| Step | Hyp | Ref
| Expression |
| 1 | | dvmptfprod.i |
. 2
⊢ (𝜑 → 𝐼 ∈ Fin) |
| 2 | | ssid 4006 |
. . 3
⊢ 𝐼 ⊆ 𝐼 |
| 3 | 2 | jctr 524 |
. 2
⊢ (𝜑 → (𝜑 ∧ 𝐼 ⊆ 𝐼)) |
| 4 | | sseq1 4009 |
. . . . 5
⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼)) |
| 5 | 4 | anbi2d 630 |
. . . 4
⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼))) |
| 6 | | prodeq1 15943 |
. . . . . . 7
⊢ (𝑎 = ∅ → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴) |
| 7 | 6 | mpteq2dv 5244 |
. . . . . 6
⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) |
| 8 | 7 | oveq2d 7447 |
. . . . 5
⊢ (𝑎 = ∅ → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))) |
| 9 | | sumeq1 15725 |
. . . . . . 7
⊢ (𝑎 = ∅ → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) |
| 10 | | difeq1 4119 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗})) |
| 11 | 10 | prodeq1d 15956 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) |
| 12 | 11 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)) |
| 13 | 12 | sumeq2sdv 15739 |
. . . . . . 7
⊢ (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)) |
| 14 | 9, 13 | eqtrd 2777 |
. . . . . 6
⊢ (𝑎 = ∅ → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)) |
| 15 | 14 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))) |
| 16 | 8, 15 | eqeq12d 2753 |
. . . 4
⊢ (𝑎 = ∅ → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))) |
| 17 | 5, 16 | imbi12d 344 |
. . 3
⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))))) |
| 18 | | sseq1 4009 |
. . . . 5
⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼)) |
| 19 | 18 | anbi2d 630 |
. . . 4
⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ 𝑏 ⊆ 𝐼))) |
| 20 | | prodeq1 15943 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ 𝑏 𝐴) |
| 21 | 20 | mpteq2dv 5244 |
. . . . . 6
⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) |
| 22 | 21 | oveq2d 7447 |
. . . . 5
⊢ (𝑎 = 𝑏 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴))) |
| 23 | | sumeq1 15725 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) |
| 24 | | difeq1 4119 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗})) |
| 25 | 24 | prodeq1d 15956 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴) |
| 26 | 25 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 27 | 26 | sumeq2sdv 15739 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 28 | 23, 27 | eqtrd 2777 |
. . . . . 6
⊢ (𝑎 = 𝑏 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 29 | 28 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 30 | 22, 29 | eqeq12d 2753 |
. . . 4
⊢ (𝑎 = 𝑏 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))) |
| 31 | 19, 30 | imbi12d 344 |
. . 3
⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))))) |
| 32 | | sseq1 4009 |
. . . . 5
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) |
| 33 | 32 | anbi2d 630 |
. . . 4
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼))) |
| 34 | | prodeq1 15943 |
. . . . . . 7
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) |
| 35 | 34 | mpteq2dv 5244 |
. . . . . 6
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) |
| 36 | 35 | oveq2d 7447 |
. . . . 5
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))) |
| 37 | | sumeq1 15725 |
. . . . . . 7
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) |
| 38 | | difeq1 4119 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗})) |
| 39 | 38 | prodeq1d 15956 |
. . . . . . . . 9
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴) |
| 40 | 39 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)) |
| 41 | 40 | sumeq2sdv 15739 |
. . . . . . 7
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)) |
| 42 | 37, 41 | eqtrd 2777 |
. . . . . 6
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)) |
| 43 | 42 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))) |
| 44 | 36, 43 | eqeq12d 2753 |
. . . 4
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))) |
| 45 | 33, 44 | imbi12d 344 |
. . 3
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))))) |
| 46 | | sseq1 4009 |
. . . . 5
⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼)) |
| 47 | 46 | anbi2d 630 |
. . . 4
⊢ (𝑎 = 𝐼 → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ 𝐼 ⊆ 𝐼))) |
| 48 | | prodeq1 15943 |
. . . . . . 7
⊢ (𝑎 = 𝐼 → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ 𝐼 𝐴) |
| 49 | 48 | mpteq2dv 5244 |
. . . . . 6
⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) |
| 50 | 49 | oveq2d 7447 |
. . . . 5
⊢ (𝑎 = 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴))) |
| 51 | | sumeq1 15725 |
. . . . . . 7
⊢ (𝑎 = 𝐼 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) |
| 52 | | difeq1 4119 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗})) |
| 53 | 52 | prodeq1d 15956 |
. . . . . . . . 9
⊢ (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴) |
| 54 | 53 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑎 = 𝐼 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)) |
| 55 | 54 | sumeq2sdv 15739 |
. . . . . . 7
⊢ (𝑎 = 𝐼 → Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)) |
| 56 | 51, 55 | eqtrd 2777 |
. . . . . 6
⊢ (𝑎 = 𝐼 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)) |
| 57 | 56 | mpteq2dv 5244 |
. . . . 5
⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))) |
| 58 | 50, 57 | eqeq12d 2753 |
. . . 4
⊢ (𝑎 = 𝐼 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))) |
| 59 | 47, 58 | imbi12d 344 |
. . 3
⊢ (𝑎 = 𝐼 → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ 𝐼 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))))) |
| 60 | | prod0 15979 |
. . . . . . . 8
⊢
∏𝑖 ∈
∅ 𝐴 =
1 |
| 61 | 60 | mpteq2i 5247 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥 ∈ 𝑋 ↦ 1) |
| 62 | 61 | oveq2i 7442 |
. . . . . 6
⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 1)) |
| 63 | 62 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 1))) |
| 64 | | dvmptfprod.s |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
| 65 | | dvmptfprod.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝐽) |
| 66 | | dvmptfprod.j |
. . . . . . . 8
⊢ 𝐽 = (𝐾 ↾t 𝑆) |
| 67 | | dvmptfprod.k |
. . . . . . . . 9
⊢ 𝐾 =
(TopOpen‘ℂfld) |
| 68 | 67 | oveq1i 7441 |
. . . . . . . 8
⊢ (𝐾 ↾t 𝑆) =
((TopOpen‘ℂfld) ↾t 𝑆) |
| 69 | 66, 68 | eqtri 2765 |
. . . . . . 7
⊢ 𝐽 =
((TopOpen‘ℂfld) ↾t 𝑆) |
| 70 | 65, 69 | eleqtrdi 2851 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
| 71 | | 1cnd 11256 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℂ) |
| 72 | 64, 70, 71 | dvmptconst 45930 |
. . . . 5
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 1)) = (𝑥 ∈ 𝑋 ↦ 0)) |
| 73 | | sum0 15757 |
. . . . . . . 8
⊢
Σ𝑗 ∈
∅ (𝐶 ·
∏𝑖 ∈ (∅
∖ {𝑗})𝐴) = 0 |
| 74 | 73 | eqcomi 2746 |
. . . . . . 7
⊢ 0 =
Σ𝑗 ∈ ∅
(𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) |
| 75 | 74 | mpteq2i 5247 |
. . . . . 6
⊢ (𝑥 ∈ 𝑋 ↦ 0) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)) |
| 76 | 75 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))) |
| 77 | 63, 72, 76 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))) |
| 78 | 77 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))) |
| 79 | | simp3 1139 |
. . . . 5
⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) |
| 80 | | simp1r 1199 |
. . . . 5
⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐 ∈ 𝑏) |
| 81 | | ssun1 4178 |
. . . . . . . . . 10
⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐}) |
| 82 | | sstr2 3990 |
. . . . . . . . . 10
⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑏 ⊆ 𝐼)) |
| 83 | 81, 82 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑏 ⊆ 𝐼) |
| 84 | 83 | anim2i 617 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝜑 ∧ 𝑏 ⊆ 𝐼)) |
| 85 | 84 | adantl 481 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑 ∧ 𝑏 ⊆ 𝐼)) |
| 86 | | simpl 482 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))) |
| 87 | 85, 86 | mpd 15 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 88 | 87 | 3adant1 1131 |
. . . . 5
⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 89 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) |
| 90 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑥𝑆 |
| 91 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑥
D |
| 92 | | nfmpt1 5250 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴) |
| 93 | 90, 91, 92 | nfov 7461 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) |
| 94 | | nfmpt1 5250 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 95 | 93, 94 | nfeq 2919 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 96 | 89, 95 | nfan 1899 |
. . . . . 6
⊢
Ⅎ𝑥(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 97 | | dvmptfprod.iph |
. . . . . . . . 9
⊢
Ⅎ𝑖𝜑 |
| 98 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼 |
| 99 | 97, 98 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) |
| 100 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑖 ¬ 𝑐 ∈ 𝑏 |
| 101 | 99, 100 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑖((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) |
| 102 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑖𝑆 |
| 103 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑖
D |
| 104 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑖𝑋 |
| 105 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖𝑏 |
| 106 | 105 | nfcprod1 15944 |
. . . . . . . . . 10
⊢
Ⅎ𝑖∏𝑖 ∈ 𝑏 𝐴 |
| 107 | 104, 106 | nfmpt 5249 |
. . . . . . . . 9
⊢
Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴) |
| 108 | 102, 103,
107 | nfov 7461 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) |
| 109 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖𝐶 |
| 110 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖
· |
| 111 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝑏 ∖ {𝑗}) |
| 112 | 111 | nfcprod1 15944 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴 |
| 113 | 109, 110,
112 | nfov 7461 |
. . . . . . . . . 10
⊢
Ⅎ𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴) |
| 114 | 105, 113 | nfsum 15727 |
. . . . . . . . 9
⊢
