849 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ≠ 0)
5218, 26, 40divcld 10680 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐵 / 𝐶) ∈ ℂ)
53 eqid 2610 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐵 / 𝐶)) = (𝑘𝐴 ↦ (𝐵 / 𝐶))
5453fvmpt2 6200 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐵 / 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
5537, 52, 54syl2anc 691 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
5619fvmpt2 6200 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5737, 18, 56syl2anc 691 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5857, 39oveq12d 6567 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 / 𝐶))
5955, 58eqtr4d 2647 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6059ralrimiva 2949 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6160ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
62 nffvmpt1 6111 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛))
63 nffvmpt1 6111 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
64 nfcv 2751 . . . . . . . . . . . . 13 𝑘 /
6563, 64, 44nfov 6575 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6662, 65nfeq 2762 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
67 fveq2 6103 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
68 fveq2 6103 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6968, 47oveq12d 6567 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
7067, 69eqeq12d 2625 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7166, 70rspc 3276 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7236, 61, 71sylc 63 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
73 fvco3 6185 . . . . . . . . . 10 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
7433, 73sylan 487 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
75 fvco3 6185 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7633, 75sylan 487 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7776, 35oveq12d 6567 . . . . . . . . 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 2654 . . . . . . . 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, 25, 31, 51, 78prodfdiv 14467 . . . . . . 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 6103 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
8152, 53fmptd 6292 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐵 / 𝐶)):𝐴⟶ℂ)
8281adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 / 𝐶)):𝐴⟶ℂ)
8382ffvelrnda 6267 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) ∈ ℂ)
8480, 15, 32, 83, 74fprod 14510 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (seq1( · , ((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓))‘(#‘𝐴)))
85 fveq2 6103 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
8620adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
8786ffvelrnda 6267 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8885, 15, 32, 87, 76fprod 14510 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴)))
89 fveq2 6103 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9028adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
9190ffvelrnda 6267 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
9289, 15, 32, 91, 35fprod 14510 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(#‘𝐴)))
9388, 92oveq12d 6567 . . . . . . 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 2654 . . . . . 6 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
95 prodfc 14514 . . . . . 6 𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 / 𝐶)
96 prodfc 14514 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
97 prodfc 14514 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶
9896, 97oveq12i 6561 . . . . . 6 (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)
9994, 95, 983eqtr3g 2667 . . . . 5 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
10099expr 641 . . . 4 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
101100exlimdv 1848 . . 3 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
102101expimpd 627 . 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 14288 . . 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 395 1 (𝜑 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  0cc0 9815  1c1 9816   · cmul 9820   / cdiv 10563  ℕcn 10897  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-prod 14475 This theorem is referenced by:  fproddivf  14557  bcprod  30877
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