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Theorem fproddiv 14735
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 10755 . . . . 5 (1 / 1) = 1
21eqcomi 2660 . . . 4 1 = (1 / 1)
3 prodeq1 14683 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = ∏𝑘 ∈ ∅ (𝐵 / 𝐶))
4 prod0 14717 . . . . 5 𝑘 ∈ ∅ (𝐵 / 𝐶) = 1
53, 4syl6eq 2701 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = 1)
6 prodeq1 14683 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
7 prod0 14717 . . . . . 6 𝑘 ∈ ∅ 𝐵 = 1
86, 7syl6eq 2701 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = 1)
9 prodeq1 14683 . . . . . 6 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
10 prod0 14717 . . . . . 6 𝑘 ∈ ∅ 𝐶 = 1
119, 10syl6eq 2701 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = 1)
128, 11oveq12d 6708 . . . 4 (𝐴 = ∅ → (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶) = (1 / 1))
132, 5, 123eqtr4a 2711 . . 3 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
1413a1i 11 . 2 (𝜑 → (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
15 simprl 809 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ ℕ)
16 nnuz 11761 . . . . . . . . 9 ℕ = (ℤ‘1)
1715, 16syl6eleq 2740 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
18 fprodmul.2 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
19 eqid 2651 . . . . . . . . . . 11 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
2018, 19fmptd 6425 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
21 f1of 6175 . . . . . . . . . . 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 6096 . . . . . . . . . 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 6399 . . . . . . . 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 2651 . . . . . . . . . . 11 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
2826, 27fmptd 6425 . . . . . . . . . 10 (𝜑 → (𝑘𝐴𝐶):𝐴⟶ℂ)
29 fco 6096 . . . . . . . . . 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 6399 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
32 simprr 811 . . . . . . . . . . 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 6314 . . . . . . . . . 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 6399 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
37 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
3827fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
3937, 26, 38syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
40 fproddiv.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → 𝐶 ≠ 0)
4139, 40eqnetrd 2890 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4241ralrimiva 2995 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
4342ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0)
44 nffvmpt1 6237 . . . . . . . . . . . 12 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
45 nfcv 2793 . . . . . . . . . . . 12 𝑘0
4644, 45nfne 2923 . . . . . . . . . . 11 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0
47 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
4847neeq1d 2882 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐶)‘𝑘) ≠ 0 ↔ ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
4946, 48rspc 3334 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴𝐶)‘𝑘) ≠ 0 → ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0))
5036, 43, 49sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ((𝑘𝐴𝐶)‘(𝑓𝑛)) ≠ 0)
5135, 50eqnetrd 2890 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ≠ 0)
5218, 26, 40divcld 10839 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐵 / 𝐶) ∈ ℂ)
53 eqid 2651 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐵 / 𝐶)) = (𝑘𝐴 ↦ (𝐵 / 𝐶))
5453fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐵 / 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
5537, 52, 54syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (𝐵 / 𝐶))
5619fvmpt2 6330 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5737, 18, 56syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5857, 39oveq12d 6708 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 / 𝐶))
5955, 58eqtr4d 2688 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6059ralrimiva 2995 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
6160ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)))
62 nffvmpt1 6237 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛))
63 nffvmpt1 6237 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
64 nfcv 2793 . . . . . . . . . . . . 13 𝑘 /
6563, 64, 44nfov 6716 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6662, 65nfeq 2805 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))
67 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
68 fveq2 6229 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6968, 47oveq12d 6708 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛))))
7067, 69eqeq12d 2666 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7166, 70rspc 3334 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) / ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) / ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
7236, 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 6314 . . . . . . . . . 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 6314 . . . . . . . . . . 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 6708 . . . . . . . . 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 2695 . . . . . . . 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 14672 . . . . . . 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 6229 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘(𝑓𝑛)))
8152, 53fmptd 6425 . . . . . . . . . 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 6399 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) ∈ ℂ)
8480, 15, 32, 83, 74fprod 14715 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (seq1( · , ((𝑘𝐴 ↦ (𝐵 / 𝐶)) ∘ 𝑓))‘(#‘𝐴)))
85 fveq2 6229 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
8620adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
8786ffvelrnda 6399 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8885, 15, 32, 87, 76fprod 14715 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴)))
89 fveq2 6229 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9028adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
9190ffvelrnda 6399 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
9289, 15, 32, 91, 35fprod 14715 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(#‘𝐴)))
9388, 92oveq12d 6708 . . . . . . 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 2695 . . . . . 6 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
95 prodfc 14719 . . . . . 6 𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 / 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 / 𝐶)
96 prodfc 14719 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
97 prodfc 14719 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶
9896, 97oveq12i 6702 . . . . . 6 (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) / ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)
9994, 95, 983eqtr3g 2708 . . . . 5 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
10099expr 642 . . . 4 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
101100exlimdv 1901 . . 3 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶)))
102101expimpd 628 . 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 14485 . . 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 1523  wex 1744  wcel 2030  wne 2823  wral 2941  c0 3948  cmpt 4762  ccom 5147  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  0cc0 9974  1c1 9975   · cmul 9979   / cdiv 10722  cn 11058  cuz 11725  ...cfz 12364  seqcseq 12841  #chash 13157  cprod 14679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-prod 14680
This theorem is referenced by:  fproddivf  14762  bcprod  31750
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