Ⅎ𝑖Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴) |
| 115 | 104, 114 | nfmpt 5249 |
. . . . . . . 8
⊢
Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 116 | 108, 115 | nfeq 2919 |
. . . . . . 7
⊢
Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 117 | 101, 116 | nfan 1899 |
. . . . . 6
⊢
Ⅎ𝑖(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 118 | | dvmptfprod.jph |
. . . . . . . . 9
⊢
Ⅎ𝑗𝜑 |
| 119 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼 |
| 120 | 118, 119 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) |
| 121 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑗 ¬ 𝑐 ∈ 𝑏 |
| 122 | 120, 121 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑗((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) |
| 123 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) |
| 124 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑗𝑋 |
| 125 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑗𝑏 |
| 126 | 125 | nfsum1 15726 |
. . . . . . . . 9
⊢
Ⅎ𝑗Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴) |
| 127 | 124, 126 | nfmpt 5249 |
. . . . . . . 8
⊢
Ⅎ𝑗(𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 128 | 123, 127 | nfeq 2919 |
. . . . . . 7
⊢
Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)) |
| 129 | 122, 128 | nfan 1899 |
. . . . . 6
⊢
Ⅎ𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 130 | | nfcsb1v 3923 |
. . . . . 6
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐴 |
| 131 | | nfcsb1v 3923 |
. . . . . 6
⊢
Ⅎ𝑗⦋𝑐 / 𝑗⦌𝐶 |
| 132 | | simpl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑) |
| 133 | 132 | ad2antrr 726 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑) |
| 134 | | dvmptfprod.a |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
| 135 | 133, 134 | syl3an1 1164 |
. . . . . 6
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
| 136 | 1 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝐼 ∈ Fin) |
| 137 | 83 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏 ⊆ 𝐼) |
| 138 | 137 | ad2antrr 726 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ⊆ 𝐼) |
| 139 | 136, 138 | ssfid 9301 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin) |
| 140 | | vex 3484 |
. . . . . . 7
⊢ 𝑐 ∈ V |
| 141 | 140 | a1i 11 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑐 ∈ V) |
| 142 | | simplr 769 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐 ∈ 𝑏) |
| 143 | | simpllr 776 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼) |
| 144 | 64 | ad3antrrr 730 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ}) |
| 145 | 133 | ad2antrr 726 |
. . . . . . 7
⊢
((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝜑) |
| 146 | 138 | ad2antrr 726 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑏 ⊆ 𝐼) |
| 147 | | simpr 484 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝑏) |
| 148 | 146, 147 | sseldd 3984 |
. . . . . . 7
⊢
((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝐼) |
| 149 | | simplr 769 |
. . . . . . 7
⊢
((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑥 ∈ 𝑋) |
| 150 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑖 𝑗 ∈ 𝐼 |
| 151 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑖 𝑥 ∈ 𝑋 |
| 152 | 97, 150, 151 | nf3an 1901 |
. . . . . . . . 9
⊢
Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) |
| 153 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑖 𝐶 ∈ ℂ |
| 154 | 152, 153 | nfim 1896 |
. . . . . . . 8
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) |
| 155 | | eleq1w 2824 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝐼 ↔ 𝑗 ∈ 𝐼)) |
| 156 | 155 | 3anbi2d 1443 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋))) |
| 157 | | dvmptfprod.bc |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶) |
| 158 | 157 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
| 159 | 156, 158 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ))) |
| 160 | | dvmptfprod.b |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
| 161 | 154, 159,
160 | chvarfv 2240 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) |
| 162 | 145, 148,
149, 161 | syl3anc 1373 |
. . . . . 6
⊢
((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝐶 ∈ ℂ) |
| 163 | | simpr 484 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) |
| 164 | 132 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝜑) |
| 165 | | id 22 |
. . . . . . . . . 10
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼) |
| 166 | | vsnid 4663 |
. . . . . . . . . . 11
⊢ 𝑐 ∈ {𝑐} |
| 167 | | elun2 4183 |
. . . . . . . . . . 11
⊢ (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐})) |
| 168 | 166, 167 | mp1i 13 |
. . . . . . . . . 10
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ (𝑏 ∪ {𝑐})) |
| 169 | 165, 168 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ 𝐼) |
| 170 | 169 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝑐 ∈ 𝐼) |
| 171 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
| 172 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗 𝑐 ∈ 𝐼 |
