Step | Hyp | Ref
| Expression |
1 | | ssid 3587 |
. 2
⊢ 𝐴 ⊆ 𝐴 |
2 | | fsumcn.5 |
. . 3
⊢ (𝜑 → 𝐴 ∈ Fin) |
3 | | sseq1 3589 |
. . . . . 6
⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
4 | | sumeq1 14267 |
. . . . . . . 8
⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵) |
5 | 4 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝑤 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵)) |
6 | 5 | eleq1d 2672 |
. . . . . 6
⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))) |
7 | 3, 6 | imbi12d 333 |
. . . . 5
⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))) |
8 | 7 | imbi2d 329 |
. . . 4
⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))))) |
9 | | sseq1 3589 |
. . . . . 6
⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) |
10 | | sumeq1 14267 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑦 𝐵) |
11 | 10 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵)) |
12 | 11 | eleq1d 2672 |
. . . . . 6
⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) |
13 | 9, 12 | imbi12d 333 |
. . . . 5
⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)))) |
14 | 13 | imbi2d 329 |
. . . 4
⊢ (𝑤 = 𝑦 → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))))) |
15 | | sseq1 3589 |
. . . . . 6
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) |
16 | | sumeq1 14267 |
. . . . . . . 8
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) |
17 | 16 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) |
18 | 17 | eleq1d 2672 |
. . . . . 6
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))) |
19 | 15, 18 | imbi12d 333 |
. . . . 5
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))) |
20 | 19 | imbi2d 329 |
. . . 4
⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))) |
21 | | sseq1 3589 |
. . . . . 6
⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) |
22 | | sumeq1 14267 |
. . . . . . . 8
⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
23 | 22 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝑤 = 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) |
24 | 23 | eleq1d 2672 |
. . . . . 6
⊢ (𝑤 = 𝐴 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))) |
25 | 21, 24 | imbi12d 333 |
. . . . 5
⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))) |
26 | 25 | imbi2d 329 |
. . . 4
⊢ (𝑤 = 𝐴 → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))))) |
27 | | sum0 14299 |
. . . . . . 7
⊢
Σ𝑘 ∈
∅ 𝐵 =
0 |
28 | 27 | mpteq2i 4669 |
. . . . . 6
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 0) |
29 | | fsumcn.4 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
30 | | fsumcn.3 |
. . . . . . . . 9
⊢ 𝐾 =
(TopOpen‘ℂfld) |
31 | 30 | cnfldtopon 22396 |
. . . . . . . 8
⊢ 𝐾 ∈
(TopOn‘ℂ) |
32 | 31 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈
(TopOn‘ℂ)) |
33 | | 0cnd 9912 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
ℂ) |
34 | 29, 32, 33 | cnmptc 21275 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝐽 Cn 𝐾)) |
35 | 28, 34 | syl5eqel 2692 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)) |
36 | 35 | a1d 25 |
. . . 4
⊢ (𝜑 → (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))) |
37 | | ssun1 3738 |
. . . . . . . . . 10
⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧}) |
38 | | sstr 3576 |
. . . . . . . . . 10
⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑦 ⊆ 𝐴) |
39 | 37, 38 | mpan 702 |
. . . . . . . . 9
⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑦 ⊆ 𝐴) |
40 | 39 | imim1i 61 |
. . . . . . . 8
⊢ ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) |
41 | | simplr 788 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ¬ 𝑧 ∈ 𝑦) |
42 | | disjsn 4192 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦) |
43 | 41, 42 | sylibr 223 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∩ {𝑧}) = ∅) |
44 | | eqidd 2611 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧})) |
45 | 2 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ Fin) |
46 | | simprl 790 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴) |
47 | | ssfi 8065 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin) |
48 | 45, 46, 47 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ∈ Fin) |
49 | | simplll 794 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑) |
50 | 46 | sselda 3568 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴) |
51 | | simplrr 797 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ 𝑋) |
52 | 29 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋)) |
53 | 31 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐾 ∈
(TopOn‘ℂ)) |
54 | | fsumcn.6 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) |
55 | | cnf2 20863 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ) |
56 | 52, 53, 54, 55 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ) |
57 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) |
58 | 57 | fmpt 6289 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑥 ∈
𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ) |
59 | 56, 58 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ) |
60 | | rsp 2913 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑥 ∈
𝑋 𝐵 ∈ ℂ → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℂ)) |
61 | 59, 60 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℂ)) |
62 | 61 | imp 444 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
63 | 49, 50, 51, 62 | syl21anc 1317 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℂ) |
64 | 43, 44, 48, 63 | fsumsplit 14318 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + Σ𝑘 ∈ {𝑧}𝐵)) |
65 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑦 ∪ {𝑧}) ⊆ 𝐴) |
66 | 65 | unssbd 3753 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → {𝑧} ⊆ 𝐴) |
67 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑧 ∈ V |
68 | 67 | snss 4259 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴) |
69 | 66, 68 | sylibr 223 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑧 ∈ 𝐴) |
70 | 69 | adantrr 749 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝐴) |
71 | 61 | impancom 455 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐴 → 𝐵 ∈ ℂ)) |
72 | 71 | ralrimiv 2948 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
73 | 72 | ad2ant2rl 781 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
74 | | nfcsb1v 3515 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 |
75 | 74 | nfel1 2765 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ |
76 | | csbeq1a 3508 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵) |
77 | 76 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)) |
78 | 75, 77 | rspc 3276 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)) |
79 | 70, 73, 78 | sylc 63 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) |
80 | | sumsns 14323 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵) |
81 | 70, 79, 80 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵) |
82 | 81 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (Σ𝑘 ∈ 𝑦 𝐵 + Σ𝑘 ∈ {𝑧}𝐵) = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
83 | 64, 82 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
84 | 83 | anassrs 678 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
85 | 84 | mpteq2dva 4672 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))) |
86 | 85 | adantrr 749 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))) |
87 | | nfcv 2751 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) |
88 | | nfcv 2751 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥𝑦 |
89 | | nfcsb1v 3515 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵 |
90 | 88, 89 | nfsum 14269 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 |
91 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥
+ |
92 | | nfcv 2751 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥𝑧 |
93 | 92, 89 | nfcsb 3517 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵 |
94 | 90, 91, 93 | nfov 6575 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵) |
95 | | csbeq1a 3508 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵) |
96 | 95 | sumeq2sdv 14282 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵) |
97 | 95 | csbeq2dv 3944 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵) |
98 | 96, 97 | oveq12d 6567 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)) |
99 | 87, 94, 98 | cbvmpt 4677 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) = (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)) |
100 | 86, 99 | syl6eq 2660 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))) |
101 | 29 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝐽 ∈ (TopOn‘𝑋)) |
102 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐵 |
103 | 102, 90, 96 | cbvmpt 4677 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) = (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵) |
104 | | simprr 792 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) |
105 | 103, 104 | syl5eqelr 2693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
106 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐵 |
107 | 106, 93, 97 | cbvmpt 4677 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) = (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵) |
108 | 69 | adantrr 749 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝑧 ∈ 𝐴) |
109 | 54 | ralrimiva 2949 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) |
110 | 109 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) |
111 | | nfcv 2751 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘𝑋 |
112 | 111, 74 | nfmpt 4674 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) |
113 | 112 | nfel1 2765 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾) |
114 | 76 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵)) |
115 | 114 | eleq1d 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))) |
116 | 113, 115 | rspc 3276 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))) |
117 | 108, 110,
116 | sylc 63 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
118 | 107, 117 | syl5eqelr 2693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
119 | 30 | addcn 22476 |
. . . . . . . . . . . . 13
⊢ + ∈
((𝐾 ×t
𝐾) Cn 𝐾) |
120 | 119 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
121 | 101, 105,
118, 120 | cnmpt12f 21279 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)) ∈ (𝐽 Cn 𝐾)) |
122 | 100, 121 | eqeltrd 2688 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)) |
123 | 122 | exp32 629 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))) |
124 | 123 | a2d 29 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))) |
125 | 40, 124 | syl5 33 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))) |
126 | 125 | expcom 450 |
. . . . . 6
⊢ (¬
𝑧 ∈ 𝑦 → (𝜑 → ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))) |
127 | 126 | adantl 481 |
. . . . 5
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜑 → ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))) |
128 | 127 | a2d 29 |
. . . 4
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))) |
129 | 8, 14, 20, 26, 36, 128 | findcard2s 8086 |
. . 3
⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))) |
130 | 2, 129 | mpcom 37 |
. 2
⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))) |
131 | 1, 130 | mpi 20 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |