| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑦(𝐴 · 𝐶) | 
| 2 |  | nfcsb1v 3923 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 | 
| 3 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑥
· | 
| 4 |  | nfcsb1v 3923 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | 
| 5 | 2, 3, 4 | nfov 7461 | . . . . 5
⊢
Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐴 · ⦋𝑦 / 𝑥⦌𝐶) | 
| 6 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | 
| 7 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | 
| 8 | 6, 7 | oveq12d 7449 | . . . . 5
⊢ (𝑥 = 𝑦 → (𝐴 · 𝐶) = (⦋𝑦 / 𝑥⦌𝐴 · ⦋𝑦 / 𝑥⦌𝐶)) | 
| 9 | 1, 5, 8 | cbvmpt 5253 | . . . 4
⊢ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶)) = (𝑦 ∈ 𝑋 ↦ (⦋𝑦 / 𝑥⦌𝐴 · ⦋𝑦 / 𝑥⦌𝐶)) | 
| 10 | 9 | oveq2i 7442 | . . 3
⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑆 D (𝑦 ∈ 𝑋 ↦ (⦋𝑦 / 𝑥⦌𝐴 · ⦋𝑦 / 𝑥⦌𝐶))) | 
| 11 | 10 | a1i 11 | . 2
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑆 D (𝑦 ∈ 𝑋 ↦ (⦋𝑦 / 𝑥⦌𝐴 · ⦋𝑦 / 𝑥⦌𝐶)))) | 
| 12 |  | dvmptmulf.s | . . 3
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | 
| 13 |  | dvmptmulf.ph | . . . . . 6
⊢
Ⅎ𝑥𝜑 | 
| 14 |  | nfv 1914 | . . . . . 6
⊢
Ⅎ𝑥 𝑦 ∈ 𝑋 | 
| 15 | 13, 14 | nfan 1899 | . . . . 5
⊢
Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝑋) | 
| 16 | 2 | nfel1 2922 | . . . . 5
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ | 
| 17 | 15, 16 | nfim 1896 | . . . 4
⊢
Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ) | 
| 18 |  | eleq1w 2824 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑋 ↔ 𝑦 ∈ 𝑋)) | 
| 19 | 18 | anbi2d 630 | . . . . 5
⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑦 ∈ 𝑋))) | 
| 20 | 6 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)) | 
| 21 | 19, 20 | imbi12d 344 | . . . 4
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ))) | 
| 22 |  | dvmptmulf.a | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | 
| 23 | 17, 21, 22 | chvarfv 2240 | . . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ) | 
| 24 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑥𝑦 | 
| 25 | 24 | nfcsb1 3922 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | 
| 26 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑥𝑉 | 
| 27 | 25, 26 | nfel 2920 | . . . . 5
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉 | 
| 28 | 15, 27 | nfim 1896 | . . . 4
⊢
Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉) | 
| 29 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | 
| 30 | 29 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑉 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉)) | 
| 31 | 19, 30 | imbi12d 344 | . . . 4
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) ↔ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉))) | 
| 32 |  | dvmptmulf.b | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | 
| 33 | 28, 31, 32 | chvarfv 2240 | . . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑉) | 
| 34 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑦𝐴 | 
| 35 |  | csbeq1a 3913 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑥 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐴) | 
| 36 |  | csbcow 3914 | . . . . . . . . . 10
⊢
⦋𝑥 /
𝑦⦌⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑥 / 𝑥⦌𝐴 | 
| 37 |  | csbid 3912 | . . . . . . . . . 10
⊢
⦋𝑥 /
𝑥⦌𝐴 = 𝐴 | 
| 38 | 36, 37 | eqtri 2765 | . . . . . . . . 9
⊢
⦋𝑥 /
𝑦⦌⦋𝑦 / 𝑥⦌𝐴 = 𝐴 | 
| 39 | 38 | a1i 11 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → ⦋𝑥 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐴 = 𝐴) | 
| 40 | 35, 39 | eqtrd 2777 | . . . . . . 7
⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐴) | 
| 41 | 2, 34, 40 | cbvmpt 5253 | . . . . . 6
⊢ (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | 
| 42 | 41 | oveq2i 7442 | . . . . 5
⊢ (𝑆 D (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) | 
| 43 | 42 | a1i 11 | . . . 4
⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) | 
| 44 |  | dvmptmulf.ab | . . . 4
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | 
| 45 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑦𝐵 | 
| 46 | 45, 25, 29 | cbvmpt 5253 | . . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) | 
| 47 | 46 | a1i 11 | . . . 4
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵)) | 
| 48 | 43, 44, 47 | 3eqtrd 2781 | . . 3
⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵)) | 
| 49 | 4 | nfel1 2922 | . . . . 5
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ | 
| 50 | 15, 49 | nfim 1896 | . . . 4
⊢
Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ) | 
| 51 | 7 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ)) | 
| 52 | 19, 51 | imbi12d 344 | . . . 4
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ))) | 
| 53 |  | dvmptmulf.c | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) | 
| 54 | 50, 52, 53 | chvarfv 2240 | . . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ) | 
| 55 | 24 | nfcsb1 3922 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷 | 
| 56 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑥𝑊 | 
| 57 | 55, 56 | nfel 2920 | . . . . 5
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷 ∈ 𝑊 | 
