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Theorem fprodmul 15996
Description: The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)
Hypotheses
Ref Expression
fprodmul.1 (𝜑𝐴 ∈ Fin)
fprodmul.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fprodmul.3 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
Assertion
Ref Expression
fprodmul (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 fprodmul
Dummy variables 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1t1e1 12428 . . . . 5 (1 · 1) = 1
2 prod0 15979 . . . . . 6 𝑘 ∈ ∅ 𝐵 = 1
3 prod0 15979 . . . . . 6 𝑘 ∈ ∅ 𝐶 = 1
42, 3oveq12i 7443 . . . . 5 (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5 prod0 15979 . . . . 5 𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
61, 4, 53eqtr4ri 2776 . . . 4 𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7 prodeq1 15943 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8 prodeq1 15943 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9 prodeq1 15943 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
108, 9oveq12d 7449 . . . 4 (𝐴 = ∅ → (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶))
116, 7, 103eqtr4a 2803 . . 3 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
1211a1i 11 . 2 (𝜑 → (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
13 simprl 771 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 nnuz 12921 . . . . . . . . 9 ℕ = (ℤ‘1)
1513, 14eleqtrdi 2851 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
16 fprodmul.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1716fmpttd 7135 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
1817adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
19 f1of 6848 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2019ad2antll 729 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
21 fco 6760 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2218, 20, 21syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2322ffvelcdmda 7104 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
24 fprodmul.3 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2524fmpttd 7135 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐶):𝐴⟶ℂ)
2625adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
27 fco 6760 . . . . . . . . . 10 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2826, 20, 27syl2anc 584 . . . . . . . . 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)
3020ffvelcdmda 7104 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
31 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝑘𝐴)
3216, 24mulcld 11281 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → (𝐵 · 𝐶) ∈ ℂ)
33 eqid 2737 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐵 · 𝐶)) = (𝑘𝐴 ↦ (𝐵 · 𝐶))
3433fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐵 · 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (𝐵 · 𝐶))
3531, 32, 34syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (𝐵 · 𝐶))
36 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
3736fvmpt2 7027 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
3831, 16, 37syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
39 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
4039fvmpt2 7027 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
4131, 24, 40syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
4238, 41oveq12d 7449 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 · 𝐶))
4335, 42eqtr4d 2780 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)))
4443ralrimiva 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)))
4544ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)))
46 nffvmpt1 6917 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛))
47 nffvmpt1 6917 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
48 nfcv 2905 . . . . . . . . . . . . 13 𝑘 ·
49 nffvmpt1 6917 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
5047, 48, 49nfov 7461 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛)))
5146, 50nfeq 2919 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛)))
52 fveq2 6906 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
53 fveq2 6906 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
54 fveq2 6906 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
5553, 54oveq12d 7449 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛))))
5652, 55eqeq12d 2753 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
5751, 56rspc 3610 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) · ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
5830, 45, 57sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛))))
59 fvco3 7008 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
6020, 59sylan 580 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
61 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6220, 61sylan 580 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
63 fvco3 7008 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6420, 63sylan 580 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6562, 64oveq12d 7449 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) · (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) · ((𝑘𝐴𝐶)‘(𝑓𝑛))))
6658, 60, 653eqtr4d 2787 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) · (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
6715, 23, 29, 66prodfmul 15926 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( · , ((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓))‘(♯‘𝐴)) = ((seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)) · (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴))))
68 fveq2 6906 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
69 simprr 773 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
7032fmpttd 7135 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
7170adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
7271ffvelcdmda 7104 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) ∈ ℂ)
7368, 13, 69, 72, 60fprod 15977 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (seq1( · , ((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓))‘(♯‘𝐴)))
74 fveq2 6906 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7518ffvelcdmda 7104 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7674, 13, 69, 75, 62fprod 15977 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
77 fveq2 6906 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
7826ffvelcdmda 7104 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
7977, 13, 69, 78, 64fprod 15977 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( · , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴)))
8076, 79oveq12d 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))))
8167, 73, 803eqtr4d 2787 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) · ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
82 prodfc 15981 . . . . . 6 𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 · 𝐶)
83 prodfc 15981 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
84 prodfc 15981 . . . . . . 7 𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶
8583, 84oveq12i 7443 . . . . . 6 (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) · ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)
8681, 82, 853eqtr3g 2800 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
8786expr 456 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
8887exlimdv 1933 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
8988expimpd 453 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
90 fprodmul.1 . . 3 (𝜑𝐴 ∈ Fin)
91 fz1f1o 15746 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9290, 91syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9312, 89, 92mpjaod 861 1 (𝜑 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wral 3061  c0 4333  cmpt 5225  ccom 5689  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  1c1 11156   · cmul 11160  cn 12266  cuz 12878  ...cfz 13547  seqcseq 14042  chash 14369  cprod 15939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-prod 15940
This theorem is referenced by:  fprodsplit  16002  risefallfac  16060  gausslemma2dlem5  27415  gausslemma2dlem6  27416  bcprod  35738
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