| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝜓) |
| 2 | | fsum2d.7 |
. . . 4
⊢ (𝜓 ↔ Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) |
| 3 | 1, 2 | sylib 218 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) |
| 4 | | csbeq1a 3913 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵) |
| 5 | | csbeq1a 3913 |
. . . . . . . 8
⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶) |
| 6 | 5 | adantr 480 |
. . . . . . 7
⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶) |
| 7 | 4, 6 | sumeq12dv 15742 |
. . . . . 6
⊢ (𝑗 = 𝑚 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶) |
| 8 | | nfcv 2905 |
. . . . . 6
⊢
Ⅎ𝑚Σ𝑘 ∈ 𝐵 𝐶 |
| 9 | | nfcsb1v 3923 |
. . . . . . 7
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 |
| 10 | | nfcsb1v 3923 |
. . . . . . 7
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶 |
| 11 | 9, 10 | nfsum 15727 |
. . . . . 6
⊢
Ⅎ𝑗Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 |
| 12 | 7, 8, 11 | cbvsum 15731 |
. . . . 5
⊢
Σ𝑗 ∈
{𝑦}Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑚 ∈ {𝑦}Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 |
| 13 | | fsum2d.6 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴) |
| 14 | 13 | unssbd 4194 |
. . . . . . . 8
⊢ (𝜑 → {𝑦} ⊆ 𝐴) |
| 15 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
| 16 | 15 | snss 4785 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴) |
| 17 | 14, 16 | sylibr 234 |
. . . . . . 7
⊢ (𝜑 → 𝑦 ∈ 𝐴) |
| 18 | | fsum2d.3 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) |
| 19 | 18 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin) |
| 20 | | nfcsb1v 3923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 |
| 21 | 20 | nfel1 2922 |
. . . . . . . . . 10
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin |
| 22 | | csbeq1a 3913 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
| 23 | 22 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)) |
| 24 | 21, 23 | rspc 3610 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)) |
| 25 | 17, 19, 24 | sylc 65 |
. . . . . . . 8
⊢ (𝜑 → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) |
| 26 | | fsum2d.4 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ) |
| 27 | 26 | ralrimivva 3202 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
| 28 | | nfcsb1v 3923 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 |
| 29 | 28 | nfel1 2922 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
| 30 | 20, 29 | nfralw 3311 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
| 31 | | csbeq1a 3913 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
| 32 | 31 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
| 33 | 22, 32 | raleqbidv 3346 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
| 34 | 30, 33 | rspc 3610 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
| 35 | 17, 27, 34 | sylc 65 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
| 36 | 35 | r19.21bi 3251 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
| 37 | 25, 36 | fsumcl 15769 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
| 38 | | csbeq1 3902 |
. . . . . . . . 9
⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
| 39 | | csbeq1 3902 |
. . . . . . . . . 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 | sumeq12dv 15742 |
. . . . . . . 8
⊢ (𝑚 = 𝑦 → Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = Σ𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
| 42 | 41 | sumsn 15782 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐴 ∧ Σ𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) → Σ𝑚 ∈ {𝑦}Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = Σ𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
| 43 | 17, 37, 42 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → Σ𝑚 ∈ {𝑦}Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = Σ𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
| 44 | | csbeq1a 3913 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 45 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶 |
| 46 | | nfcsb1v 3923 |
. . . . . . . 8
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
| 47 | 44, 45, 46 | cbvsum 15731 |
. . . . . . 7
⊢
Σ𝑘 ∈
⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = Σ𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
| 48 | | csbeq1 3902 |
. . . . . . . . 9
⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 49 | | snfi 9083 |
. . . . . . . . . 10
⊢ {𝑦} ∈ Fin |
| 50 | | xpfi 9358 |
. . . . . . . . . 10
⊢ (({𝑦} ∈ Fin ∧
⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin) |
| 51 | 49, 25, 50 | sylancr 587 |
. . . . . . . . 9
⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin) |
| 52 | | 2ndconst 8126 |
. . . . . . . . . 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 6925 |
. . . . . . . . . 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 | 46 | nfel1 2922 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
| 57 | 44 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
| 58 | 56, 57 | rspc 3610 |
. . . . . . . . . 10
⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
| 59 | 35, 58 | mpan9 506 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
| 60 | 48, 51, 53, 55, 59 | fsumf1o 15759 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 61 | | elxp 5708 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))) |
| 62 | | nfv 1914 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑧 = 〈𝑚, 𝑘〉 |
| 63 | | nfv 1914 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗 𝑚 ∈ {𝑦} |
| 64 | 20 | nfcri 2897 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵 |
| 65 | 63, 64 | nfan 1899 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) |
| 66 | 62, 65 | nfan 1899 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
| 67 | 66 | nfex 2324 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
| 68 | | nfv 1914 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑚∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) |
| 69 | | opeq1 4873 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → 〈𝑚, 𝑘〉 = 〈𝑗, 𝑘〉) |
| 70 | 69 | eqeq2d 2748 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑗 → (𝑧 = 〈𝑚, 𝑘〉 ↔ 𝑧 = 〈𝑗, 𝑘〉)) |
| 71 | | velsn 4642 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ {𝑦} ↔ 𝑚 = 𝑦) |
| 72 | 71 | anbi1i 624 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
| 73 | | eqtr2 2761 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑚 = 𝑗 ∧ 𝑚 = 𝑦) → 𝑗 = 𝑦) |
| 74 | 73, 22 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑚 = 𝑗 ∧ 𝑚 = 𝑦) → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
| 75 | 74 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑚 = 𝑗 ∧ 𝑚 = 𝑦) → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
| 76 | 75 | pm5.32da 579 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑗 → ((𝑚 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))) |
| 77 | 72, 76 | bitr4id 290 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑚 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
| 78 | | equequ1 2024 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑗 → (𝑚 = 𝑦 ↔ 𝑗 = 𝑦)) |
| 79 | 78 | anbi1d 631 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → ((𝑚 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
| 80 | 77, 79 | bitrd 279 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
| 81 | 70, 80 | anbi12d 632 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑗 → ((𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))) |
| 82 | 81 | exbidv 1921 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑗 → (∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))) |
| 83 | 67, 68, 82 | cbvexv1 2344 |
. . . . . . . . . . . 12
⊢
(∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
| 84 | 61, 83 | bitri 275 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
| 85 | | nfv 1914 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝜑 |
| 86 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(2nd ‘𝑧) |
| 87 | 86, 28 | nfcsbw 3925 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
| 88 | 87 | nfeq2 2923 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝐷 =
⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
| 89 | | nfv 1914 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘𝜑 |
| 90 | | nfcsb1v 3923 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
| 91 | 90 | nfeq2 2923 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘 𝐷 =
⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
| 92 | | fsum2d.1 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶) |
| 93 | 92 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶) |
| 94 | 31 | ad2antrl 728 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
| 95 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 〈𝑗, 𝑘〉 → (2nd ‘𝑧) = (2nd
‘〈𝑗, 𝑘〉)) |
| 96 | | vex 3484 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑗 ∈ V |
| 97 | | vex 3484 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑘 ∈ V |
| 98 | 96, 97 | op2nd 8023 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘〈𝑗, 𝑘〉) = 𝑘 |
| 99 | 95, 98 | eqtr2di 2794 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝑘 = (2nd ‘𝑧)) |
| 100 | 99 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧)) |
| 101 | | csbeq1a 3913 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (2nd ‘𝑧) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 102 | 100, 101 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 103 | 93, 94, 102 | 3eqtrd 2781 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 104 | 103 | expl 457 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
| 105 | 89, 91, 104 | exlimd 2218 |
. . . . . . . . . . . 12
⊢ (𝜑 → (∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
| 106 | 85, 88, 105 | exlimd 2218 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
| 107 | 84, 106 | biimtrid 242 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
| 108 | 107 | imp 406 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 109 | 108 | sumeq2dv 15738 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
| 110 | 60, 109 | eqtr4d 2780 |
. . . . . . 7
⊢ (𝜑 → Σ𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
| 111 | 47, 110 | eqtrid 2789 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
| 112 | 43, 111 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → Σ𝑚 ∈ {𝑦}Σ𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
| 113 | 12, 112 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → Σ𝑗 ∈ {𝑦}Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
| 114 | 113 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → Σ𝑗 ∈ {𝑦}Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
| 115 | 3, 114 | oveq12d 7449 |
. 2
⊢ ((𝜑 ∧ 𝜓) → (Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 + Σ𝑗 ∈ {𝑦}Σ𝑘 ∈ 𝐵 𝐶) = (Σ𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
| 116 | | fsum2d.5 |
. . . . 5
⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥) |
| 117 | | disjsn 4711 |
. . . . 5
⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥) |
| 118 | 116, 117 | sylibr 234 |
. . . 4
⊢ (𝜑 → (𝑥 ∩ {𝑦}) = ∅) |
| 119 | | eqidd 2738 |
. . . 4
⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦})) |
| 120 | | fsum2d.2 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ Fin) |
| 121 | 120, 13 | ssfid 9301 |
. . . 4
⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin) |
| 122 | 13 | sselda 3983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴) |
| 123 | 26 | anassrs 467 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
| 124 | 18, 123 | fsumcl 15769 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Σ𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
| 125 | 122, 124 | syldan 591 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → Σ𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
| 126 | 118, 119,
121, 125 | fsumsplit 15777 |
. . 3
⊢ (𝜑 → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘 ∈ 𝐵 𝐶 = (Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 + Σ𝑗 ∈ {𝑦}Σ𝑘 ∈ 𝐵 𝐶)) |
| 127 | 126 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘 ∈ 𝐵 𝐶 = (Σ𝑗 ∈ 𝑥 Σ𝑘 ∈ 𝐵 𝐶 + Σ𝑗 ∈ {𝑦}Σ𝑘 ∈ 𝐵 𝐶)) |
| 128 | | eliun 4995 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵)) |
| 129 | | xp1st 8046 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗}) |
| 130 | | elsni 4643 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗) |
| 131 | 129, 130 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗) |
| 132 | 131 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) = 𝑗) |
| 133 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑗 ∈ 𝑥) |
| 134 | 132, 133 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥) |
| 135 | 134 | rexlimiva 3147 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥) |
| 136 | 128, 135 | sylbi 217 |
. . . . . . . . 9
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥) |
| 137 | | xp1st 8046 |
. . . . . . . . 9
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦}) |
| 138 | 136, 137 | anim12i 613 |
. . . . . . . 8
⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦})) |
| 139 | | elin 3967 |
. . . . . . . 8
⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))) |
| 140 | | elin 3967 |
. . . . . . . 8
⊢
((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦})) |
| 141 | 138, 139,
140 | 3imtr4i 292 |
. . . . . . 7
⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → (1st ‘𝑧) ∈ (𝑥 ∩ {𝑦})) |
| 142 | 118 | eleq2d 2827 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈
∅)) |
| 143 | | noel 4338 |
. . . . . . . . 9
⊢ ¬
(1st ‘𝑧)
∈ ∅ |
| 144 | 143 | pm2.21i 119 |
. . . . . . . 8
⊢
((1st ‘𝑧) ∈ ∅ → 𝑧 ∈ ∅) |
| 145 | 142, 144 | biimtrdi 253 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅)) |
| 146 | 141, 145 | syl5 34 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝑧 ∈ ∅)) |
| 147 | 146 | ssrdv 3989 |
. . . . 5
⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅) |
| 148 | | ss0 4402 |
. . . . 5
⊢
((∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅ → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅) |
| 149 | 147, 148 | syl 17 |
. . . 4
⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅) |
| 150 | | iunxun 5094 |
. . . . . 6
⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) |
| 151 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑚({𝑗} × 𝐵) |
| 152 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑗{𝑚} |
| 153 | 152, 9 | nfxp 5718 |
. . . . . . . . 9
⊢
Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
| 154 | | sneq 4636 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚}) |
| 155 | 154, 4 | xpeq12d 5716 |
. . . . . . . . 9
⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)) |
| 156 | 151, 153,
155 | cbviun 5036 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪
𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
| 157 | | sneq 4636 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦}) |
| 158 | 157, 38 | xpeq12d 5716 |
. . . . . . . . 9
⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
| 159 | 15, 158 | iunxsn 5091 |
. . . . . . . 8
⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) |
| 160 | 156, 159 | eqtri 2765 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) |
| 161 | 160 | uneq2i 4165 |
. . . . . 6
⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
| 162 | 150, 161 | eqtri 2765 |
. . . . 5
⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
| 163 | 162 | a1i 11 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))) |
| 164 | | snfi 9083 |
. . . . . . 7
⊢ {𝑗} ∈ Fin |
| 165 | 122, 18 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin) |
| 166 | | xpfi 9358 |
. . . . . . 7
⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin) |
| 167 | 164, 165,
166 | sylancr 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin) |
| 168 | 167 | ralrimiva 3146 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
| 169 | | iunfi 9383 |
. . . . 5
⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
| 170 | 121, 168,
169 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
| 171 | | eliun 4995 |
. . . . . 6
⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵)) |
| 172 | | elxp 5708 |
. . . . . . . 8
⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) |
| 173 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = 〈𝑚, 𝑘〉) |
| 174 | | simprrl 781 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 ∈ {𝑗}) |
| 175 | | elsni 4643 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗) |
| 176 | 174, 175 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 = 𝑗) |
| 177 | 176 | opeq1d 4879 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 〈𝑚, 𝑘〉 = 〈𝑗, 𝑘〉) |
| 178 | 173, 177 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = 〈𝑗, 𝑘〉) |
| 179 | 178, 92 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 = 𝐶) |
| 180 | | simpll 767 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑) |
| 181 | 122 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴) |
| 182 | | simprrr 782 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵) |
| 183 | 180, 181,
182, 26 | syl12anc 837 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ) |
| 184 | 179, 183 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ) |
| 185 | 184 | ex 412 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ)) |
| 186 | 185 | exlimdvv 1934 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ)) |
| 187 | 172, 186 | biimtrid 242 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
| 188 | 187 | rexlimdva 3155 |
. . . . . 6
⊢ (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
| 189 | 171, 188 | biimtrid 242 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
| 190 | 189 | imp 406 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ) |
| 191 | 149, 163,
170, 190 | fsumsplit 15777 |
. . 3
⊢ (𝜑 → Σ𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
| 192 | 191 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝜓) → Σ𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (Σ𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 + Σ𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
| 193 | 115, 127,
192 | 3eqtr4d 2787 |
1
⊢ ((𝜑 ∧ 𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷) |