| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | iprodmul.1 | . 2
⊢ 𝑍 =
(ℤ≥‘𝑀) | 
| 2 |  | iprodmul.2 | . 2
⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 3 |  | iprodmul.3 | . . . 4
⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) | 
| 4 |  | iprodmul.4 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | 
| 5 |  | iprodmul.5 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | 
| 6 | 4, 5 | eqeltrd 2841 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | 
| 7 |  | iprodmul.6 | . . . 4
⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∃𝑧(𝑧 ≠ 0 ∧ seq𝑚( · , 𝐺) ⇝ 𝑧)) | 
| 8 |  | iprodmul.7 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵) | 
| 9 |  | iprodmul.8 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) | 
| 10 | 8, 9 | eqeltrd 2841 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) | 
| 11 |  | fveq2 6906 | . . . . . . 7
⊢ (𝑎 = 𝑘 → (𝐹‘𝑎) = (𝐹‘𝑘)) | 
| 12 |  | fveq2 6906 | . . . . . . 7
⊢ (𝑎 = 𝑘 → (𝐺‘𝑎) = (𝐺‘𝑘)) | 
| 13 | 11, 12 | oveq12d 7449 | . . . . . 6
⊢ (𝑎 = 𝑘 → ((𝐹‘𝑎) · (𝐺‘𝑎)) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 14 |  | eqid 2737 | . . . . . 6
⊢ (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎))) = (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎))) | 
| 15 |  | ovex 7464 | . . . . . 6
⊢ ((𝐹‘𝑘) · (𝐺‘𝑘)) ∈ V | 
| 16 | 13, 14, 15 | fvmpt 7016 | . . . . 5
⊢ (𝑘 ∈ 𝑍 → ((𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 17 | 16 | adantl 481 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 18 | 1, 3, 6, 7, 10, 17 | ntrivcvgmul 15938 | . . 3
⊢ (𝜑 → ∃𝑝 ∈ 𝑍 ∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))) ⇝ 𝑤)) | 
| 19 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑚 = 𝑎 → (𝐹‘𝑚) = (𝐹‘𝑎)) | 
| 20 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑚 = 𝑎 → (𝐺‘𝑚) = (𝐺‘𝑎)) | 
| 21 | 19, 20 | oveq12d 7449 | . . . . . . . . 9
⊢ (𝑚 = 𝑎 → ((𝐹‘𝑚) · (𝐺‘𝑚)) = ((𝐹‘𝑎) · (𝐺‘𝑎))) | 
| 22 | 21 | cbvmptv 5255 | . . . . . . . 8
⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚))) = (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎))) | 
| 23 |  | seqeq3 14047 | . . . . . . . 8
⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚))) = (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎))) → seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) = seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎))))) | 
| 24 | 22, 23 | ax-mp 5 | . . . . . . 7
⊢ seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) = seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))) | 
| 25 | 24 | breq1i 5150 | . . . . . 6
⊢ (seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ⇝ 𝑤 ↔ seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))) ⇝ 𝑤) | 
| 26 | 25 | anbi2i 623 | . . . . 5
⊢ ((𝑤 ≠ 0 ∧ seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ⇝ 𝑤) ↔ (𝑤 ≠ 0 ∧ seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))) ⇝ 𝑤)) | 
| 27 | 26 | exbii 1848 | . . . 4
⊢
(∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ⇝ 𝑤) ↔ ∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))) ⇝ 𝑤)) | 
| 28 | 27 | rexbii 3094 | . . 3
⊢
(∃𝑝 ∈
𝑍 ∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ⇝ 𝑤) ↔ ∃𝑝 ∈ 𝑍 ∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , (𝑎 ∈ 𝑍 ↦ ((𝐹‘𝑎) · (𝐺‘𝑎)))) ⇝ 𝑤)) | 
| 29 | 18, 28 | sylibr 234 | . 2
⊢ (𝜑 → ∃𝑝 ∈ 𝑍 ∃𝑤(𝑤 ≠ 0 ∧ seq𝑝( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ⇝ 𝑤)) | 
| 30 |  | eqid 2737 | . . . 4
⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚))) | 
| 31 |  | fveq2 6906 | . . . . 5
⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | 
| 32 |  | fveq2 6906 | . . . . 5
⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘)) | 
| 33 | 31, 32 | oveq12d 7449 | . . . 4
⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) · (𝐺‘𝑚)) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 34 |  | simpr 484 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | 
| 35 | 6, 10 | mulcld 11281 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) · (𝐺‘𝑘)) ∈ ℂ) | 
| 36 | 30, 33, 34, 35 | fvmptd3 7039 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 37 | 4, 8 | oveq12d 7449 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) · (𝐺‘𝑘)) = (𝐴 · 𝐵)) | 
| 38 | 36, 37 | eqtrd 2777 | . 2
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))‘𝑘) = (𝐴 · 𝐵)) | 
| 39 | 5, 9 | mulcld 11281 | . 2
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 · 𝐵) ∈ ℂ) | 
| 40 | 1, 2, 3, 4, 5 | iprodclim2 16035 | . . 3
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ∏𝑘 ∈ 𝑍 𝐴) | 
| 41 |  | seqex 14044 | . . . 4
⊢ seq𝑀( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ∈ V | 
| 42 | 41 | a1i 11 | . . 3
⊢ (𝜑 → seq𝑀( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ∈ V) | 
| 43 | 1, 2, 7, 8, 9 | iprodclim2 16035 | . . 3
⊢ (𝜑 → seq𝑀( · , 𝐺) ⇝ ∏𝑘 ∈ 𝑍 𝐵) | 
| 44 | 1, 2, 6 | prodf 15923 | . . . 4
⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) | 
| 45 | 44 | ffvelcdmda 7104 | . . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , 𝐹)‘𝑗) ∈ ℂ) | 
| 46 | 1, 2, 10 | prodf 15923 | . . . 4
⊢ (𝜑 → seq𝑀( · , 𝐺):𝑍⟶ℂ) | 
| 47 | 46 | ffvelcdmda 7104 | . . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , 𝐺)‘𝑗) ∈ ℂ) | 
| 48 |  | simpr 484 | . . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | 
| 49 | 48, 1 | eleqtrdi 2851 | . . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) | 
| 50 |  | elfzuz 13560 | . . . . . . 7
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | 
| 51 | 50, 1 | eleqtrrdi 2852 | . . . . . 6
⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) | 
| 52 | 51, 6 | sylan2 593 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) | 
| 53 | 52 | adantlr 715 | . . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) | 
| 54 | 51, 10 | sylan2 593 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ) | 
| 55 | 54 | adantlr 715 | . . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ) | 
| 56 | 36 | adantlr 715 | . . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 57 | 51, 56 | sylan2 593 | . . . 4
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) | 
| 58 | 49, 53, 55, 57 | prodfmul 15926 | . . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚))))‘𝑗) = ((seq𝑀( · , 𝐹)‘𝑗) · (seq𝑀( · , 𝐺)‘𝑗))) | 
| 59 | 1, 2, 40, 42, 43, 45, 47, 58 | climmul 15669 | . 2
⊢ (𝜑 → seq𝑀( · , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) · (𝐺‘𝑚)))) ⇝ (∏𝑘 ∈ 𝑍 𝐴 · ∏𝑘 ∈ 𝑍 𝐵)) | 
| 60 | 1, 2, 29, 38, 39, 59 | iprodclim 16034 | 1
⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (𝐴 · 𝐵) = (∏𝑘 ∈ 𝑍 𝐴 · ∏𝑘 ∈ 𝑍 𝐵)) |