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Theorem fprodmul 11594
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 𝑝 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1t1e1 9069 . . . . 5 (1 · 1) = 1
2 prod0 11588 . . . . . 6 𝑘 ∈ ∅ 𝐵 = 1
3 prod0 11588 . . . . . 6 𝑘 ∈ ∅ 𝐶 = 1
42, 3oveq12i 5886 . . . . 5 (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶) = (1 · 1)
5 prod0 11588 . . . . 5 𝑘 ∈ ∅ (𝐵 · 𝐶) = 1
61, 4, 53eqtr4ri 2209 . . . 4 𝑘 ∈ ∅ (𝐵 · 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶)
7 prodeq1 11556 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 · 𝐶) = ∏𝑘 ∈ ∅ (𝐵 · 𝐶))
8 prodeq1 11556 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
9 prodeq1 11556 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶)
108, 9oveq12d 5892 . . . 4 (𝐴 = ∅ → (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶) = (∏𝑘 ∈ ∅ 𝐵 · ∏𝑘 ∈ ∅ 𝐶))
116, 7, 103eqtr4a 2236 . . 3 (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
1211a1i 9 . 2 (𝜑 → (𝐴 = ∅ → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
13 simprl 529 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 nnuz 9561 . . . . . . . . 9 ℕ = (ℤ‘1)
1513, 14eleqtrdi 2270 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
16 elnnuz 9562 . . . . . . . . . . . 12 (𝑝 ∈ ℕ ↔ 𝑝 ∈ (ℤ‘1))
1716biimpri 133 . . . . . . . . . . 11 (𝑝 ∈ (ℤ‘1) → 𝑝 ∈ ℕ)
1817adantl 277 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → 𝑝 ∈ ℕ)
19 fprodmul.2 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2019fmpttd 5671 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2120adantr 276 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
22 f1of 5461 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2322ad2antll 491 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
24 fco 5381 . . . . . . . . . . . . . 14 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2521, 23, 24syl2anc 411 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2625ad2antrr 488 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
27 simpr 110 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ≤ (♯‘𝐴))
28 simplr 528 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (ℤ‘1))
2913ad2antrr 488 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
3029nnzd 9372 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
31 elfz5 10014 . . . . . . . . . . . . . 14 ((𝑝 ∈ (ℤ‘1) ∧ (♯‘𝐴) ∈ ℤ) → (𝑝 ∈ (1...(♯‘𝐴)) ↔ 𝑝 ≤ (♯‘𝐴)))
3228, 30, 31syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑝 ∈ (1...(♯‘𝐴)) ↔ 𝑝 ≤ (♯‘𝐴)))
3327, 32mpbird 167 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑝 ∈ (1...(♯‘𝐴)))
3426, 33ffvelcdmd 5652 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝) ∈ ℂ)
35 1cnd 7972 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
3618nnzd 9372 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → 𝑝 ∈ ℤ)
3713adantr 276 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℕ)
3837nnzd 9372 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
39 zdcle 9327 . . . . . . . . . . . 12 ((𝑝 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑝 ≤ (♯‘𝐴))
4036, 38, 39syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → DECID 𝑝 ≤ (♯‘𝐴))
4134, 35, 40ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) ∈ ℂ)
42 breq1 4006 . . . . . . . . . . . 12 (𝑛 = 𝑝 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑝 ≤ (♯‘𝐴)))
43 fveq2 5515 . . . . . . . . . . . 12 (𝑛 = 𝑝 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝))
4442, 43ifbieq1d 3556 . . . . . . . . . . 11 (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1))
45 eqid 2177 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1))
4644, 45fvmptg 5592 . . . . . . . . . 10 ((𝑝 ∈ ℕ ∧ if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1))
4718, 41, 46syl2anc 411 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1))
4847, 41eqeltrd 2254 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) ∈ ℂ)
49 fprodmul.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
5049fmpttd 5671 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘𝐴𝐶):𝐴⟶ℂ)
5150adantr 276 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
