Step | Hyp | Ref
| Expression |
1 | | simpr 476 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝜓) |
2 | | fprod2d.7 |
. . . 4
⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) |
3 | 1, 2 | sylib 207 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) |
4 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑚∏𝑘 ∈ 𝐵 𝐶 |
5 | | nfcsb1v 3515 |
. . . . . . 7
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 |
6 | | nfcsb1v 3515 |
. . . . . . 7
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶 |
7 | 5, 6 | nfcprod 14480 |
. . . . . 6
⊢
Ⅎ𝑗∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 |
8 | | csbeq1a 3508 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵) |
9 | | csbeq1a 3508 |
. . . . . . . 8
⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶) |
10 | 9 | adantr 480 |
. . . . . . 7
⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶) |
11 | 8, 10 | prodeq12dv 14495 |
. . . . . 6
⊢ (𝑗 = 𝑚 → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶) |
12 | 4, 7, 11 | cbvprodi 14486 |
. . . . 5
⊢
∏𝑗 ∈
{𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 |
13 | | fprod2d.6 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴) |
14 | 13 | unssbd 3753 |
. . . . . . . 8
⊢ (𝜑 → {𝑦} ⊆ 𝐴) |
15 | | vex 3176 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
16 | 15 | snss 4259 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴) |
17 | 14, 16 | sylibr 223 |
. . . . . . 7
⊢ (𝜑 → 𝑦 ∈ 𝐴) |
18 | | fprod2d.3 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) |
19 | 18 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin) |
20 | | nfcsb1v 3515 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 |
21 | 20 | nfel1 2765 |
. . . . . . . . . 10
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin |
22 | | csbeq1a 3508 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
23 | 22 | eleq1d 2672 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)) |
24 | 21, 23 | rspc 3276 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)) |
25 | 17, 19, 24 | sylc 63 |
. . . . . . . 8
⊢ (𝜑 → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) |
26 | | fprod2d.4 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ) |
27 | 26 | ralrimivva 2954 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
28 | | nfcsb1v 3515 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 |
29 | 28 | nfel1 2765 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
30 | 20, 29 | nfral 2929 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
31 | | csbeq1a 3508 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
32 | 31 | eleq1d 2672 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
33 | 22, 32 | raleqbidv 3129 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
34 | 30, 33 | rspc 3276 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
35 | 17, 27, 34 | sylc 63 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
36 | 35 | r19.21bi 2916 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
37 | 25, 36 | fprodcl 14521 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
38 | | csbeq1 3502 |
. . . . . . . . 9
⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
39 | | csbeq1 3502 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
40 | 39 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑚 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
41 | 38, 40 | prodeq12dv 14495 |
. . . . . . . 8
⊢ (𝑚 = 𝑦 → ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
42 | 41 | prodsn 14531 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐴 ∧ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
43 | 17, 37, 42 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
44 | | nfcv 2751 |
. . . . . . . 8
⊢
Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶 |
45 | | nfcsb1v 3515 |
. . . . . . . 8
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
46 | | csbeq1a 3508 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
47 | 44, 45, 46 | cbvprodi 14486 |
. . . . . . 7
⊢
∏𝑘 ∈
⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
48 | | csbeq1 3502 |
. . . . . . . . 9
⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
49 | | snfi 7923 |
. . . . . . . . . 10
⊢ {𝑦} ∈ Fin |
50 | | xpfi 8116 |
. . . . . . . . . 10
⊢ (({𝑦} ∈ Fin ∧
⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin) |
51 | 49, 25, 50 | sylancr 694 |
. . . . . . . . 9
⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin) |
52 | | 2ndconst 7153 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐴 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵) |
53 | 17, 52 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (2nd ↾
({𝑦} ×
⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵) |
54 | | fvres 6117 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧)) |
55 | 54 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧)) |
56 | 45 | nfel1 2765 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
57 | 46 | eleq1d 2672 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
58 | 56, 57 | rspc 3276 |
. . . . . . . . . 10
⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
59 | 35, 58 | mpan9 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
60 | 48, 51, 53, 55, 59 | fprodf1o 14515 |
. . . . . . . 8
⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
61 | | elxp 5055 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))) |
62 | | nfv 1830 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑧 = 〈𝑚, 𝑘〉 |
63 | | nfv 1830 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗 𝑚 ∈ {𝑦} |
64 | 20 | nfcri 2745 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵 |
65 | 63, 64 | nfan 1816 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) |
66 | 62, 65 | nfan 1816 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
67 | 66 | nfex 2140 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
68 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑚∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) |
69 | | opeq1 4340 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → 〈𝑚, 𝑘〉 = 〈𝑗, 𝑘〉) |
70 | 69 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑗 → (𝑧 = 〈𝑚, 𝑘〉 ↔ 𝑧 = 〈𝑗, 𝑘〉)) |
71 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦})) |
72 | | velsn 4141 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦) |
73 | 71, 72 | syl6bb 275 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦)) |
74 | 73 | anbi1d 737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))) |
75 | 22 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
76 | 75 | pm5.32i 667 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
77 | 74, 76 | syl6bbr 277 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
78 | 70, 77 | anbi12d 743 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑗 → ((𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))) |
79 | 78 | exbidv 1837 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑗 → (∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))) |
80 | 67, 68, 79 | cbvex 2260 |
. . . . . . . . . . . 12
⊢
(∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
81 | 61, 80 | bitri 263 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
82 | | nfv 1830 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝜑 |
83 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(2nd ‘𝑧) |
84 | 83, 28 | nfcsb 3517 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
85 | 84 | nfeq2 2766 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝐷 =
⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
86 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘𝜑 |
87 | | nfcsb1v 3515 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
88 | 87 | nfeq2 2766 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘 𝐷 =
⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
89 | | fprod2d.1 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶) |
90 | 89 | ad2antlr 759 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶) |
91 | 31 | ad2antrl 760 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
92 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 〈𝑗, 𝑘〉 → (2nd ‘𝑧) = (2nd
‘〈𝑗, 𝑘〉)) |
93 | | vex 3176 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑗 ∈ V |
94 | | vex 3176 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑘 ∈ V |
95 | 93, 94 | op2nd 7068 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘〈𝑗, 𝑘〉) = 𝑘 |
96 | 92, 95 | syl6req 2661 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝑘 = (2nd ‘𝑧)) |
97 | 96 | ad2antlr 759 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧)) |
98 | | csbeq1a 3508 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (2nd ‘𝑧) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
100 | 90, 91, 99 | 3eqtrd 2648 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
101 | 100 | expl 646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
102 | 86, 88, 101 | exlimd 2074 |
. . . . . . . . . . . 12
⊢ (𝜑 → (∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
103 | 82, 85, 102 | exlimd 2074 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
104 | 81, 103 | syl5bi 231 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
105 | 104 | imp 444 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
106 | 105 | prodeq2dv 14492 |
. . . . . . . 8
⊢ (𝜑 → ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
107 | 60, 106 | eqtr4d 2647 |
. . . . . . 7
⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
108 | 47, 107 | syl5eq 2656 |
. . . . . 6
⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
109 | 43, 108 | eqtrd 2644 |
. . . . 5
⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
110 | 12, 109 | syl5eq 2656 |
. . . 4
⊢ (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
111 | 110 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
112 | 3, 111 | oveq12d 6567 |
. 2
⊢ ((𝜑 ∧ 𝜓) → (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶) = (∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
113 | | fprod2d.5 |
. . . . 5
⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥) |
114 | | disjsn 4192 |
. . . . 5
⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥) |
115 | 113, 114 | sylibr 223 |
. . . 4
⊢ (𝜑 → (𝑥 ∩ {𝑦}) = ∅) |
116 | | eqidd 2611 |
. . . 4
⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦})) |
117 | | fprod2d.2 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ Fin) |
118 | | ssfi 8065 |
. . . . 5
⊢ ((𝐴 ∈ Fin ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin) |
119 | 117, 13, 118 | syl2anc 691 |
. . . 4
⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin) |
120 | 13 | sselda 3568 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴) |
121 | 26 | anassrs 678 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
122 | 18, 121 | fprodcl 14521 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
123 | 120, 122 | syldan 486 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
124 | 115, 116,
119, 123 | fprodsplit 14535 |
. . 3
⊢ (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶)) |
125 | 124 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶)) |
126 | | eliun 4460 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵)) |
127 | | xp1st 7089 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗}) |
