| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prodeq1 15943 | . . 3
⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ ∅ (𝐵↑𝑁)) | 
| 2 |  | prodeq1 15943 | . . . 4
⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵) | 
| 3 | 2 | oveq1d 7446 | . . 3
⊢ (𝑥 = ∅ → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ ∅ 𝐵↑𝑁)) | 
| 4 | 1, 3 | eqeq12d 2753 | . 2
⊢ (𝑥 = ∅ → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ ∅ (𝐵↑𝑁) = (∏𝑘 ∈ ∅ 𝐵↑𝑁))) | 
| 5 |  | prodeq1 15943 | . . 3
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ 𝑦 (𝐵↑𝑁)) | 
| 6 |  | prodeq1 15943 | . . . 4
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵) | 
| 7 | 6 | oveq1d 7446 | . . 3
⊢ (𝑥 = 𝑦 → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) | 
| 8 | 5, 7 | eqeq12d 2753 | . 2
⊢ (𝑥 = 𝑦 → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁))) | 
| 9 |  | prodeq1 15943 | . . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁)) | 
| 10 |  | prodeq1 15943 | . . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) | 
| 11 | 10 | oveq1d 7446 | . . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁)) | 
| 12 | 9, 11 | eqeq12d 2753 | . 2
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁))) | 
| 13 |  | prodeq1 15943 | . . 3
⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ 𝐴 (𝐵↑𝑁)) | 
| 14 |  | prodeq1 15943 | . . . 4
⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵) | 
| 15 | 14 | oveq1d 7446 | . . 3
⊢ (𝑥 = 𝐴 → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁)) | 
| 16 | 13, 15 | eqeq12d 2753 | . 2
⊢ (𝑥 = 𝐴 → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁))) | 
| 17 |  | fprodexp.n | . . . . . 6
⊢ (𝜑 → 𝑁 ∈
ℕ0) | 
| 18 | 17 | nn0zd 12639 | . . . . 5
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 19 |  | 1exp 14132 | . . . . 5
⊢ (𝑁 ∈ ℤ →
(1↑𝑁) =
1) | 
| 20 | 18, 19 | syl 17 | . . . 4
⊢ (𝜑 → (1↑𝑁) = 1) | 
| 21 | 20 | eqcomd 2743 | . . 3
⊢ (𝜑 → 1 = (1↑𝑁)) | 
| 22 |  | prod0 15979 | . . . 4
⊢
∏𝑘 ∈
∅ (𝐵↑𝑁) = 1 | 
| 23 | 22 | a1i 11 | . . 3
⊢ (𝜑 → ∏𝑘 ∈ ∅ (𝐵↑𝑁) = 1) | 
| 24 |  | prod0 15979 | . . . . 5
⊢
∏𝑘 ∈
∅ 𝐵 =
1 | 
| 25 | 24 | oveq1i 7441 | . . . 4
⊢
(∏𝑘 ∈
∅ 𝐵↑𝑁) = (1↑𝑁) | 
| 26 | 25 | a1i 11 | . . 3
⊢ (𝜑 → (∏𝑘 ∈ ∅ 𝐵↑𝑁) = (1↑𝑁)) | 
| 27 | 21, 23, 26 | 3eqtr4d 2787 | . 2
⊢ (𝜑 → ∏𝑘 ∈ ∅ (𝐵↑𝑁) = (∏𝑘 ∈ ∅ 𝐵↑𝑁)) | 
| 28 |  | fprodexp.kph | . . . . . . . . 9
⊢
Ⅎ𝑘𝜑 | 
| 29 |  | nfv 1914 | . . . . . . . . 9
⊢
Ⅎ𝑘(𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) | 
| 30 | 28, 29 | nfan 1899 | . . . . . . . 8
⊢
Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) | 
| 31 |  | fprodexp.a | . . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ Fin) | 
| 32 | 31 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ Fin) | 
| 33 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴) | 
| 34 |  | ssfi 9213 | . . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin) | 
| 35 | 32, 33, 34 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin) | 
| 36 | 35 | adantrr 717 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin) | 
| 37 |  | simpll 767 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝜑) | 
| 38 | 33 | sselda 3983 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴) | 
| 39 |  | fprodexp.b | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| 40 | 37, 38, 39 | syl2anc 584 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ) | 
| 41 | 40 | adantlrr 721 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ) | 
| 42 | 30, 36, 41 | fprodclf 16028 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ) | 
| 43 |  | simpl 482 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑) | 
| 44 |  | simprr 773 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦)) | 
| 45 | 44 | eldifad 3963 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴) | 
| 46 |  | nfv 1914 | . . . . . . . . . . 11
⊢
Ⅎ𝑘 𝑧 ∈ 𝐴 | 
| 47 | 28, 46 | nfan 1899 | . . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴) | 
| 48 |  | nfcsb1v 3923 | . . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 | 
| 49 | 48 | nfel1 2922 | . . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ | 
| 50 | 47, 49 | nfim 1896 | . . . . . . . . 9
⊢
Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) | 
| 51 |  | eleq1w 2824 | . . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | 
| 52 | 51 | anbi2d 630 | . . . . . . . . . 10
⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴))) | 
| 53 |  | csbeq1a 3913 | . . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵) | 
| 54 | 53 | eleq1d 2826 | . . . . . . . . . 10
⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)) | 
| 55 | 52, 54 | imbi12d 344 | . . . . . . . . 9
⊢ (𝑘 = 𝑧 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))) | 
| 56 | 50, 55, 39 | chvarfv 2240 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) | 
| 57 | 43, 45, 56 | syl2anc 584 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) | 
| 58 | 17 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑁 ∈
ℕ0) | 
| 59 |  | mulexp 14142 | . . . . . . 7
⊢
((∏𝑘 ∈
𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) →
((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 60 | 42, 57, 58, 59 | syl3anc 1373 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 61 | 60 | eqcomd 2743 | . . . . 5
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁)) | 
| 62 | 61 | adantr 480 | . . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁)) | 
| 63 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑘↑ | 
| 64 |  | nfcv 2905 | . . . . . . . 8
⊢
Ⅎ𝑘𝑁 | 
| 65 | 48, 63, 64 | nfov 7461 | . . . . . . 7
⊢
Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵↑𝑁) | 
| 66 | 44 | eldifbd 3964 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦) | 
| 67 | 17 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑁 ∈
ℕ0) | 
| 68 | 40, 67 | expcld 14186 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → (𝐵↑𝑁) ∈ ℂ) | 
| 69 | 68 | adantlrr 721 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (𝐵↑𝑁) ∈ ℂ) | 
| 70 | 53 | oveq1d 7446 | . . . . . . 7
⊢ (𝑘 = 𝑧 → (𝐵↑𝑁) = (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) | 
| 71 | 57, 58 | expcld 14186 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (⦋𝑧 / 𝑘⦌𝐵↑𝑁) ∈ ℂ) | 
| 72 | 30, 65, 36, 44, 66, 69, 70, 71 | fprodsplitsn 16025 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 73 | 72 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 74 |  | oveq1 7438 | . . . . . 6
⊢
(∏𝑘 ∈
𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁) → (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 75 | 74 | adantl 481 | . . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 76 | 73, 75 | eqtrd 2777 | . . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁))) | 
| 77 | 30, 48, 36, 44, 66, 41, 53, 57 | fprodsplitsn 16025 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) | 
| 78 | 77 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) | 
| 79 | 78 | oveq1d 7446 | . . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁)) | 
| 80 | 62, 76, 79 | 3eqtr4d 2787 | . . 3
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁)) | 
| 81 | 80 | ex 412 | . 2
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁))) | 
| 82 | 4, 8, 12, 16, 27, 81, 31 | findcard2d 9206 | 1
⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁)) |