| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prodeq1 15944 | . . . 4
⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵) | 
| 2 | 1 | fveq2d 6909 | . . 3
⊢ (𝑥 = ∅ →
(abs‘∏𝑘 ∈
𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵)) | 
| 3 |  | prodeq1 15944 | . . 3
⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)) | 
| 4 | 2, 3 | eqeq12d 2752 | . 2
⊢ (𝑥 = ∅ →
((abs‘∏𝑘 ∈
𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))) | 
| 5 |  | prodeq1 15944 | . . . 4
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵) | 
| 6 | 5 | fveq2d 6909 | . . 3
⊢ (𝑥 = 𝑦 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝑦 𝐵)) | 
| 7 |  | prodeq1 15944 | . . 3
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) | 
| 8 | 6, 7 | eqeq12d 2752 | . 2
⊢ (𝑥 = 𝑦 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵))) | 
| 9 |  | prodeq1 15944 | . . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) | 
| 10 | 9 | fveq2d 6909 | . . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) | 
| 11 |  | prodeq1 15944 | . . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)) | 
| 12 | 10, 11 | eqeq12d 2752 | . 2
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))) | 
| 13 |  | prodeq1 15944 | . . . 4
⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵) | 
| 14 | 13 | fveq2d 6909 | . . 3
⊢ (𝑥 = 𝐴 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝐴 𝐵)) | 
| 15 |  | prodeq1 15944 | . . 3
⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)) | 
| 16 | 14, 15 | eqeq12d 2752 | . 2
⊢ (𝑥 = 𝐴 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))) | 
| 17 |  | abs1 15337 | . . . 4
⊢
(abs‘1) = 1 | 
| 18 |  | prod0 15980 | . . . . 5
⊢
∏𝑘 ∈
∅ 𝐵 =
1 | 
| 19 | 18 | fveq2i 6908 | . . . 4
⊢
(abs‘∏𝑘
∈ ∅ 𝐵) =
(abs‘1) | 
| 20 |  | prod0 15980 | . . . 4
⊢
∏𝑘 ∈
∅ (abs‘𝐵) =
1 | 
| 21 | 17, 19, 20 | 3eqtr4i 2774 | . . 3
⊢
(abs‘∏𝑘
∈ ∅ 𝐵) =
∏𝑘 ∈ ∅
(abs‘𝐵) | 
| 22 | 21 | a1i 11 | . 2
⊢ (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)) | 
| 23 |  | eqidd 2737 | . . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 24 |  | nfv 1913 | . . . . . . . 8
⊢
Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) | 
| 25 |  | nfcsb1v 3922 | . . . . . . . 8
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 | 
| 26 |  | fprodabs2.a | . . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ Fin) | 
| 27 | 26 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ Fin) | 
| 28 |  | simpr 484 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴) | 
| 29 |  | ssfi 9214 | . . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin) | 
| 30 | 27, 28, 29 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin) | 
| 31 | 30 | adantrr 717 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin) | 
| 32 |  | simprr 772 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦)) | 
| 33 | 32 | eldifbd 3963 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦) | 
| 34 |  | simpll 766 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑) | 
| 35 | 28 | sselda 3982 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴) | 
| 36 | 35 | adantlrr 721 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴) | 
| 37 |  | fprodabs2.b | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| 38 | 34, 36, 37 | syl2anc 584 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ) | 
| 39 |  | csbeq1a 3912 | . . . . . . . 8
⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵) | 
| 40 |  | simpl 482 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑) | 
| 41 | 32 | eldifad 3962 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴) | 
| 42 |  | nfv 1913 | . . . . . . . . . . 11
⊢
Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴) | 
| 43 | 25 | nfel1 2921 | . . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ | 
| 44 | 42, 43 | nfim 1895 | . . . . . . . . . 10
⊢
Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) | 
| 45 |  | eleq1w 2823 | . . . . . . . . . . . 12
⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | 
| 46 | 45 | anbi2d 630 | . . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴))) | 
| 47 | 39 | eleq1d 2825 | . . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)) | 
| 48 | 46, 47 | imbi12d 344 | . . . . . . . . . 10
⊢ (𝑘 = 𝑧 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))) | 
| 49 | 44, 48, 37 | chvarfv 2239 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) | 
| 50 | 40, 41, 49 | syl2anc 584 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) | 
| 51 | 24, 25, 31, 32, 33, 38, 39, 50 | fprodsplitsn 16026 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) | 
| 52 | 51 | adantr 480 | . . . . . 6
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) | 
| 53 | 52 | fveq2d 6909 | . . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))) | 
| 54 | 24, 31, 38 | fprodclf 16029 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ) | 
| 55 | 54, 50 | absmuld 15494 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘(∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 56 | 55 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘(∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 57 |  | oveq1 7439 | . . . . . 6
⊢
((abs‘∏𝑘
∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵) → ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 58 | 57 | adantl 481 | . . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 59 | 53, 56, 58 | 3eqtrd 2780 | . . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 60 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑘abs | 
| 61 | 60, 25 | nffv 6915 | . . . . . 6
⊢
Ⅎ𝑘(abs‘⦋𝑧 / 𝑘⦌𝐵) | 
| 62 | 38 | abscld 15476 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℝ) | 
| 63 | 62 | recnd 11290 | . . . . . 6
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℂ) | 
| 64 | 39 | fveq2d 6909 | . . . . . 6
⊢ (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘⦋𝑧 / 𝑘⦌𝐵)) | 
| 65 | 50 | abscld 15476 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ) | 
| 66 | 65 | recnd 11290 | . . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) | 
| 67 | 24, 61, 31, 32, 33, 63, 64, 66 | fprodsplitsn 16026 | . . . . 5
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 68 | 67 | adantr 480 | . . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵))) | 
| 69 | 23, 59, 68 | 3eqtr4d 2786 | . . 3
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)) | 
| 70 | 69 | ex 412 | . 2
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))) | 
| 71 | 4, 8, 12, 16, 22, 70, 26 | findcard2d 9207 | 1
⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)) |