128 | | elsni 4142 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗) |
129 | 127, 128 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗) |
130 | 129 | eleq1d 2672 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) ∈ 𝑥 ↔ 𝑗 ∈ 𝑥)) |
131 | 130 | biimparc 503 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥) |
132 | 131 | rexlimiva 3010 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥) |
133 | 126, 132 | sylbi 206 |
. . . . . . . . 9
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥) |
134 | | xp1st 7089 |
. . . . . . . . 9
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦}) |
135 | 133, 134 | anim12i 588 |
. . . . . . . 8
⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦})) |
136 | | elin 3758 |
. . . . . . . 8
⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))) |
137 | | elin 3758 |
. . . . . . . 8
⊢
((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦})) |
138 | 135, 136,
137 | 3imtr4i 280 |
. . . . . . 7
⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → (1st ‘𝑧) ∈ (𝑥 ∩ {𝑦})) |
139 | 115 | eleq2d 2673 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈
∅)) |
140 | | noel 3878 |
. . . . . . . . 9
⊢ ¬
(1st ‘𝑧)
∈ ∅ |
141 | 140 | pm2.21i 115 |
. . . . . . . 8
⊢
((1st ‘𝑧) ∈ ∅ → 𝑧 ∈ ∅) |
142 | 139, 141 | syl6bi 242 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅)) |
143 | 138, 142 | syl5 33 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝑧 ∈ ∅)) |
144 | 143 | ssrdv 3574 |
. . . . 5
⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅) |
145 | | ss0 3926 |
. . . . 5
⊢
((∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅ → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅) |
146 | 144, 145 | syl 17 |
. . . 4
⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅) |
147 | | iunxun 4541 |
. . . . . 6
⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) |
148 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑚({𝑗} × 𝐵) |
149 | | nfcv 2751 |
. . . . . . . . . 10
⊢
Ⅎ𝑗{𝑚} |
150 | 149, 5 | nfxp 5066 |
. . . . . . . . 9
⊢
Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
151 | | sneq 4135 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚}) |
152 | 151, 8 | xpeq12d 5064 |
. . . . . . . . 9
⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)) |
153 | 148, 150,
152 | cbviun 4493 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪
𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
154 | | sneq 4135 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦}) |
155 | 154, 38 | xpeq12d 5064 |
. . . . . . . . 9
⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
156 | 15, 155 | iunxsn 4539 |
. . . . . . . 8
⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) |
157 | 153, 156 | eqtri 2632 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) |
158 | 157 | uneq2i 3726 |
. . . . . 6
⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
159 | 147, 158 | eqtri 2632 |
. . . . 5
⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
160 | 159 | a1i 11 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))) |
161 | | snfi 7923 |
. . . . . . 7
⊢ {𝑗} ∈ Fin |
162 | 120, 18 | syldan 486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin) |
163 | | xpfi 8116 |
. . . . . . 7
⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin) |
164 | 161, 162,
163 | sylancr 694 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin) |
165 | 164 | ralrimiva 2949 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
166 | | iunfi 8137 |
. . . . 5
⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
167 | 119, 165,
166 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
168 | | eliun 4460 |
. . . . . 6
⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵)) |
169 | | elxp 5055 |
. . . . . . . 8
⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) |
170 | | simprl 790 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = 〈𝑚, 𝑘〉) |
171 | | simprrl 800 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 ∈ {𝑗}) |
172 | | elsni 4142 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗) |
173 | 171, 172 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 = 𝑗) |
174 | 173 | opeq1d 4346 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 〈𝑚, 𝑘〉 = 〈𝑗, 𝑘〉) |
175 | 170, 174 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = 〈𝑗, 𝑘〉) |
176 | 175, 89 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 = 𝐶) |
177 | | simpll 786 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑) |
178 | 120 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴) |
179 | | simprrr 801 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵) |
180 | 177, 178,
179, 26 | syl12anc 1316 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ) |
181 | 176, 180 | eqeltrd 2688 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ) |
182 | 181 | ex 449 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ)) |
183 | 182 | exlimdvv 1849 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ)) |
184 | 169, 183 | syl5bi 231 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
185 | 184 | rexlimdva 3013 |
. . . . . 6
⊢ (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
186 | 168, 185 | syl5bi 231 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
187 | 186 | imp 444 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ) |
188 | 146, 160,
167, 187 | fprodsplit 14535 |
. . 3
⊢ (𝜑 → ∏𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
189 | 188 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ∏𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
190 | 112, 125,
189 | 3eqtr4d 2654 |
1
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷) |