| Step | Hyp | Ref
| Expression |
| 1 | | sumeq1 15725 |
. . 3
⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ ∅ 𝐵) |
| 2 | 1 | breq2d 5155 |
. 2
⊢ (𝑥 = ∅ → (2 ∥
Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ ∅ 𝐵)) |
| 3 | | sumeq1 15725 |
. . 3
⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝑦 𝐵) |
| 4 | 3 | breq2d 5155 |
. 2
⊢ (𝑥 = 𝑦 → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵)) |
| 5 | | sumeq1 15725 |
. . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) |
| 6 | 5 | breq2d 5155 |
. 2
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) |
| 7 | | sumeq1 15725 |
. . 3
⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
| 8 | 7 | breq2d 5155 |
. 2
⊢ (𝑥 = 𝐴 → (2 ∥ Σ𝑘 ∈ 𝑥 𝐵 ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵)) |
| 9 | | z0even 16404 |
. . . 4
⊢ 2 ∥
0 |
| 10 | | sum0 15757 |
. . . 4
⊢
Σ𝑘 ∈
∅ 𝐵 =
0 |
| 11 | 9, 10 | breqtrri 5170 |
. . 3
⊢ 2 ∥
Σ𝑘 ∈ ∅
𝐵 |
| 12 | 11 | a1i 11 |
. 2
⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ ∅ 𝐵) |
| 13 | | 2z 12649 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
| 14 | 13 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 2 ∈ ℤ) |
| 15 | | sumeven.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ Fin) |
| 16 | | ssfi 9213 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin) |
| 17 | 16 | expcom 413 |
. . . . . . . . . 10
⊢ (𝑦 ⊆ 𝐴 → (𝐴 ∈ Fin → 𝑦 ∈ Fin)) |
| 18 | 17 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝐴 ∈ Fin → 𝑦 ∈ Fin)) |
| 19 | 15, 18 | mpan9 506 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin) |
| 20 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑) |
| 21 | | ssel 3977 |
. . . . . . . . . . . 12
⊢ (𝑦 ⊆ 𝐴 → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴)) |
| 22 | 21 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴)) |
| 23 | 22 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ 𝑦 → 𝑘 ∈ 𝐴)) |
| 24 | 23 | imp 406 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴) |
| 25 | | sumeven.b |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
| 26 | 20, 24, 25 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℤ) |
| 27 | 19, 26 | fsumzcl 15771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ) |
| 28 | | eldifi 4131 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → 𝑧 ∈ 𝐴) |
| 29 | 28 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑧 ∈ 𝐴) |
| 30 | 29 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴) |
| 31 | 25 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
| 32 | 31 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
| 33 | | rspcsbela 4438 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
| 34 | 30, 32, 33 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) |
| 35 | 14, 27, 34 | 3jca 1129 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∈ ℤ ∧
Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)) |
| 36 | 35 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (2 ∈ ℤ ∧
Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ)) |
| 37 | | sumeven.e |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 2 ∥ 𝐵) |
| 38 | 37 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 2 ∥ 𝐵) |
| 39 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘2 |
| 40 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘
∥ |
| 41 | | nfcsb1v 3923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 |
| 42 | 39, 40, 41 | nfbr 5190 |
. . . . . . . . . 10
⊢
Ⅎ𝑘2 ∥
⦋𝑧 / 𝑘⦌𝐵 |
| 43 | | csbeq1a 3913 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵) |
| 44 | 43 | breq2d 5155 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑧 → (2 ∥ 𝐵 ↔ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)) |
| 45 | 42, 44 | rspc 3610 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 2 ∥ 𝐵 → 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)) |
| 46 | 28, 38, 45 | syl2imc 41 |
. . . . . . . 8
⊢ (𝜑 → (𝑧 ∈ (𝐴 ∖ 𝑦) → 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)) |
| 47 | 46 | a1d 25 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ⊆ 𝐴 → (𝑧 ∈ (𝐴 ∖ 𝑦) → 2 ∥ ⦋𝑧 / 𝑘⦌𝐵))) |
| 48 | 47 | imp32 418 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 2 ∥ ⦋𝑧 / 𝑘⦌𝐵) |
| 49 | 48 | anim1ci 616 |
. . . . 5
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 ∧ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵)) |
| 50 | | dvds2add 16327 |
. . . . 5
⊢ ((2
∈ ℤ ∧ Σ𝑘 ∈ 𝑦 𝐵 ∈ ℤ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℤ) → ((2 ∥
Σ𝑘 ∈ 𝑦 𝐵 ∧ 2 ∥ ⦋𝑧 / 𝑘⦌𝐵) → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))) |
| 51 | 36, 49, 50 | sylc 65 |
. . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → 2 ∥ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
| 52 | | vex 3484 |
. . . . . . 7
⊢ 𝑧 ∈ V |
| 53 | 52 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ V) |
| 54 | | eldif 3961 |
. . . . . . . . 9
⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦)) |
| 55 | | df-nel 3047 |
. . . . . . . . . 10
⊢ (𝑧 ∉ 𝑦 ↔ ¬ 𝑧 ∈ 𝑦) |
| 56 | 55 | biimpri 228 |
. . . . . . . . 9
⊢ (¬
𝑧 ∈ 𝑦 → 𝑧 ∉ 𝑦) |
| 57 | 54, 56 | simplbiim 504 |
. . . . . . . 8
⊢ (𝑧 ∈ (𝐴 ∖ 𝑦) → 𝑧 ∉ 𝑦) |
| 58 | 57 | adantl 481 |
. . . . . . 7
⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑧 ∉ 𝑦) |
| 59 | 58 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∉ 𝑦) |
| 60 | | simpll 767 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑) |
| 61 | | elun 4153 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧})) |
| 62 | 22 | com12 32 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ 𝑦 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴)) |
| 63 | | elsni 4643 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ {𝑧} → 𝑘 = 𝑧) |
| 64 | | eleq1w 2824 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
| 65 | 29, 64 | imbitrrid 246 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑧 → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴)) |
| 66 | 63, 65 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ {𝑧} → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴)) |
| 67 | 62, 66 | jaoi 858 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → 𝑘 ∈ 𝐴)) |
| 68 | 67 | com12 32 |
. . . . . . . . . . 11
⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → ((𝑘 ∈ 𝑦 ∨ 𝑘 ∈ {𝑧}) → 𝑘 ∈ 𝐴)) |
| 69 | 61, 68 | biimtrid 242 |
. . . . . . . . . 10
⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴)) |
| 70 | 69 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) → 𝑘 ∈ 𝐴)) |
| 71 | 70 | imp 406 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴) |
| 72 | 60, 71, 25 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℤ) |
| 73 | 72 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ ℤ) |
| 74 | | fsumsplitsnun 15791 |
. . . . . 6
⊢ ((𝑦 ∈ Fin ∧ (𝑧 ∈ V ∧ 𝑧 ∉ 𝑦) ∧ ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
| 75 | 19, 53, 59, 73, 74 | syl121anc 1377 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
| 76 | 75 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) |
| 77 | 51, 76 | breqtrrd 5171 |
. . 3
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 2 ∥ Σ𝑘 ∈ 𝑦 𝐵) → 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) |
| 78 | 77 | ex 412 |
. 2
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (2 ∥ Σ𝑘 ∈ 𝑦 𝐵 → 2 ∥ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) |
| 79 | 2, 4, 6, 8, 12, 78, 15 | findcard2d 9206 |
1
⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) |