| Step | Hyp | Ref
| Expression |
| 1 | | esumeq1 34035 |
. . 3
⊢ (𝑎 = ∅ →
Σ*𝑗 ∈
𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ ∅Σ*𝑘 ∈ 𝐵𝐶) |
| 2 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑧 𝑎 = ∅ |
| 3 | | iuneq1 5008 |
. . . 4
⊢ (𝑎 = ∅ → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪
𝑗 ∈ ∅ ({𝑗} × 𝐵)) |
| 4 | 2, 3 | esumeq1d 34036 |
. . 3
⊢ (𝑎 = ∅ →
Σ*𝑧 ∈
∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪
𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹) |
| 5 | 1, 4 | eqeq12d 2753 |
. 2
⊢ (𝑎 = ∅ →
(Σ*𝑗
∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈
∅Σ*𝑘
∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹)) |
| 6 | | esumeq1 34035 |
. . 3
⊢ (𝑎 = 𝑏 → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶) |
| 7 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑧 𝑎 = 𝑏 |
| 8 | | iuneq1 5008 |
. . . 4
⊢ (𝑎 = 𝑏 → ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) |
| 9 | 7, 8 | esumeq1d 34036 |
. . 3
⊢ (𝑎 = 𝑏 → Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) |
| 10 | 6, 9 | eqeq12d 2753 |
. 2
⊢ (𝑎 = 𝑏 → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹)) |
| 11 | | esumeq1 34035 |
. . 3
⊢ (𝑎 = (𝑏 ∪ {𝑙}) → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶) |
| 12 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑧 𝑎 = (𝑏 ∪ {𝑙}) |
| 13 | | iuneq1 5008 |
. . . 4
⊢ (𝑎 = (𝑏 ∪ {𝑙}) → ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪
𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)) |
| 14 | 12, 13 | esumeq1d 34036 |
. . 3
⊢ (𝑎 = (𝑏 ∪ {𝑙}) → Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪
𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹) |
| 15 | 11, 14 | eqeq12d 2753 |
. 2
⊢ (𝑎 = (𝑏 ∪ {𝑙}) → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹)) |
| 16 | | esumeq1 34035 |
. . 3
⊢ (𝑎 = 𝐴 → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶) |
| 17 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑧 𝑎 = 𝐴 |
| 18 | | iuneq1 5008 |
. . . 4
⊢ (𝑎 = 𝐴 → ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 19 | 17, 18 | esumeq1d 34036 |
. . 3
⊢ (𝑎 = 𝐴 → Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) |
| 20 | 16, 19 | eqeq12d 2753 |
. 2
⊢ (𝑎 = 𝐴 → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)) |
| 21 | | esumnul 34049 |
. . . 4
⊢
Σ*𝑧
∈ ∅𝐹 =
0 |
| 22 | | 0iun 5063 |
. . . . 5
⊢ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅ |
| 23 | | esumeq1 34035 |
. . . . 5
⊢ (∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅ → Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∅𝐹) |
| 24 | 22, 23 | ax-mp 5 |
. . . 4
⊢
Σ*𝑧
∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∅𝐹 |
| 25 | | esumnul 34049 |
. . . 4
⊢
Σ*𝑗
∈ ∅Σ*𝑘 ∈ 𝐵𝐶 = 0 |
| 26 | 21, 24, 25 | 3eqtr4ri 2776 |
. . 3
⊢
Σ*𝑗
∈ ∅Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹 |
| 27 | 26 | a1i 11 |
. 2
⊢ (𝜑 → Σ*𝑗 ∈
∅Σ*𝑘
∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹) |
| 28 | | simpr 484 |
. . . . 5
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) |
| 29 | | nfcsb1v 3923 |
. . . . . . . . 9
⊢
Ⅎ𝑗⦋𝑙 / 𝑗⦌𝐵 |
| 30 | | nfcsb1v 3923 |
. . . . . . . . 9
⊢
Ⅎ𝑗⦋𝑙 / 𝑗⦌𝐶 |
| 31 | 29, 30 | nfesum2 34042 |
. . . . . . . 8
⊢
Ⅎ𝑗Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 |
| 32 | | csbeq1a 3913 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑗⦌𝐵) |
| 33 | | csbeq1a 3913 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑗⦌𝐶) |
| 34 | 32, 33 | esumeq12d 34034 |
. . . . . . . . 9
⊢ (𝑗 = 𝑙 → Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶) |
| 35 | 34 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 = 𝑙) → Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶) |
| 36 | | simprr 773 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑙 ∈ (𝐴 ∖ 𝑏)) |
| 37 | 36 | eldifad 3963 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑙 ∈ 𝐴) |
| 38 | | esum2d.3 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
| 39 | 38 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
| 40 | 39 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) |
| 41 | | rspcsbela 4438 |
. . . . . . . . . 10
⊢ ((𝑙 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) → ⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊) |
| 42 | 37, 40, 41 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊) |
| 43 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝜑) |
| 44 | 37 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝑙 ∈ 𝐴) |
| 45 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) |
| 46 | | esum2d.4 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞)) |
| 47 | 46 | ex 412 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))) |
| 48 | 47 | sbcimdv 3859 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ([𝑙 / 𝑗](𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → [𝑙 / 𝑗]𝐶 ∈ (0[,]+∞))) |
| 49 | | sbcan 3838 |
. . . . . . . . . . . . . 14
⊢
([𝑙 / 𝑗](𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ ([𝑙 / 𝑗]𝑗 ∈ 𝐴 ∧ [𝑙 / 𝑗]𝑘 ∈ 𝐵)) |
| 50 | | sbcel1v 3856 |
. . . . . . . . . . . . . . 15
⊢
([𝑙 / 𝑗]𝑗 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴) |
| 51 | | sbcel2 4418 |
. . . . . . . . . . . . . . 15
⊢
([𝑙 / 𝑗]𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) |
| 52 | 50, 51 | anbi12i 628 |
. . . . . . . . . . . . . 14
⊢
(([𝑙 / 𝑗]𝑗 ∈ 𝐴 ∧ [𝑙 / 𝑗]𝑘 ∈ 𝐵) ↔ (𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵)) |
| 53 | 49, 52 | bitri 275 |
. . . . . . . . . . . . 13
⊢
([𝑙 / 𝑗](𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵)) |
| 54 | | vex 3484 |
. . . . . . . . . . . . . 14
⊢ 𝑙 ∈ V |
| 55 | | sbcel1g 4416 |
. . . . . . . . . . . . . 14
⊢ (𝑙 ∈ V → ([𝑙 / 𝑗]𝐶 ∈ (0[,]+∞) ↔
⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))) |
| 56 | 54, 55 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
([𝑙 / 𝑗]𝐶 ∈ (0[,]+∞) ↔
⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) |
| 57 | 48, 53, 56 | 3imtr3g 295 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))) |
| 58 | 57 | imp 406 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵)) → ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) |
| 59 | 43, 44, 45, 58 | syl12anc 837 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) |
| 60 | 59 | ralrimiva 3146 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∀𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) |
| 61 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑙 / 𝑗⦌𝐵 |
| 62 | 61 | esumcl 34031 |
. . . . . . . . 9
⊢
((⦋𝑙 /
𝑗⦌𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) →
Σ*𝑘 ∈
⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) |
| 63 | 42, 60, 62 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) |
| 64 | 31, 35, 36, 63 | esumsnf 34065 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶) |
| 65 | | esum2d.0 |
. . . . . . . . 9
⊢
Ⅎ𝑘𝐹 |
| 66 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑘(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) |
| 67 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗 𝑧 = 〈𝑙, 𝑘〉 |
| 68 | 30 | nfeq2 2923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗 𝐹 = ⦋𝑙 / 𝑗⦌𝐶 |
| 69 | 67, 68 | nfim 1896 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝑧 = 〈𝑙, 𝑘〉 → 𝐹 = ⦋𝑙 / 𝑗⦌𝐶) |
| 70 | | opeq1 4873 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑙 → 〈𝑗, 𝑘〉 = 〈𝑙, 𝑘〉) |
| 71 | 70 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → (𝑧 = 〈𝑗, 𝑘〉 ↔ 𝑧 = 〈𝑙, 𝑘〉)) |
| 72 | 33 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → (𝐹 = 𝐶 ↔ 𝐹 = ⦋𝑙 / 𝑗⦌𝐶)) |
| 73 | 71, 72 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑙 → ((𝑧 = 〈𝑗, 𝑘〉 → 𝐹 = 𝐶) ↔ (𝑧 = 〈𝑙, 𝑘〉 → 𝐹 = ⦋𝑙 / 𝑗⦌𝐶))) |
| 74 | | esum2d.1 |
. . . . . . . . . 10
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐹 = 𝐶) |
| 75 | 69, 73, 74 | chvarfv 2240 |
. . . . . . . . 9
⊢ (𝑧 = 〈𝑙, 𝑘〉 → 𝐹 = ⦋𝑙 / 𝑗⦌𝐶) |
| 76 | | vsnid 4663 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑗 ∈ {𝑗} |
| 77 | 76 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑗 ∈ {𝑗}) |
| 78 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵) |
| 79 | 77, 78 | opelxpd 5724 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 〈𝑗, 𝑘〉 ∈ ({𝑗} × 𝐵)) |
| 80 | | xp2nd 8047 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵) |
| 81 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗}) |
| 82 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(1st ‘𝑧) ∈ V |
| 83 | 82 | elsn 4641 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑧) ∈ {𝑗} ↔ (1st ‘𝑧) = 𝑗) |
| 84 | 81, 83 | sylib 218 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗) |
| 85 | | eqop 8056 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (𝑧 = 〈𝑗, 𝑘〉 ↔ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) = 𝑘))) |
| 86 | 84, 85 | mpbirand 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (𝑧 = 〈𝑗, 𝑘〉 ↔ (2nd ‘𝑧) = 𝑘)) |
| 87 | | eqcom 2744 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑧) = 𝑘 ↔ 𝑘 = (2nd ‘𝑧)) |
| 88 | 86, 87 | bitrdi 287 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (𝑧 = 〈𝑗, 𝑘〉 ↔ 𝑘 = (2nd ‘𝑧))) |
| 89 | 88 | ad2antlr 727 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) → (𝑧 = 〈𝑗, 𝑘〉 ↔ 𝑘 = (2nd ‘𝑧))) |
| 90 | 89 | ralrimiva 3146 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑧 = 〈𝑗, 𝑘〉 ↔ 𝑘 = (2nd ‘𝑧))) |
| 91 | | reu6i 3734 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 (𝑧 = 〈𝑗, 𝑘〉 ↔ 𝑘 = (2nd ‘𝑧))) → ∃!𝑘 ∈ 𝐵 𝑧 = 〈𝑗, 𝑘〉) |
| 92 | 80, 90, 91 | syl2an2 686 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃!𝑘 ∈ 𝐵 𝑧 = 〈𝑗, 𝑘〉) |
| 93 | 79, 92 | f1mptrn 32645 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Fun ◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉)) |
| 94 | 93 | ex 412 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑗 ∈ 𝐴 → Fun ◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉))) |
| 95 | 94 | sbcimdv 3859 |
. . . . . . . . . . . 12
⊢ (𝜑 → ([𝑙 / 𝑗]𝑗 ∈ 𝐴 → [𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉))) |
| 96 | | sbcfung 6590 |
. . . . . . . . . . . . . 14
⊢ (𝑙 ∈ V → ([𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) ↔ Fun ⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉))) |
| 97 | | csbcnv 5894 |
. . . . . . . . . . . . . . . 16
⊢ ◡⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) = ⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) |
| 98 | | csbmpt12 5562 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) = (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⦋𝑙 / 𝑗⦌〈𝑗, 𝑘〉)) |
| 99 | | csbopg 4891 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌〈𝑗, 𝑘〉 = 〈⦋𝑙 / 𝑗⦌𝑗, ⦋𝑙 / 𝑗⦌𝑘〉) |
| 100 | | csbvarg 4434 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌𝑗 = 𝑙) |
| 101 | | csbconstg 3918 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌𝑘 = 𝑘) |
| 102 | 100, 101 | opeq12d 4881 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑙 ∈ V →
〈⦋𝑙 /
𝑗⦌𝑗, ⦋𝑙 / 𝑗⦌𝑘〉 = 〈𝑙, 𝑘〉) |
| 103 | 99, 102 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌〈𝑗, 𝑘〉 = 〈𝑙, 𝑘〉) |
| 104 | 103 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑙 ∈ V → (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⦋𝑙 / 𝑗⦌〈𝑗, 𝑘〉) = (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 105 | 98, 104 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) = (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 106 | 105 | cnveqd 5886 |
. . . . . . . . . . . . . . . 16
⊢ (𝑙 ∈ V → ◡⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) = ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 107 | 97, 106 | eqtr3id 2791 |
. . . . . . . . . . . . . . 15
⊢ (𝑙 ∈ V →
⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) = ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 108 | 107 | funeqd 6588 |
. . . . . . . . . . . . . 14
⊢ (𝑙 ∈ V → (Fun
⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) ↔ Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉))) |
| 109 | 96, 108 | bitrd 279 |
. . . . . . . . . . . . 13
⊢ (𝑙 ∈ V → ([𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) ↔ Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉))) |
| 110 | 54, 109 | ax-mp 5 |
. . . . . . . . . . . 12
⊢
([𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ 〈𝑗, 𝑘〉) ↔ Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 111 | 95, 50, 110 | 3imtr3g 295 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑙 ∈ 𝐴 → Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉))) |
| 112 | 111 | imp 406 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 113 | 37, 112 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ 〈𝑙, 𝑘〉)) |
| 114 | | vsnid 4663 |
. . . . . . . . . . 11
⊢ 𝑙 ∈ {𝑙} |
| 115 | 114 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝑙 ∈ {𝑙}) |
| 116 | 115, 45 | opelxpd 5724 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 〈𝑙, 𝑘〉 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 117 | 65, 66, 61, 75, 42, 113, 59, 116 | esumc 34052 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 = Σ*𝑧 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉}𝐹) |
| 118 | | nfab1 2907 |
. . . . . . . . . 10
⊢
Ⅎ𝑡{𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉} |
| 119 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑡({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) |
| 120 | | opeq1 4873 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑙 → 〈𝑖, 𝑘〉 = 〈𝑙, 𝑘〉) |
| 121 | 120 | eqeq2d 2748 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑙 → (𝑡 = 〈𝑖, 𝑘〉 ↔ 𝑡 = 〈𝑙, 𝑘〉)) |
| 122 | 121 | rexbidv 3179 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑙 → (∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑖, 𝑘〉 ↔ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉)) |
| 123 | 54, 122 | rexsn 4682 |
. . . . . . . . . . 11
⊢
(∃𝑖 ∈
{𝑙}∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑖, 𝑘〉 ↔ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉) |
| 124 | | elxp2 5709 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ↔ ∃𝑖 ∈ {𝑙}∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑖, 𝑘〉) |
| 125 | | abid 2718 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉} ↔ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉) |
| 126 | 123, 124,
125 | 3bitr4ri 304 |
. . . . . . . . . 10
⊢ (𝑡 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉} ↔ 𝑡 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 127 | 118, 119,
126 | eqri 4004 |
. . . . . . . . 9
⊢ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉} = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) |
| 128 | | esumeq1 34035 |
. . . . . . . . 9
⊢ ({𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉} = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) → Σ*𝑧 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉}𝐹 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹) |
| 129 | 127, 128 | ax-mp 5 |
. . . . . . . 8
⊢
Σ*𝑧
∈ {𝑡 ∣
∃𝑘 ∈
⦋ 𝑙 / 𝑗⦌𝐵𝑡 = 〈𝑙, 𝑘〉}𝐹 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹 |
| 130 | 117, 129 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹) |
| 131 | 64, 130 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹) |
| 132 | 131 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹) |
| 133 | 28, 132 | oveq12d 7449 |
. . . 4
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 +𝑒
Σ*𝑗 ∈
{𝑙}Σ*𝑘 ∈ 𝐵𝐶) = (Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒
Σ*𝑧 ∈
({𝑙} ×
⦋𝑙 / 𝑗⦌𝐵)𝐹)) |
| 134 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑗(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) |
| 135 | | nfcv 2905 |
. . . . . 6
⊢
Ⅎ𝑗𝑏 |
| 136 | | nfcv 2905 |
. . . . . 6
⊢
Ⅎ𝑗{𝑙} |
| 137 | | vex 3484 |
. . . . . . 7
⊢ 𝑏 ∈ V |
| 138 | 137 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V) |
| 139 | | vsnex 5434 |
. . . . . . 7
⊢ {𝑙} ∈ V |
| 140 | 139 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → {𝑙} ∈ V) |
| 141 | 36 | eldifbd 3964 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑙 ∈ 𝑏) |
| 142 | | disjsn 4711 |
. . . . . . 7
⊢ ((𝑏 ∩ {𝑙}) = ∅ ↔ ¬ 𝑙 ∈ 𝑏) |
| 143 | 141, 142 | sylibr 234 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑙}) = ∅) |
| 144 | | simpll 767 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝜑) |
| 145 | | simprl 771 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴) |
| 146 | 145 | sselda 3983 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝐴) |
| 147 | 46 | anassrs 467 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)) |
| 148 | 147 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) |
| 149 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑘𝐵 |
| 150 | 149 | esumcl 34031 |
. . . . . . . 8
⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) →
Σ*𝑘 ∈
𝐵𝐶 ∈ (0[,]+∞)) |
| 151 | 38, 148, 150 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞)) |
| 152 | 144, 146,
151 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞)) |
| 153 | | simpll 767 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ {𝑙}) → 𝜑) |
| 154 | 37 | snssd 4809 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → {𝑙} ⊆ 𝐴) |
| 155 | 154 | sselda 3983 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ {𝑙}) → 𝑗 ∈ 𝐴) |
| 156 | 153, 155,
151 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ {𝑙}) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞)) |
| 157 | 134, 135,
136, 138, 140, 143, 152, 156 | esumsplit 34054 |
. . . . 5
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 +𝑒
Σ*𝑗 ∈
{𝑙}Σ*𝑘 ∈ 𝐵𝐶)) |
| 158 | 157 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 +𝑒
Σ*𝑗 ∈
{𝑙}Σ*𝑘 ∈ 𝐵𝐶)) |
| 159 | | iunxun 5094 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑙} ({𝑗} × 𝐵)) |
| 160 | 136, 29 | nfxp 5718 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) |
| 161 | | sneq 4636 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑙 → {𝑗} = {𝑙}) |
| 162 | 161, 32 | xpeq12d 5716 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → ({𝑗} × 𝐵) = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 163 | 160, 162 | iunxsngf 5092 |
. . . . . . . . . 10
⊢ (𝑙 ∈ V → ∪ 𝑗 ∈ {𝑙} ({𝑗} × 𝐵) = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 164 | 54, 163 | ax-mp 5 |
. . . . . . . . 9
⊢ ∪ 𝑗 ∈ {𝑙} ({𝑗} × 𝐵) = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) |
| 165 | 164 | uneq2i 4165 |
. . . . . . . 8
⊢ (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑙} ({𝑗} × 𝐵)) = (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 166 | 159, 165 | eqtri 2765 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 167 | | esumeq1 34035 |
. . . . . . 7
⊢ (∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))𝐹) |
| 168 | 166, 167 | ax-mp 5 |
. . . . . 6
⊢
Σ*𝑧
∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))𝐹 |
| 169 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑧(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) |
| 170 | | nfcv 2905 |
. . . . . . 7
⊢
Ⅎ𝑧∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) |
| 171 | | nfcv 2905 |
. . . . . . 7
⊢
Ⅎ𝑧({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) |
| 172 | | vsnex 5434 |
. . . . . . . . . 10
⊢ {𝑗} ∈ V |
| 173 | 146, 39 | syldan 591 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝐵 ∈ 𝑊) |
| 174 | | xpexg 7770 |
. . . . . . . . . 10
⊢ (({𝑗} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝑗} × 𝐵) ∈ V) |
| 175 | 172, 173,
174 | sylancr 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → ({𝑗} × 𝐵) ∈ V) |
| 176 | 175 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∀𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V) |
| 177 | | iunexg 7988 |
. . . . . . . 8
⊢ ((𝑏 ∈ V ∧ ∀𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V) |
| 178 | 137, 176,
177 | sylancr 587 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V) |
| 179 | | xpexg 7770 |
. . . . . . . 8
⊢ (({𝑙} ∈ V ∧
⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊) → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ∈ V) |
| 180 | 139, 42, 179 | sylancr 587 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ∈ V) |
| 181 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝑏) |
| 182 | 141 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → ¬ 𝑙 ∈ 𝑏) |
| 183 | | nelne2 3040 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝑏 ∧ ¬ 𝑙 ∈ 𝑏) → 𝑗 ≠ 𝑙) |
| 184 | 181, 182,
183 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝑗 ≠ 𝑙) |
| 185 | | disjsn2 4712 |
. . . . . . . . . 10
⊢ (𝑗 ≠ 𝑙 → ({𝑗} ∩ {𝑙}) = ∅) |
| 186 | | xpdisj1 6181 |
. . . . . . . . . 10
⊢ (({𝑗} ∩ {𝑙}) = ∅ → (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∅) |
| 187 | 184, 185,
186 | 3syl 18 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∅) |
| 188 | 187 | iuneq2dv 5016 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∪ 𝑗 ∈ 𝑏 (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∪
𝑗 ∈ 𝑏 ∅) |
| 189 | 160 | iunin1f 32570 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ 𝑏 (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = (∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) |
| 190 | | iun0 5062 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ 𝑏 ∅ = ∅ |
| 191 | 188, 189,
190 | 3eqtr3g 2800 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∅) |
| 192 | | simpll 767 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) → 𝜑) |
| 193 | | iunss1 5006 |
. . . . . . . . . 10
⊢ (𝑏 ⊆ 𝐴 → ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 194 | 145, 193 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 195 | 194 | sselda 3983 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 196 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑗𝜑 |
| 197 | | nfiu1 5027 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
| 198 | 197 | nfcri 2897 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
| 199 | 196, 198 | nfan 1899 |
. . . . . . . . 9
⊢
Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 200 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) |
| 201 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘(0[,]+∞) |
| 202 | 65, 201 | nfel 2920 |
. . . . . . . . . 10
⊢
Ⅎ𝑘 𝐹 ∈
(0[,]+∞) |
| 203 | 74 | adantl 481 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = 〈𝑗, 𝑘〉) → 𝐹 = 𝐶) |
| 204 | | simp-5l 785 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = 〈𝑗, 𝑘〉) → 𝜑) |
| 205 | | simp-4r 784 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = 〈𝑗, 𝑘〉) → 𝑗 ∈ 𝐴) |
| 206 | | simplr 769 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = 〈𝑗, 𝑘〉) → 𝑘 ∈ 𝐵) |
| 207 | 204, 205,
206, 46 | syl12anc 837 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = 〈𝑗, 𝑘〉) → 𝐶 ∈ (0[,]+∞)) |
| 208 | 203, 207 | eqeltrd 2841 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = 〈𝑗, 𝑘〉) → 𝐹 ∈ (0[,]+∞)) |
| 209 | | elsnxp 6311 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ 𝐴 → (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐵 𝑧 = 〈𝑗, 𝑘〉)) |
| 210 | 209 | biimpa 476 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = 〈𝑗, 𝑘〉) |
| 211 | 210 | adantll 714 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = 〈𝑗, 𝑘〉) |
| 212 | 200, 202,
208, 211 | r19.29af2 3267 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞)) |
| 213 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 214 | | eliun 4995 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵)) |
| 215 | 213, 214 | sylib 218 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵)) |
| 216 | 199, 212,
215 | r19.29af 3268 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞)) |
| 217 | 192, 195,
216 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞)) |
| 218 | | simpll 767 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → 𝜑) |
| 219 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗𝐴 |
| 220 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗𝑙 |
| 221 | 219, 220,
160, 162 | ssiun2sf 32572 |
. . . . . . . . . 10
⊢ (𝑙 ∈ 𝐴 → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 222 | 37, 221 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 223 | 222 | sselda 3983 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → 𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) |
| 224 | 218, 223,
216 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → 𝐹 ∈ (0[,]+∞)) |
| 225 | 169, 170,
171, 178, 180, 191, 217, 224 | esumsplit 34054 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑧 ∈ (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))𝐹 = (Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒
Σ*𝑧 ∈
({𝑙} ×
⦋𝑙 / 𝑗⦌𝐵)𝐹)) |
| 226 | 168, 225 | eqtrid 2789 |
. . . . 5
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = (Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒
Σ*𝑧 ∈
({𝑙} ×
⦋𝑙 / 𝑗⦌𝐵)𝐹)) |
| 227 | 226 | adantr 480 |
. . . 4
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = (Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒
Σ*𝑧 ∈
({𝑙} ×
⦋𝑙 / 𝑗⦌𝐵)𝐹)) |
| 228 | 133, 158,
227 | 3eqtr4d 2787 |
. . 3
⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹) |
| 229 | 228 | ex 412 |
. 2
⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹)) |
| 230 | | esum2dlem.e |
. 2
⊢ (𝜑 → 𝐴 ∈ Fin) |
| 231 | 5, 10, 15, 20, 27, 229, 230 | findcard2d 9206 |
1
⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪
𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) |