| 58 | 15, 57 | nfim 1896 | . . . 4
⊢
Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐷 ∈ 𝑊) | 
| 59 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷) | 
| 60 | 59 | eleq1d 2826 | . . . . 5
⊢ (𝑥 = 𝑦 → (𝐷 ∈ 𝑊 ↔ ⦋𝑦 / 𝑥⦌𝐷 ∈ 𝑊)) | 
| 61 | 19, 60 | imbi12d 344 | . . . 4
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊) ↔ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐷 ∈ 𝑊))) | 
| 62 |  | dvmptmulf.d | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊) | 
| 63 | 58, 61, 62 | chvarfv 2240 | . . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐷 ∈ 𝑊) | 
| 64 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑦𝐶 | 
| 65 |  | eqcom 2744 | . . . . . . . . . 10
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | 
| 66 | 65 | imbi1i 349 | . . . . . . . . 9
⊢ ((𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) ↔ (𝑦 = 𝑥 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)) | 
| 67 |  | eqcom 2744 | . . . . . . . . . 10
⊢ (𝐶 = ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶) | 
| 68 | 67 | imbi2i 336 | . . . . . . . . 9
⊢ ((𝑦 = 𝑥 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶)) | 
| 69 | 66, 68 | bitri 275 | . . . . . . . 8
⊢ ((𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶)) | 
| 70 | 7, 69 | mpbi 230 | . . . . . . 7
⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶) | 
| 71 | 4, 64, 70 | cbvmpt 5253 | . . . . . 6
⊢ (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶) | 
| 72 | 71 | oveq2i 7442 | . . . . 5
⊢ (𝑆 D (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐶)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) | 
| 73 | 72 | a1i 11 | . . . 4
⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐶)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶))) | 
| 74 |  | dvmptmulf.cd | . . . 4
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) | 
| 75 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑦𝐷 | 
| 76 | 75, 55, 59 | cbvmpt 5253 | . . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ 𝐷) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐷) | 
| 77 | 76 | a1i 11 | . . . 4
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐷)) | 
| 78 | 73, 74, 77 | 3eqtrd 2781 | . . 3
⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐶)) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐷)) | 
| 79 | 12, 23, 33, 48, 54, 63, 78 | dvmptmul 25999 | . 2
⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑋 ↦ (⦋𝑦 / 𝑥⦌𝐴 · ⦋𝑦 / 𝑥⦌𝐶))) = (𝑦 ∈ 𝑋 ↦ ((⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) + (⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴)))) | 
| 80 | 25, 3, 4 | nfov 7461 | . . . . 5
⊢
Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) | 
| 81 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑥
+ | 
| 82 | 55, 3, 2 | nfov 7461 | . . . . 5
⊢
Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴) | 
| 83 | 80, 81, 82 | nfov 7461 | . . . 4
⊢
Ⅎ𝑥((⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) + (⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴)) | 
| 84 |  | nfcv 2905 | . . . 4
⊢
Ⅎ𝑦((𝐵 · 𝐶) + (𝐷 · 𝐴)) | 
| 85 | 65 | imbi1i 349 | . . . . . . . 8
⊢ ((𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)) | 
| 86 |  | eqcom 2744 | . . . . . . . . 9
⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) | 
| 87 | 86 | imbi2i 336 | . . . . . . . 8
⊢ ((𝑦 = 𝑥 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)) | 
| 88 | 85, 87 | bitri 275 | . . . . . . 7
⊢ ((𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)) | 
| 89 | 29, 88 | mpbi 230 | . . . . . 6
⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) | 
| 90 | 89, 70 | oveq12d 7449 | . . . . 5
⊢ (𝑦 = 𝑥 → (⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) = (𝐵 · 𝐶)) | 
| 91 | 65 | imbi1i 349 | . . . . . . . 8
⊢ ((𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷) ↔ (𝑦 = 𝑥 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷)) | 
| 92 |  | eqcom 2744 | . . . . . . . . 9
⊢ (𝐷 = ⦋𝑦 / 𝑥⦌𝐷 ↔ ⦋𝑦 / 𝑥⦌𝐷 = 𝐷) | 
| 93 | 92 | imbi2i 336 | . . . . . . . 8
⊢ ((𝑦 = 𝑥 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐷 = 𝐷)) | 
| 94 | 91, 93 | bitri 275 | . . . . . . 7
⊢ ((𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐷 = 𝐷)) | 
| 95 | 59, 94 | mpbi 230 | . . . . . 6
⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐷 = 𝐷) | 
| 96 | 95, 40 | oveq12d 7449 | . . . . 5
⊢ (𝑦 = 𝑥 → (⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴) = (𝐷 · 𝐴)) | 
| 97 | 90, 96 | oveq12d 7449 | . . . 4
⊢ (𝑦 = 𝑥 → ((⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) + (⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴)) = ((𝐵 · 𝐶) + (𝐷 · 𝐴))) | 
| 98 | 83, 84, 97 | cbvmpt 5253 | . . 3
⊢ (𝑦 ∈ 𝑋 ↦ ((⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) + (⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))) | 
| 99 | 98 | a1i 11 | . 2
⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ ((⦋𝑦 / 𝑥⦌𝐵 · ⦋𝑦 / 𝑥⦌𝐶) + (⦋𝑦 / 𝑥⦌𝐷 · ⦋𝑦 / 𝑥⦌𝐴))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴)))) | 
| 100 | 11, 79, 99 | 3eqtrd 2781 | 1
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴)))) |