Step | Hyp | Ref
| Expression |
1 | | relxp 5150 |
. . . . . . . . 9
⊢ Rel
({𝑗} × 𝐵) |
2 | 1 | rgenw 2908 |
. . . . . . . 8
⊢
∀𝑗 ∈
𝐴 Rel ({𝑗} × 𝐵) |
3 | | reliun 5162 |
. . . . . . . 8
⊢ (Rel
∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐵)) |
4 | 2, 3 | mpbir 220 |
. . . . . . 7
⊢ Rel
∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
5 | | relcnv 5422 |
. . . . . . 7
⊢ Rel ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) |
6 | | ancom 465 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑗 ∧ 𝑦 = 𝑘) ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗)) |
7 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑥 ∈ V |
8 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
9 | 7, 8 | opth 4871 |
. . . . . . . . . . . 12
⊢
(〈𝑥, 𝑦〉 = 〈𝑗, 𝑘〉 ↔ (𝑥 = 𝑗 ∧ 𝑦 = 𝑘)) |
10 | 8, 7 | opth 4871 |
. . . . . . . . . . . 12
⊢
(〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ↔ (𝑦 = 𝑘 ∧ 𝑥 = 𝑗)) |
11 | 6, 9, 10 | 3bitr4i 291 |
. . . . . . . . . . 11
⊢
(〈𝑥, 𝑦〉 = 〈𝑗, 𝑘〉 ↔ 〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉) |
12 | 11 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (〈𝑥, 𝑦〉 = 〈𝑗, 𝑘〉 ↔ 〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉)) |
13 | | fsumcom2.4 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) |
14 | 12, 13 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝜑 → ((〈𝑥, 𝑦〉 = 〈𝑗, 𝑘〉 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ (〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))) |
15 | 14 | 2exbidv 1839 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑗∃𝑘(〈𝑥, 𝑦〉 = 〈𝑗, 𝑘〉 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) ↔ ∃𝑗∃𝑘(〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)))) |
16 | | eliunxp 5181 |
. . . . . . . 8
⊢
(〈𝑥, 𝑦〉 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗∃𝑘(〈𝑥, 𝑦〉 = 〈𝑗, 𝑘〉 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))) |
17 | 7, 8 | opelcnv 5226 |
. . . . . . . . 9
⊢
(〈𝑥, 𝑦〉 ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ 〈𝑦, 𝑥〉 ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)) |
18 | | eliunxp 5181 |
. . . . . . . . 9
⊢
(〈𝑦, 𝑥〉 ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑘∃𝑗(〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) |
19 | | excom 2029 |
. . . . . . . . 9
⊢
(∃𝑘∃𝑗(〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷)) ↔ ∃𝑗∃𝑘(〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) |
20 | 17, 18, 19 | 3bitri 285 |
. . . . . . . 8
⊢
(〈𝑥, 𝑦〉 ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ ∃𝑗∃𝑘(〈𝑦, 𝑥〉 = 〈𝑘, 𝑗〉 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) |
21 | 15, 16, 20 | 3bitr4g 302 |
. . . . . . 7
⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷))) |
22 | 4, 5, 21 | eqrelrdv 5139 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)) |
23 | | nfcv 2751 |
. . . . . . 7
⊢
Ⅎ𝑚({𝑗} × 𝐵) |
24 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑗{𝑚} |
25 | | nfcsb1v 3515 |
. . . . . . . 8
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 |
26 | 24, 25 | nfxp 5066 |
. . . . . . 7
⊢
Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
27 | | sneq 4135 |
. . . . . . . 8
⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚}) |
28 | | csbeq1a 3508 |
. . . . . . . 8
⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵) |
29 | 27, 28 | xpeq12d 5064 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)) |
30 | 23, 26, 29 | cbviun 4493 |
. . . . . 6
⊢ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ∪
𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
31 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑛({𝑘} × 𝐷) |
32 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑘{𝑛} |
33 | | nfcsb1v 3515 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐷 |
34 | 32, 33 | nfxp 5066 |
. . . . . . . 8
⊢
Ⅎ𝑘({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) |
35 | | sneq 4135 |
. . . . . . . . 9
⊢ (𝑘 = 𝑛 → {𝑘} = {𝑛}) |
36 | | csbeq1a 3508 |
. . . . . . . . 9
⊢ (𝑘 = 𝑛 → 𝐷 = ⦋𝑛 / 𝑘⦌𝐷) |
37 | 35, 36 | xpeq12d 5064 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → ({𝑘} × 𝐷) = ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
38 | 31, 34, 37 | cbviun 4493 |
. . . . . . 7
⊢ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) |
39 | 38 | cnveqi 5219 |
. . . . . 6
⊢ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) |
40 | 22, 30, 39 | 3eqtr3g 2667 |
. . . . 5
⊢ (𝜑 → ∪ 𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ◡∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
41 | 40 | sumeq1d 14279 |
. . . 4
⊢ (𝜑 → Σ𝑧 ∈ ∪
𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸 = Σ𝑧 ∈ ◡ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸) |
42 | | vex 3176 |
. . . . . . . 8
⊢ 𝑛 ∈ V |
43 | | vex 3176 |
. . . . . . . 8
⊢ 𝑚 ∈ V |
44 | 42, 43 | op1std 7069 |
. . . . . . 7
⊢ (𝑤 = 〈𝑛, 𝑚〉 → (1st ‘𝑤) = 𝑛) |
45 | 44 | csbeq1d 3506 |
. . . . . 6
⊢ (𝑤 = 〈𝑛, 𝑚〉 → ⦋(1st
‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸) |
46 | 42, 43 | op2ndd 7070 |
. . . . . . . 8
⊢ (𝑤 = 〈𝑛, 𝑚〉 → (2nd ‘𝑤) = 𝑚) |
47 | 46 | csbeq1d 3506 |
. . . . . . 7
⊢ (𝑤 = 〈𝑛, 𝑚〉 → ⦋(2nd
‘𝑤) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸) |
48 | 47 | csbeq2dv 3944 |
. . . . . 6
⊢ (𝑤 = 〈𝑛, 𝑚〉 → ⦋𝑛 / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
49 | 45, 48 | eqtrd 2644 |
. . . . 5
⊢ (𝑤 = 〈𝑛, 𝑚〉 → ⦋(1st
‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
50 | 43, 42 | op2ndd 7070 |
. . . . . . 7
⊢ (𝑧 = 〈𝑚, 𝑛〉 → (2nd ‘𝑧) = 𝑛) |
51 | 50 | csbeq1d 3506 |
. . . . . 6
⊢ (𝑧 = 〈𝑚, 𝑛〉 → ⦋(2nd
‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸) |
52 | 43, 42 | op1std 7069 |
. . . . . . . 8
⊢ (𝑧 = 〈𝑚, 𝑛〉 → (1st ‘𝑧) = 𝑚) |
53 | 52 | csbeq1d 3506 |
. . . . . . 7
⊢ (𝑧 = 〈𝑚, 𝑛〉 → ⦋(1st
‘𝑧) / 𝑗⦌𝐸 = ⦋𝑚 / 𝑗⦌𝐸) |
54 | 53 | csbeq2dv 3944 |
. . . . . 6
⊢ (𝑧 = 〈𝑚, 𝑛〉 → ⦋𝑛 / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
55 | 51, 54 | eqtrd 2644 |
. . . . 5
⊢ (𝑧 = 〈𝑚, 𝑛〉 → ⦋(2nd
‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
56 | | fsumcom2.2 |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ Fin) |
57 | | snfi 7923 |
. . . . . . . 8
⊢ {𝑛} ∈ Fin |
58 | | fsumcom2.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ Fin) |
59 | 58 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐴 ∈ Fin) |
60 | 43, 42 | opelcnv 5226 |
. . . . . . . . . . . . . . . 16
⊢
(〈𝑚, 𝑛〉 ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ 〈𝑛, 𝑚〉 ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)) |
61 | 33, 36 | opeliunxp2f 7223 |
. . . . . . . . . . . . . . . 16
⊢
(〈𝑛, 𝑚〉 ∈ ∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷) ↔ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) |
62 | 60, 61 | sylbbr 225 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷) → 〈𝑚, 𝑛〉 ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)) |
63 | 62 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 〈𝑚, 𝑛〉 ∈ ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)) |
64 | 22 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) = ◡∪ 𝑘 ∈ 𝐶 ({𝑘} × 𝐷)) |
65 | 63, 64 | eleqtrrd 2691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 〈𝑚, 𝑛〉 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
66 | | eliun 4460 |
. . . . . . . . . . . . 13
⊢
(〈𝑚, 𝑛〉 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) |
67 | 65, 66 | sylib 207 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑗 ∈ 𝐴 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) |
68 | | simpr 476 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑗 ∈ 𝐴 ∧ 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) → 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) |
69 | | opelxp 5070 |
. . . . . . . . . . . . . . . . 17
⊢
(〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵) ↔ (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵)) |
70 | 68, 69 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 ∈ 𝐴 ∧ 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) → (𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵)) |
71 | 70 | simpld 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝑗 ∈ 𝐴 ∧ 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ {𝑗}) |
72 | | elsni 4142 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗) |
73 | 71, 72 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ 𝐴 ∧ 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) → 𝑚 = 𝑗) |
74 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ 𝐴 ∧ 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝐴) |
75 | 73, 74 | eqeltrd 2688 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ 𝐴 ∧ 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵)) → 𝑚 ∈ 𝐴) |
76 | 75 | rexlimiva 3010 |
. . . . . . . . . . . 12
⊢
(∃𝑗 ∈
𝐴 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵) → 𝑚 ∈ 𝐴) |
77 | 67, 76 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑚 ∈ 𝐴) |
78 | 77 | expr 641 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷 → 𝑚 ∈ 𝐴)) |
79 | 78 | ssrdv 3574 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ⊆ 𝐴) |
80 | 59, 79 | ssfid 8068 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ⦋𝑛 / 𝑘⦌𝐷 ∈ Fin) |
81 | | xpfi 8116 |
. . . . . . . 8
⊢ (({𝑛} ∈ Fin ∧
⦋𝑛 / 𝑘⦌𝐷 ∈ Fin) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) |
82 | 57, 80, 81 | sylancr 694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) |
83 | 82 | ralrimiva 2949 |
. . . . . 6
⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) |
84 | | iunfi 8137 |
. . . . . 6
⊢ ((𝐶 ∈ Fin ∧ ∀𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) |
85 | 56, 83, 84 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ∈ Fin) |
86 | | reliun 5162 |
. . . . . . 7
⊢ (Rel
∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∀𝑛 ∈ 𝐶 Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
87 | | relxp 5150 |
. . . . . . . 8
⊢ Rel
({𝑛} ×
⦋𝑛 / 𝑘⦌𝐷) |
88 | 87 | a1i 11 |
. . . . . . 7
⊢ (𝑛 ∈ 𝐶 → Rel ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
89 | 86, 88 | mprgbir 2911 |
. . . . . 6
⊢ Rel
∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) |
90 | 89 | a1i 11 |
. . . . 5
⊢ (𝜑 → Rel ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
91 | | simpr 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
92 | | eliun 4460 |
. . . . . . . 8
⊢ (𝑤 ∈ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) ↔ ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
93 | 91, 92 | sylib 207 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∃𝑛 ∈ 𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) |
94 | | xp2nd 7090 |
. . . . . . . . . 10
⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷) |
95 | 94 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈ ⦋𝑛 / 𝑘⦌𝐷) |
96 | | xp1st 7089 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ {𝑛}) |
97 | 96 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ {𝑛}) |
98 | | elsni 4142 |
. . . . . . . . . . 11
⊢
((1st ‘𝑤) ∈ {𝑛} → (1st ‘𝑤) = 𝑛) |
99 | 97, 98 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) = 𝑛) |
100 | 99 | csbeq1d 3506 |
. . . . . . . . 9
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st
‘𝑤) / 𝑘⦌𝐷 = ⦋𝑛 / 𝑘⦌𝐷) |
101 | 95, 100 | eleqtrrd 2691 |
. . . . . . . 8
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈
⦋(1st ‘𝑤) / 𝑘⦌𝐷) |
102 | 101 | rexlimiva 3010 |
. . . . . . 7
⊢
(∃𝑛 ∈
𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (2nd ‘𝑤) ∈
⦋(1st ‘𝑤) / 𝑘⦌𝐷) |
103 | 93, 102 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (2nd ‘𝑤) ∈
⦋(1st ‘𝑤) / 𝑘⦌𝐷) |
104 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ 𝐶) |
105 | 99, 104 | eqeltrd 2688 |
. . . . . . . . 9
⊢ ((𝑛 ∈ 𝐶 ∧ 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶) |
106 | 105 | rexlimiva 3010 |
. . . . . . . 8
⊢
(∃𝑛 ∈
𝐶 𝑤 ∈ ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷) → (1st ‘𝑤) ∈ 𝐶) |
107 | 93, 106 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → (1st ‘𝑤) ∈ 𝐶) |
108 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝜑) |
109 | 25 | nfcri 2745 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 |
110 | 72 | equcomd 1933 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ {𝑗} → 𝑗 = 𝑚) |
111 | 110, 28 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ {𝑗} → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵) |
112 | 111 | eleq2d 2673 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ {𝑗} → (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)) |
113 | 112 | biimpa 500 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 ∈ {𝑗} ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵) |
114 | 69, 113 | sylbi 206 |
. . . . . . . . . . . . 13
⊢
(〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵) |
115 | 114 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ 𝐴 → (〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)) |
116 | 109, 115 | rexlimi 3006 |
. . . . . . . . . . 11
⊢
(∃𝑗 ∈
𝐴 〈𝑚, 𝑛〉 ∈ ({𝑗} × 𝐵) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵) |
117 | 67, 116 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵) |
118 | | fsumcom2.5 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ) |
119 | 118 | ralrimivva 2954 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ) |
120 | | nfcsb1v 3515 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸 |
121 | 120 | nfel1 2765 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ |
122 | 25, 121 | nfral 2929 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ |
123 | | csbeq1a 3508 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑚 → 𝐸 = ⦋𝑚 / 𝑗⦌𝐸) |
124 | 123 | eleq1d 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑚 → (𝐸 ∈ ℂ ↔ ⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
125 | 28, 124 | raleqbidv 3129 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑚 → (∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
126 | 122, 125 | rspc 3276 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐸 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
127 | 119, 126 | mpan9 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ) |
128 | | nfcsb1v 3515 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 |
129 | 128 | nfel1 2765 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ |
130 | | csbeq1a 3508 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
131 | 130 | eleq1d 2672 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → (⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
132 | 129, 131 | rspc 3276 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
133 | 127, 132 | syl5com 31 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵 → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
134 | 133 | impr 647 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ 𝐴 ∧ 𝑛 ∈ ⦋𝑚 / 𝑗⦌𝐵)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ) |
135 | 108, 77, 117, 134 | syl12anc 1316 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ 𝐶 ∧ 𝑚 ∈ ⦋𝑛 / 𝑘⦌𝐷)) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ) |
136 | 135 | ralrimivva 2954 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ) |
137 | 136 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ) |
138 | | csbeq1 3502 |
. . . . . . . . 9
⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌𝐷 = ⦋(1st
‘𝑤) / 𝑘⦌𝐷) |
139 | | csbeq1 3502 |
. . . . . . . . . 10
⊢ (𝑛 = (1st ‘𝑤) → ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st
‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
140 | 139 | eleq1d 2672 |
. . . . . . . . 9
⊢ (𝑛 = (1st ‘𝑤) → (⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔
⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
141 | 138, 140 | raleqbidv 3129 |
. . . . . . . 8
⊢ (𝑛 = (1st ‘𝑤) → (∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔ ∀𝑚 ∈ ⦋
(1st ‘𝑤) /
𝑘⦌𝐷⦋(1st
‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
142 | 141 | rspcv 3278 |
. . . . . . 7
⊢
((1st ‘𝑤) ∈ 𝐶 → (∀𝑛 ∈ 𝐶 ∀𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ → ∀𝑚 ∈ ⦋
(1st ‘𝑤) /
𝑘⦌𝐷⦋(1st
‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ)) |
143 | 107, 137,
142 | sylc 63 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ∀𝑚 ∈ ⦋ (1st
‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ) |
144 | | csbeq1 3502 |
. . . . . . . . 9
⊢ (𝑚 = (2nd ‘𝑤) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋(2nd
‘𝑤) / 𝑗⦌𝐸) |
145 | 144 | csbeq2dv 3944 |
. . . . . . . 8
⊢ (𝑚 = (2nd ‘𝑤) →
⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = ⦋(1st
‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸) |
146 | 145 | eleq1d 2672 |
. . . . . . 7
⊢ (𝑚 = (2nd ‘𝑤) →
(⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ ↔
⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)) |
147 | 146 | rspcv 3278 |
. . . . . 6
⊢
((2nd ‘𝑤) ∈ ⦋(1st
‘𝑤) / 𝑘⦌𝐷 → (∀𝑚 ∈ ⦋ (1st
‘𝑤) / 𝑘⦌𝐷⦋(1st ‘𝑤) / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 ∈ ℂ →
⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 ∈ ℂ)) |
148 | 103, 143,
147 | sylc 63 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)) → ⦋(1st
‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 ∈ ℂ) |
149 | 49, 55, 85, 90, 148 | fsumcnv 14346 |
. . . 4
⊢ (𝜑 → Σ𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸 = Σ𝑧 ∈ ◡ ∪ 𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸) |
150 | 41, 149 | eqtr4d 2647 |
. . 3
⊢ (𝜑 → Σ𝑧 ∈ ∪
𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸 = Σ𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸) |
151 | | fsumcom2.3 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) |
152 | 151 | ralrimiva 2949 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin) |
153 | 25 | nfel1 2765 |
. . . . . 6
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 ∈ Fin |
154 | 28 | eleq1d 2672 |
. . . . . 6
⊢ (𝑗 = 𝑚 → (𝐵 ∈ Fin ↔ ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin)) |
155 | 153, 154 | rspc 3276 |
. . . . 5
⊢ (𝑚 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin)) |
156 | 152, 155 | mpan9 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑗⦌𝐵 ∈ Fin) |
157 | 55, 58, 156, 134 | fsum2d 14344 |
. . 3
⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑧 ∈ ∪
𝑚 ∈ 𝐴 ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋(1st
‘𝑧) / 𝑗⦌𝐸) |
158 | 49, 56, 80, 135 | fsum2d 14344 |
. . 3
⊢ (𝜑 → Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑤 ∈ ∪
𝑛 ∈ 𝐶 ({𝑛} × ⦋𝑛 / 𝑘⦌𝐷)⦋(1st ‘𝑤) / 𝑘⦌⦋(2nd
‘𝑤) / 𝑗⦌𝐸) |
159 | 150, 157,
158 | 3eqtr4d 2654 |
. 2
⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
160 | | nfcv 2751 |
. . 3
⊢
Ⅎ𝑚Σ𝑘 ∈ 𝐵 𝐸 |
161 | | nfcv 2751 |
. . . . 5
⊢
Ⅎ𝑗𝑛 |
162 | 161, 120 | nfcsb 3517 |
. . . 4
⊢
Ⅎ𝑗⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 |
163 | 25, 162 | nfsum 14269 |
. . 3
⊢
Ⅎ𝑗Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 |
164 | | nfcv 2751 |
. . . . 5
⊢
Ⅎ𝑛𝐸 |
165 | | nfcsb1v 3515 |
. . . . 5
⊢
Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐸 |
166 | | csbeq1a 3508 |
. . . . 5
⊢ (𝑘 = 𝑛 → 𝐸 = ⦋𝑛 / 𝑘⦌𝐸) |
167 | 164, 165,
166 | cbvsumi 14275 |
. . . 4
⊢
Σ𝑘 ∈
𝐵 𝐸 = Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸 |
168 | 123 | csbeq2dv 3944 |
. . . . . 6
⊢ (𝑗 = 𝑚 → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
169 | 168 | adantr 480 |
. . . . 5
⊢ ((𝑗 = 𝑚 ∧ 𝑛 ∈ 𝐵) → ⦋𝑛 / 𝑘⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
170 | 28, 169 | sumeq12dv 14284 |
. . . 4
⊢ (𝑗 = 𝑚 → Σ𝑛 ∈ 𝐵 ⦋𝑛 / 𝑘⦌𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
171 | 167, 170 | syl5eq 2656 |
. . 3
⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
172 | 160, 163,
171 | cbvsumi 14275 |
. 2
⊢
Σ𝑗 ∈
𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑚 ∈ 𝐴 Σ𝑛 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 |
173 | | nfcv 2751 |
. . 3
⊢
Ⅎ𝑛Σ𝑗 ∈ 𝐷 𝐸 |
174 | 33, 128 | nfsum 14269 |
. . 3
⊢
Ⅎ𝑘Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 |
175 | | nfcv 2751 |
. . . . 5
⊢
Ⅎ𝑚𝐸 |
176 | 175, 120,
123 | cbvsumi 14275 |
. . . 4
⊢
Σ𝑗 ∈
𝐷 𝐸 = Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸 |
177 | 130 | adantr 480 |
. . . . 5
⊢ ((𝑘 = 𝑛 ∧ 𝑚 ∈ 𝐷) → ⦋𝑚 / 𝑗⦌𝐸 = ⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
178 | 36, 177 | sumeq12dv 14284 |
. . . 4
⊢ (𝑘 = 𝑛 → Σ𝑚 ∈ 𝐷 ⦋𝑚 / 𝑗⦌𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
179 | 176, 178 | syl5eq 2656 |
. . 3
⊢ (𝑘 = 𝑛 → Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸) |
180 | 173, 174,
179 | cbvsumi 14275 |
. 2
⊢
Σ𝑘 ∈
𝐶 Σ𝑗 ∈ 𝐷 𝐸 = Σ𝑛 ∈ 𝐶 Σ𝑚 ∈ ⦋ 𝑛 / 𝑘⦌𝐷⦋𝑛 / 𝑘⦌⦋𝑚 / 𝑗⦌𝐸 |
181 | 159, 172,
180 | 3eqtr4g 2669 |
1
⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸) |