Step | Hyp | Ref
| Expression |
1 | | ssid 3848 |
. 2
⊢ 𝐼 ⊆ 𝐼 |
2 | | dvmptfsum.i |
. . 3
⊢ (𝜑 → 𝐼 ∈ Fin) |
3 | | sseq1 3851 |
. . . . . 6
⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼)) |
4 | | sumeq1 14796 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ ∅ 𝐴) |
5 | 4 | mpteq2dv 4968 |
. . . . . . . 8
⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) |
6 | 5 | oveq2d 6921 |
. . . . . . 7
⊢ (𝑎 = ∅ → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴))) |
7 | | sumeq1 14796 |
. . . . . . . 8
⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ ∅ 𝐵) |
8 | 7 | mpteq2dv 4968 |
. . . . . . 7
⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵)) |
9 | 6, 8 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑎 = ∅ → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))) |
10 | 3, 9 | imbi12d 336 |
. . . . 5
⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ (∅ ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵)))) |
11 | 10 | imbi2d 332 |
. . . 4
⊢ (𝑎 = ∅ → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))))) |
12 | | sseq1 3851 |
. . . . . 6
⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼)) |
13 | | sumeq1 14796 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝑏 𝐴) |
14 | 13 | mpteq2dv 4968 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) |
15 | 14 | oveq2d 6921 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴))) |
16 | | sumeq1 14796 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝑏 𝐵) |
17 | 16 | mpteq2dv 4968 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) |
18 | 15, 17 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑎 = 𝑏 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) |
19 | 12, 18 | imbi12d 336 |
. . . . 5
⊢ (𝑎 = 𝑏 → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ (𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)))) |
20 | 19 | imbi2d 332 |
. . . 4
⊢ (𝑎 = 𝑏 → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → (𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))))) |
21 | | sseq1 3851 |
. . . . . 6
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) |
22 | | sumeq1 14796 |
. . . . . . . . 9
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) |
23 | 22 | mpteq2dv 4968 |
. . . . . . . 8
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) |
24 | 23 | oveq2d 6921 |
. . . . . . 7
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))) |
25 | | sumeq1 14796 |
. . . . . . . 8
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) |
26 | 25 | mpteq2dv 4968 |
. . . . . . 7
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)) |
27 | 24, 26 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))) |
28 | 21, 27 | imbi12d 336 |
. . . . 5
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))) |
29 | 28 | imbi2d 332 |
. . . 4
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))) |
30 | | sseq1 3851 |
. . . . . 6
⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼)) |
31 | | sumeq1 14796 |
. . . . . . . . 9
⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝐼 𝐴) |
32 | 31 | mpteq2dv 4968 |
. . . . . . . 8
⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) |
33 | 32 | oveq2d 6921 |
. . . . . . 7
⊢ (𝑎 = 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴))) |
34 | | sumeq1 14796 |
. . . . . . . 8
⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝐼 𝐵) |
35 | 34 | mpteq2dv 4968 |
. . . . . . 7
⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)) |
36 | 33, 35 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑎 = 𝐼 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))) |
37 | 30, 36 | imbi12d 336 |
. . . . 5
⊢ (𝑎 = 𝐼 → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)))) |
38 | 37 | imbi2d 332 |
. . . 4
⊢ (𝑎 = 𝐼 → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))))) |
39 | | dvmptfsum.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
40 | | 0cnd 10349 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ) |
41 | | 0cnd 10349 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℂ) |
42 | 39, 41 | dvmptc 24120 |
. . . . . . 7
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 0)) = (𝑥 ∈ 𝑆 ↦ 0)) |
43 | | dvmptfsum.j |
. . . . . . . . 9
⊢ 𝐽 = (𝐾 ↾t 𝑆) |
44 | | dvmptfsum.k |
. . . . . . . . . . 11
⊢ 𝐾 =
(TopOpen‘ℂfld) |
45 | 44 | cnfldtopon 22956 |
. . . . . . . . . 10
⊢ 𝐾 ∈
(TopOn‘ℂ) |
46 | | recnprss 24067 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
47 | 39, 46 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
48 | | resttopon 21336 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐾
↾t 𝑆)
∈ (TopOn‘𝑆)) |
49 | 45, 47, 48 | sylancr 583 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
50 | 43, 49 | syl5eqel 2910 |
. . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
51 | | dvmptfsum.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ 𝐽) |
52 | | toponss 21102 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑆) ∧ 𝑋 ∈ 𝐽) → 𝑋 ⊆ 𝑆) |
53 | 50, 51, 52 | syl2anc 581 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
54 | 39, 40, 40, 42, 53, 43, 44, 51 | dvmptres 24125 |
. . . . . 6
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0)) |
55 | | sum0 14829 |
. . . . . . . 8
⊢
Σ𝑖 ∈
∅ 𝐴 =
0 |
56 | 55 | mpteq2i 4964 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴) = (𝑥 ∈ 𝑋 ↦ 0) |
57 | 56 | oveq2i 6916 |
. . . . . 6
⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) |
58 | | sum0 14829 |
. . . . . . 7
⊢
Σ𝑖 ∈
∅ 𝐵 =
0 |
59 | 58 | mpteq2i 4964 |
. . . . . 6
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 0) |
60 | 54, 57, 59 | 3eqtr4g 2886 |
. . . . 5
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵)) |
61 | 60 | a1d 25 |
. . . 4
⊢ (𝜑 → (∅ ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))) |
62 | | ssun1 4003 |
. . . . . . . . . 10
⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐}) |
63 | | sstr 3835 |
. . . . . . . . . 10
⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏 ⊆ 𝐼) |
64 | 62, 63 | mpan 683 |
. . . . . . . . 9
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑏 ⊆ 𝐼) |
65 | 64 | imim1i 63 |
. . . . . . . 8
⊢ ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) |
66 | | simpll 785 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝜑) |
67 | 66, 39 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝑆 ∈ {ℝ, ℂ}) |
68 | 2 | ad3antrrr 723 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝐼 ∈ Fin) |
69 | 64 | ad2antlr 720 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ⊆ 𝐼) |
70 | | ssfi 8449 |
. . . . . . . . . . . . . . 15
⊢ ((𝐼 ∈ Fin ∧ 𝑏 ⊆ 𝐼) → 𝑏 ∈ Fin) |
71 | 68, 69, 70 | syl2anc 581 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ∈ Fin) |
72 | | simp-4l 803 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝜑) |
73 | 69 | sselda 3827 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑖 ∈ 𝐼) |
74 | | simplr 787 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑎 ∈ 𝑋) |
75 | | nfv 2015 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) |
76 | | nfcsb1v 3773 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴 |
77 | 76 | nfel1 2984 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ |
78 | 75, 77 | nfim 2001 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
79 | | eleq1w 2889 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝑋 ↔ 𝑎 ∈ 𝑋)) |
80 | 79 | 3anbi3d 1572 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋))) |
81 | | csbeq1a 3766 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑥⦌𝐴) |
82 | 81 | eleq1d 2891 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → (𝐴 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)) |
83 | 80, 82 | imbi12d 336 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))) |
84 | | dvmptfsum.a |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
85 | 78, 83, 84 | chvar 2416 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
86 | 72, 73, 74, 85 | syl3anc 1496 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
87 | 71, 86 | fsumcl 14841 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
88 | 87 | adantlrr 714 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
89 | | sumex 14795 |
. . . . . . . . . . . . 13
⊢
Σ𝑖 ∈
𝑏 ⦋𝑎 / 𝑥⦌𝐵 ∈ V |
90 | 89 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 ∈ V) |
91 | | nfcv 2969 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐴 |
92 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥𝑏 |
93 | 92, 76 | nfsum 14798 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 |
94 | 81 | sumeq2sdv 14812 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐴 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴) |
95 | 91, 93, 94 | cbvmpt 4972 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴) |
96 | 95 | oveq2i 6916 |
. . . . . . . . . . . . . . 15
⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) |
97 | | nfcv 2969 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐵 |
98 | | nfcsb1v 3773 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 |
99 | 92, 98 | nfsum 14798 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 |
100 | | csbeq1a 3766 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵) |
101 | 100 | sumeq2sdv 14812 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐵 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵) |
102 | 97, 99, 101 | cbvmpt 4972 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵) |
103 | 96, 102 | eqeq12i 2839 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) ↔ (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)) |
104 | 103 | biimpi 208 |
. . . . . . . . . . . . 13
⊢ ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) → (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)) |
105 | 104 | ad2antll 722 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)) |
106 | | simplll 793 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝜑) |
107 | | ssun2 4004 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑐} ⊆ (𝑏 ∪ {𝑐}) |
108 | | sstr 3835 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑐} ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → {𝑐} ⊆ 𝐼) |
109 | 107, 108 | mpan 683 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → {𝑐} ⊆ 𝐼) |
110 | | vex 3417 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑐 ∈ V |
111 | 110 | snss 4535 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 ∈ 𝐼 ↔ {𝑐} ⊆ 𝐼) |
112 | 109, 111 | sylibr 226 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ 𝐼) |
113 | 112 | ad2antlr 720 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑐 ∈ 𝐼) |
114 | | simpr 479 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ 𝑋) |
115 | 84 | 3expb 1155 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ ℂ) |
116 | 115 | ancom2s 642 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐴 ∈ ℂ) |
117 | 116 | ralrimivva 3180 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ) |
118 | | nfcsb1v 3773 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 |
119 | 118 | nfel1 2984 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ |
120 | | csbeq1a 3766 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴) |
121 | 120 | eleq1d 2891 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)) |
122 | 77, 119, 82, 121 | rspc2 3537 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)) |
123 | 122 | ancoms 452 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)) |
124 | 117, 123 | mpan9 504 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
125 | 106, 113,
114, 124 | syl12anc 872 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
126 | 125 | adantlrr 714 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
127 | | dvmptfsum.b |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
128 | 127 | 3expb 1155 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐵 ∈ ℂ) |
129 | 128 | ancom2s 642 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐵 ∈ ℂ) |
130 | 129 | ralrimivva 3180 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ) |
131 | 98 | nfel1 2984 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ |
132 | | nfcsb1v 3773 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 |
133 | 132 | nfel1 2984 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ |
134 | 100 | eleq1d 2891 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → (𝐵 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)) |
135 | | csbeq1a 3766 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵) |
136 | 135 | eleq1d 2891 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)) |
137 | 131, 133,
134, 136 | rspc2 3537 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)) |
138 | 137 | ancoms 452 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)) |
139 | 130, 138 | mpan9 504 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) |
140 | 106, 113,
114, 139 | syl12anc 872 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) |
141 | 140 | adantlrr 714 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) |
142 | 112 | ad2antrl 721 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝑐 ∈ 𝐼) |
143 | | nfv 2015 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑖(𝜑 ∧ 𝑐 ∈ 𝐼) |
144 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑖𝑆 |
145 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑖
D |
146 | | nfcv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑖𝑋 |
147 | | nfcsb1v 3773 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐴 |
148 | 146, 147 | nfmpt 4969 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴) |
149 | 144, 145,
148 | nfov 6935 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) |
150 | | nfcsb1v 3773 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐵 |
151 | 146, 150 | nfmpt 4969 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵) |
152 | 149, 151 | nfeq 2981 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵) |
153 | 143, 152 | nfim 2001 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)) |
154 | | eleq1w 2889 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑐 → (𝑖 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼)) |
155 | 154 | anbi2d 624 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑐 → ((𝜑 ∧ 𝑖 ∈ 𝐼) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼))) |
156 | | csbeq1a 3766 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑐 → 𝐴 = ⦋𝑐 / 𝑖⦌𝐴) |
157 | 156 | mpteq2dv 4968 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) |
158 | 157 | oveq2d 6921 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑐 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))) |
159 | | csbeq1a 3766 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑐 → 𝐵 = ⦋𝑐 / 𝑖⦌𝐵) |
160 | 159 | mpteq2dv 4968 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)) |
161 | 158, 160 | eqeq12d 2840 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑐 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))) |
162 | 155, 161 | imbi12d 336 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑐 → (((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)))) |
163 | | dvmptfsum.d |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
164 | 153, 162,
163 | chvar 2416 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)) |
165 | | nfcv 2969 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐴 |
166 | | nfcv 2969 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥𝑐 |
167 | 166, 76 | nfcsb 3775 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 |
168 | 81 | csbeq2dv 4216 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴) |
169 | 165, 167,
168 | cbvmpt 4972 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴) |
170 | 169 | oveq2i 6916 |
. . . . . . . . . . . . . 14
⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) |
171 | | nfcv 2969 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐵 |
172 | 166, 98 | nfcsb 3775 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 |
173 | 100 | csbeq2dv 4216 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵) |
174 | 171, 172,
173 | cbvmpt 4972 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵) |
175 | 164, 170,
174 | 3eqtr3g 2884 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)) |
176 | 66, 142, 175 | syl2anc 581 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)) |
177 | 67, 88, 90, 105, 126, 141, 176 | dvmptadd 24122 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))) |
178 | | nfcv 2969 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴 |
179 | | nfcv 2969 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥(𝑏 ∪ {𝑐}) |
180 | 179, 76 | nfsum 14798 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴 |
181 | 81 | sumeq2sdv 14812 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴) |
182 | 178, 180,
181 | cbvmpt 4972 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴) |
183 | | simpllr 795 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ¬ 𝑐 ∈ 𝑏) |
184 | | disjsn 4465 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏) |
185 | 183, 184 | sylibr 226 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∩ {𝑐}) = ∅) |
186 | | eqidd 2826 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) = (𝑏 ∪ {𝑐})) |
187 | | simplr 787 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ⊆ 𝐼) |
188 | | ssfi 8449 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∈ Fin ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑏 ∪ {𝑐}) ∈ Fin) |
189 | 68, 187, 188 | syl2anc 581 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ∈ Fin) |
190 | | simp-4l 803 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝜑) |
191 | 187 | sselda 3827 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑖 ∈ 𝐼) |
192 | | simplr 787 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑎 ∈ 𝑋) |
193 | 190, 191,
192, 85 | syl3anc 1496 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) |
194 | 185, 186,
189, 193 | fsumsplit 14848 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴)) |
195 | | sumsns 14856 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑐 ∈ V ∧
⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴) |
196 | 110, 125,
195 | sylancr 583 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴) |
197 | 196 | oveq2d 6921 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴) = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) |
198 | 194, 197 | eqtrd 2861 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) |
199 | 198 | mpteq2dva 4967 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))) |
200 | 182, 199 | syl5eq 2873 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))) |
201 | 200 | adantrr 710 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))) |
202 | 201 | oveq2d 6921 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)))) |
203 | | nfcv 2969 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵 |
204 | 179, 98 | nfsum 14798 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵 |
205 | 100 | sumeq2sdv 14812 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵) |
206 | 203, 204,
205 | cbvmpt 4972 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵) |
207 | 75, 131 | nfim 2001 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) |
208 | 80, 134 | imbi12d 336 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))) |
209 | 207, 208,
127 | chvar 2416 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) |
210 | 190, 191,
192, 209 | syl3anc 1496 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ ¬
𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) |
211 | 185, 186,
189, 210 | fsumsplit 14848 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵)) |
212 | | sumsns 14856 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑐 ∈ V ∧
⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵) |
213 | 110, 140,
212 | sylancr 583 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵) |
214 | 213 | oveq2d 6921 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵) = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)) |
215 | 211, 214 | eqtrd 2861 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)) |
216 | 215 | mpteq2dva 4967 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))) |
217 | 206, 216 | syl5eq 2873 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))) |
218 | 217 | adantrr 710 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))) |
219 | 177, 202,
218 | 3eqtr4d 2871 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)) |
220 | 219 | exp32 413 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))) |
221 | 220 | a2d 29 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) → (((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))) |
222 | 65, 221 | syl5 34 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) → ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))) |
223 | 222 | expcom 404 |
. . . . . 6
⊢ (¬
𝑐 ∈ 𝑏 → (𝜑 → ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))) |
224 | 223 | adantl 475 |
. . . . 5
⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (𝜑 → ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))) |
225 | 224 | a2d 29 |
. . . 4
⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → ((𝜑 → (𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝜑 → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))) |
226 | 11, 20, 29, 38, 61, 225 | findcard2s 8470 |
. . 3
⊢ (𝐼 ∈ Fin → (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)))) |
227 | 2, 226 | mpcom 38 |
. 2
⊢ (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))) |
228 | 1, 227 | mpi 20 |
1
⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)) |