| 173 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗 𝑥 ∈ 𝑋 |
| 174 | 118, 172,
173 | nf3an 1901 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) |
| 175 | 131 | nfel1 2922 |
. . . . . . . . . 10
⊢
Ⅎ𝑗⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ |
| 176 | 174, 175 | nfim 1896 |
. . . . . . . . 9
⊢
Ⅎ𝑗((𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ) |
| 177 | | eleq1w 2824 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑐 → (𝑗 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼)) |
| 178 | 177 | 3anbi2d 1443 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑐 → ((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋))) |
| 179 | | csbeq1a 3913 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑗⦌𝐶) |
| 180 | 179 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)) |
| 181 | 178, 180 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑗 = 𝑐 → (((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ))) |
| 182 | 176, 181,
161 | chvarfv 2240 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ) |
| 183 | 164, 170,
171, 182 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ) |
| 184 | 183 | ad4ant14 752 |
. . . . . 6
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ) |
| 185 | 118, 172 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝜑 ∧ 𝑐 ∈ 𝐼) |
| 186 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) |
| 187 | 124, 131 | nfmpt 5249 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶) |
| 188 | 186, 187 | nfeq 2919 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶) |
| 189 | 185, 188 | nfim 1896 |
. . . . . . . . 9
⊢
Ⅎ𝑗((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)) |
| 190 | 177 | anbi2d 630 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑐 → ((𝜑 ∧ 𝑗 ∈ 𝐼) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼))) |
| 191 | | csbeq1 3902 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑐 → ⦋𝑗 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌𝐴) |
| 192 | 191 | mpteq2dv 5244 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑐 → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) |
| 193 | 192 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑐 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))) |
| 194 | 179 | mpteq2dv 5244 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)) |
| 195 | 193, 194 | eqeq12d 2753 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑐 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))) |
| 196 | 190, 195 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑗 = 𝑐 → (((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) ↔ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)))) |
| 197 | 97, 150 | nfan 1899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝐼) |
| 198 | | nfcsb1v 3923 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐴 |
| 199 | 104, 198 | nfmpt 5249 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴) |
| 200 | 102, 103,
199 | nfov 7461 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) |
| 201 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ 𝐶) |
| 202 | 200, 201 | nfeq 2919 |
. . . . . . . . . . 11
⊢
Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶) |
| 203 | 197, 202 | nfim 1896 |
. . . . . . . . . 10
⊢
Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) |
| 204 | 155 | anbi2d 630 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ 𝐼) ↔ (𝜑 ∧ 𝑗 ∈ 𝐼))) |
| 205 | | csbeq1a 3913 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑖⦌𝐴) |
| 206 | 205 | mpteq2dv 5244 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) |
| 207 | 206 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴))) |
| 208 | 157 | mpteq2dv 5244 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐶)) |
| 209 | 207, 208 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))) |
| 210 | 204, 209 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)))) |
| 211 | | dvmptfprod.d |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
| 212 | 203, 210,
211 | chvarfv 2240 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) |
| 213 | 189, 196,
212 | chvarfv 2240 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)) |
| 214 | 169, 213 | sylan2 593 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)) |
| 215 | 214 | ad2antrr 726 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)) |
| 216 | | csbeq1a 3913 |
. . . . . 6
⊢ (𝑖 = 𝑐 → 𝐴 = ⦋𝑐 / 𝑖⦌𝐴) |
| 217 | 96, 117, 129, 130, 131, 135, 139, 141, 142, 143, 144, 162, 163, 184, 215, 216, 179 | dvmptfprodlem 45959 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))) |
| 218 | 79, 80, 88, 217 | syl21anc 838 |
. . . 4
⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))) |
| 219 | 218 | 3exp 1120 |
. . 3
⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))))) |
| 220 | 17, 31, 45, 59, 78, 219 | findcard2s 9205 |
. 2
⊢ (𝐼 ∈ Fin → ((𝜑 ∧ 𝐼 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))) |
| 221 | 1, 3, 220 | sylc 65 |
1
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))) |