52 fco 5381 . . . . . . . . . . . . . 14 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
5351, 23, 52syl2anc 411 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
5453ad2antrr 488 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
5554, 33ffvelcdmd 5652 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝) ∈ ℂ)
5655, 35, 40ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1) ∈ ℂ)
57 fveq2 5515 . . . . . . . . . . . 12 (𝑛 = 𝑝 → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝))
5842, 57ifbieq1d 3556 . . . . . . . . . . 11 (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1))
59 eqid 2177 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1))
6058, 59fvmptg 5592 . . . . . . . . . 10 ((𝑝 ∈ ℕ ∧ if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1))
6118, 56, 60syl2anc 411 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1))
6261, 56eqeltrd 2254 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝) ∈ ℂ)
6323ad2antrr 488 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
6463, 33ffvelcdmd 5652 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓𝑝) ∈ 𝐴)
65 csbov12g 5913 . . . . . . . . . . . . . 14 ((𝑓𝑝) ∈ 𝐴(𝑓𝑝) / 𝑘(𝐵 · 𝐶) = ((𝑓𝑝) / 𝑘𝐵 · (𝑓𝑝) / 𝑘𝐶))
6664, 65syl 14 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓𝑝) / 𝑘(𝐵 · 𝐶) = ((𝑓𝑝) / 𝑘𝐵 · (𝑓𝑝) / 𝑘𝐶))
6719, 49mulcld 7976 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (𝐵 · 𝐶) ∈ ℂ)
6867ralrimiva 2550 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘𝐴 (𝐵 · 𝐶) ∈ ℂ)
6968ad3antrrr 492 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘𝐴 (𝐵 · 𝐶) ∈ ℂ)
70 nfcsb1v 3090 . . . . . . . . . . . . . . . . 17 𝑘(𝑓𝑝) / 𝑘(𝐵 · 𝐶)
7170nfel1 2330 . . . . . . . . . . . . . . . 16 𝑘(𝑓𝑝) / 𝑘(𝐵 · 𝐶) ∈ ℂ
72 csbeq1a 3066 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑝) → (𝐵 · 𝐶) = (𝑓𝑝) / 𝑘(𝐵 · 𝐶))
7372eleq1d 2246 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑝) → ((𝐵 · 𝐶) ∈ ℂ ↔ (𝑓𝑝) / 𝑘(𝐵 · 𝐶) ∈ ℂ))
7471, 73rspc 2835 . . . . . . . . . . . . . . 15 ((𝑓𝑝) ∈ 𝐴 → (∀𝑘𝐴 (𝐵 · 𝐶) ∈ ℂ → (𝑓𝑝) / 𝑘(𝐵 · 𝐶) ∈ ℂ))
7564, 69, 74sylc 62 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓𝑝) / 𝑘(𝐵 · 𝐶) ∈ ℂ)
76 eqid 2177 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐵 · 𝐶)) = (𝑘𝐴 ↦ (𝐵 · 𝐶))
7776fvmpts 5594 . . . . . . . . . . . . . 14 (((𝑓𝑝) ∈ 𝐴(𝑓𝑝) / 𝑘(𝐵 · 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑝)) = (𝑓𝑝) / 𝑘(𝐵 · 𝐶))
7864, 75, 77syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑝)) = (𝑓𝑝) / 𝑘(𝐵 · 𝐶))
7919ralrimiva 2550 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
8079ad3antrrr 492 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
81 nfcsb1v 3090 . . . . . . . . . . . . . . . . . 18 𝑘(𝑓𝑝) / 𝑘𝐵
8281nfel1 2330 . . . . . . . . . . . . . . . . 17 𝑘(𝑓𝑝) / 𝑘𝐵 ∈ ℂ
83 csbeq1a 3066 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑓𝑝) → 𝐵 = (𝑓𝑝) / 𝑘𝐵)
8483eleq1d 2246 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑝) → (𝐵 ∈ ℂ ↔ (𝑓𝑝) / 𝑘𝐵 ∈ ℂ))
8582, 84rspc 2835 . . . . . . . . . . . . . . . 16 ((𝑓𝑝) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑝) / 𝑘𝐵 ∈ ℂ))
8664, 80, 85sylc 62 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓𝑝) / 𝑘𝐵 ∈ ℂ)
87 eqid 2177 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
8887fvmpts 5594 . . . . . . . . . . . . . . 15 (((𝑓𝑝) ∈ 𝐴(𝑓𝑝) / 𝑘𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘(𝑓𝑝)) = (𝑓𝑝) / 𝑘𝐵)
8964, 86, 88syl2anc 411 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵)‘(𝑓𝑝)) = (𝑓𝑝) / 𝑘𝐵)
9049ralrimiva 2550 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)
9190ad3antrrr 492 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ∀𝑘𝐴 𝐶 ∈ ℂ)
92 nfcsb1v 3090 . . . . . . . . . . . . . . . . . 18 𝑘(𝑓𝑝) / 𝑘𝐶
9392nfel1 2330 . . . . . . . . . . . . . . . . 17 𝑘(𝑓𝑝) / 𝑘𝐶 ∈ ℂ
94 csbeq1a 3066 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑓𝑝) → 𝐶 = (𝑓𝑝) / 𝑘𝐶)
9594eleq1d 2246 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑝) → (𝐶 ∈ ℂ ↔ (𝑓𝑝) / 𝑘𝐶 ∈ ℂ))
9693, 95rspc 2835 . . . . . . . . . . . . . . . 16 ((𝑓𝑝) ∈ 𝐴 → (∀𝑘𝐴 𝐶 ∈ ℂ → (𝑓𝑝) / 𝑘𝐶 ∈ ℂ))
9764, 91, 96sylc 62 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (𝑓𝑝) / 𝑘𝐶 ∈ ℂ)
98 eqid 2177 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
9998fvmpts 5594 . . . . . . . . . . . . . . 15 (((𝑓𝑝) ∈ 𝐴(𝑓𝑝) / 𝑘𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘(𝑓𝑝)) = (𝑓𝑝) / 𝑘𝐶)
10064, 97, 99syl2anc 411 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐶)‘(𝑓𝑝)) = (𝑓𝑝) / 𝑘𝐶)
10189, 100oveq12d 5892 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵)‘(𝑓𝑝)) · ((𝑘𝐴𝐶)‘(𝑓𝑝))) = ((𝑓𝑝) / 𝑘𝐵 · (𝑓𝑝) / 𝑘𝐶))
10266, 78, 1013eqtr4d 2220 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑝)) = (((𝑘𝐴𝐵)‘(𝑓𝑝)) · ((𝑘𝐴𝐶)‘(𝑓𝑝))))
103 fvco3 5587 . . . . . . . . . . . . 13 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑝)))
10463, 33, 103syl2anc 411 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑝)))
105 fvco3 5587 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝) = ((𝑘𝐴𝐵)‘(𝑓𝑝)))
10663, 33, 105syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝) = ((𝑘𝐴𝐵)‘(𝑓𝑝)))
107 fvco3 5587 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑝 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝) = ((𝑘𝐴𝐶)‘(𝑓𝑝)))
10863, 33, 107syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝) = ((𝑘𝐴𝐶)‘(𝑓𝑝)))
109106, 108oveq12d 5892 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝) · (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝)) = (((𝑘𝐴𝐵)‘(𝑓𝑝)) · ((𝑘𝐴𝐶)‘(𝑓𝑝))))
110102, 104, 1093eqtr4d 2220 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝) · (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝)))
11127iftrued 3541 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝))
11227iftrued 3541 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝))
11327iftrued 3541 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝))
114112, 113oveq12d 5892 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1)) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝) · (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝)))
115110, 111, 1143eqtr4d 2220 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1)))
1161eqcomi 2181 . . . . . . . . . . 11 1 = (1 · 1)
117 simpr 110 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → ¬ 𝑝 ≤ (♯‘𝐴))
118117iffalsed 3544 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = 1)
119117iffalsed 3544 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) = 1)
120117iffalsed 3544 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1) = 1)
121119, 120oveq12d 5892 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → (if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1)) = (1 · 1))
122116, 118, 1213eqtr4a 2236 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ ¬ 𝑝 ≤ (♯‘𝐴)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1)))
123 exmiddc 836 . . . . . . . . . . 11 (DECID 𝑝 ≤ (♯‘𝐴) → (𝑝 ≤ (♯‘𝐴) ∨ ¬ 𝑝 ≤ (♯‘𝐴)))
12440, 123syl 14 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → (𝑝 ≤ (♯‘𝐴) ∨ ¬ 𝑝 ≤ (♯‘𝐴)))
125115, 122, 124mpjaodan 798 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1)))
12678, 75eqeltrd 2254 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑝)) ∈ ℂ)
127104, 126eqeltrd 2254 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) ∧ 𝑝 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝) ∈ ℂ)
128127, 35, 40ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) ∈ ℂ)
129 fveq2 5515 . . . . . . . . . . . 12 (𝑛 = 𝑝 → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝))
13042, 129ifbieq1d 3556 . . . . . . . . . . 11 (𝑛 = 𝑝 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1))
131 eqid 2177 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))
132130, 131fvmptg 5592 . . . . . . . . . 10 ((𝑝 ∈ ℕ ∧ if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1))
13318, 128, 132syl2anc 411 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))‘𝑝) = if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑝), 1))
13447, 61oveq12d 5892 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → (((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝)) = (if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑝), 1) · if(𝑝 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑝), 1)))
135125, 133, 1343eqtr4d 2220 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑝 ∈ (ℤ‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1))‘𝑝) = (((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1))‘𝑝) · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1))‘𝑝)))
13615, 48, 62, 135prod3fmul 11544 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴))))
137 fveq2 5515 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
138 simprr 531 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
13967fmpttd 5671 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
140139adantr 276 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 · 𝐶)):𝐴⟶ℂ)
141140ffvelcdmda 5651 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) ∈ ℂ)
142 fvco3 5587 . . . . . . . . 9 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
14323, 142sylan 283 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘(𝑓𝑛)))
144137, 13, 138, 141, 143fprodseq 11586 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 · 𝐶)) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)))
145 fveq2 5515 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
14621ffvelcdmda 5651 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
147 fvco3 5587 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
14823, 147sylan 283 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
149145, 13, 138, 146, 148fprodseq 11586 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)))
150 fveq2 5515 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
15151ffvelcdmda 5651 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
152 fvco3 5587 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
15323, 152sylan 283 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
154150, 13, 138, 151, 153fprodseq 11586 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)))
155149, 154oveq12d 5892 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) · ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 1)))‘(♯‘𝐴))))
156136, 144, 1553eqtr4d 2220 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) · ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
157 prodfct 11590 . . . . . . . 8 (∀𝑘𝐴 (𝐵 · 𝐶) ∈ ℂ → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 · 𝐶))
15868, 157syl 14 . . . . . . 7 (𝜑 → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 · 𝐶))
159158adantr 276 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 · 𝐶))‘𝑚) = ∏𝑘𝐴 (𝐵 · 𝐶))
160 prodfct 11590 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵)
16179, 160syl 14 . . . . . . . 8 (𝜑 → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵)
162 prodfct 11590 . . . . . . . . 9 (∀𝑘𝐴 𝐶 ∈ ℂ → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶)
16390, 162syl 14 . . . . . . . 8 (𝜑 → ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = ∏𝑘𝐴 𝐶)
164161, 163oveq12d 5892 . . . . . . 7 (𝜑 → (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) · ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
165164adantr 276 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) · ∏𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
166156, 159, 1653eqtr3d 2218 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
167166expr 375 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
168167exlimdv 1819 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
169168expimpd 363 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶)))
170 fprodmul.1 . . 3 (𝜑𝐴 ∈ Fin)
171 fz1f1o 11378 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
172170, 171syl 14 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
17312, 169, 172mpjaod 718 1 (𝜑 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wral 2455  csb 3057  c0 3422  ifcif 3534   class class class wbr 4003  cmpt 4064  ccom 4630  wf 5212  1-1-ontowf1o 5215  cfv 5216  (class class class)co 5874  Fincfn 6739  cc 7808  1c1 7811   · cmul 7815  cle 7991  cn 8917  cz 9251  cuz 9526  ...cfz 10006  seqcseq 10442  chash 10750  cprod 11553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-proddc 11554
This theorem is referenced by:  fprodsplitdc  11599  fproddivap  11633
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