| Step | Hyp | Ref
| Expression |
| 1 | | fsumiunle.1 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ Fin) |
| 2 | | fsumiunle.2 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) |
| 3 | 1, 2 | aciunf1 32673 |
. . 3
⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)) |
| 4 | | f1f1orn 6859 |
. . . . . 6
⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓) |
| 5 | 4 | anim1i 615 |
. . . . 5
⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)) |
| 6 | | f1f 6804 |
. . . . . . 7
⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵⟶∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 7 | 6 | frnd 6744 |
. . . . . 6
⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 8 | 7 | adantr 480 |
. . . . 5
⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 9 | 5, 8 | jca 511 |
. . . 4
⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
| 10 | 9 | eximi 1835 |
. . 3
⊢
(∃𝑓(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1→∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ∃𝑓((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
| 11 | 3, 10 | syl 17 |
. 2
⊢ (𝜑 → ∃𝑓((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
| 12 | | csbeq1a 3913 |
. . . . . . 7
⊢ (𝑘 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑘⦌𝐶) |
| 13 | | nfcv 2905 |
. . . . . . 7
⊢
Ⅎ𝑦𝐶 |
| 14 | | nfcsb1v 3923 |
. . . . . . 7
⊢
Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 |
| 15 | 12, 13, 14 | cbvsum 15731 |
. . . . . 6
⊢
Σ𝑘 ∈
∪ 𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶 |
| 16 | | csbeq1 3902 |
. . . . . . 7
⊢ (𝑦 = (2nd ‘𝑧) → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌𝐶) |
| 17 | | snfi 9083 |
. . . . . . . . . . . 12
⊢ {𝑥} ∈ Fin |
| 18 | | xpfi 9358 |
. . . . . . . . . . . 12
⊢ (({𝑥} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑥} × 𝐵) ∈ Fin) |
| 19 | 17, 2, 18 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × 𝐵) ∈ Fin) |
| 20 | 19 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) |
| 21 | | iunfi 9383 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) |
| 22 | 1, 20, 21 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) |
| 23 | 22 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∈ Fin) |
| 24 | | simprr 773 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 25 | 23, 24 | ssfid 9301 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ran 𝑓 ∈ Fin) |
| 26 | | simprl 771 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓) |
| 27 | | f1ocnv 6860 |
. . . . . . . . 9
⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑥 ∈ 𝐴 𝐵) |
| 28 | 26, 27 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑥 ∈ 𝐴 𝐵) |
| 29 | 28 | adantrlr 723 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑥 ∈ 𝐴 𝐵) |
| 30 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝜑 |
| 31 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝑓 |
| 32 | | nfiu1 5027 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
| 33 | 31 | nfrn 5963 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥ran
𝑓 |
| 34 | 31, 32, 33 | nff1o 6846 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 |
| 35 | | nfv 1914 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥(2nd ‘(𝑓‘𝑙)) = 𝑙 |
| 36 | 32, 35 | nfralw 3311 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙 |
| 37 | 34, 36 | nfan 1899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) |
| 38 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥ran
𝑓 |
| 39 | | nfiu1 5027 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| 40 | 38, 39 | nfss 3976 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥ran 𝑓 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| 41 | 37, 40 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 42 | 30, 41 | nfan 1899 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
| 43 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑧 ∈ ran 𝑓 |
| 44 | 42, 43 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑥((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) |
| 45 | | simpr 484 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (𝑓‘𝑘) = 𝑧) |
| 46 | 45 | fveq2d 6910 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = (2nd ‘𝑧)) |
| 47 | | simplr 769 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
| 48 | | simp-4r 784 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) |
| 49 | 48 | simpld 494 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)) |
| 50 | 49 | simprd 495 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) |
| 51 | 50 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) |
| 52 | | 2fveq3 6911 |
. . . . . . . . . . . . . 14
⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘))) |
| 53 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘) |
| 54 | 52, 53 | eqeq12d 2753 |
. . . . . . . . . . . . 13
⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘)) |
| 55 | 54 | rspcva 3620 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪
𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘) |
| 56 | 47, 51, 55 | syl2anc 584 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = 𝑘) |
| 57 | 46, 56 | eqtr3d 2779 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘𝑧) = 𝑘) |
| 58 | 49 | simpld 494 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓) |
| 59 | 58 | ad2antrr 726 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓) |
| 60 | | f1ocnvfv1 7296 |
. . . . . . . . . . 11
⊢ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘) |
| 61 | 59, 47, 60 | syl2anc 584 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘) |
| 62 | 45 | fveq2d 6910 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = (◡𝑓‘𝑧)) |
| 63 | 57, 61, 62 | 3eqtr2rd 2784 |
. . . . . . . . 9
⊢
(((((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘𝑧) = (2nd ‘𝑧)) |
| 64 | | f1ofn 6849 |
. . . . . . . . . . 11
⊢ (𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 → 𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵) |
| 65 | 58, 64 | syl 17 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵) |
| 66 | | simpllr 776 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 𝑧 ∈ ran 𝑓) |
| 67 | | fvelrnb 6969 |
. . . . . . . . . . 11
⊢ (𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)) |
| 68 | 67 | biimpa 476 |
. . . . . . . . . 10
⊢ ((𝑓 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧) |
| 69 | 65, 66, 68 | syl2anc 584 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∃𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧) |
| 70 | 63, 69 | r19.29a 3162 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (◡𝑓‘𝑧) = (2nd ‘𝑧)) |
| 71 | 24 | sselda 3983 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 72 | | eliun 4995 |
. . . . . . . . 9
⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵)) |
| 73 | 71, 72 | sylib 218 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵)) |
| 74 | 44, 70, 73 | r19.29af 3268 |
. . . . . . 7
⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (◡𝑓‘𝑧) = (2nd ‘𝑧)) |
| 75 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
| 76 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘ℂ |
| 77 | 14, 76 | nfel 2920 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ |
| 78 | 75, 77 | nfim 1896 |
. . . . . . . . 9
⊢
Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ) |
| 79 | | eleq1w 2824 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑦 → (𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵)) |
| 80 | 79 | anbi2d 630 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵))) |
| 81 | 12 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ)) |
| 82 | 80, 81 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ))) |
| 83 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥𝑘 |
| 84 | 83, 32 | nfel 2920 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 |
| 85 | 30, 84 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
| 86 | | fsumiunle.3 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ) |
| 87 | 86 | adantllr 719 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℝ) |
| 88 | 87 | recnd 11289 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
| 89 | | eliun 4995 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵) |
| 90 | 89 | biimpi 216 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵) |
| 91 | 90 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → ∃𝑥 ∈ 𝐴 𝑘 ∈ 𝐵) |
| 92 | 85, 88, 91 | r19.29af 3268 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → 𝐶 ∈ ℂ) |
| 93 | 78, 82, 92 | chvarfv 2240 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ) |
| 94 | 93 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ) |
| 95 | 16, 25, 29, 74, 94 | fsumf1o 15759 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 96 | 15, 95 | eqtrid 2789 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝐶 = Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 97 | 96 | eqcomd 2743 |
. . . 4
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = Σ𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝐶) |
| 98 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑥𝑧 |
| 99 | 98, 39 | nfel 2920 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) |
| 100 | 30, 99 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑥(𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 101 | | xp2nd 8047 |
. . . . . . . . 9
⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵) |
| 102 | 101 | adantl 481 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑧) ∈ 𝐵) |
| 103 | 86 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ) |
| 104 | 103 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ) |
| 105 | 104 | adantr 480 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ) |
| 106 | | nfcsb1v 3923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶 |
| 107 | 106 | nfel1 2922 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ |
| 108 | | csbeq1a 3913 |
. . . . . . . . . . 11
⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌𝐶) |
| 109 | 108 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑘 = (2nd ‘𝑧) → (𝐶 ∈ ℝ ↔
⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)) |
| 110 | 107, 109 | rspc 3610 |
. . . . . . . . 9
⊢
((2nd ‘𝑧) ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ →
⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ)) |
| 111 | 110 | imp 406 |
. . . . . . . 8
⊢
(((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ ℝ) →
⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ ℝ) |
| 112 | 102, 105,
111 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ⦋(2nd
‘𝑧) / 𝑘⦌𝐶 ∈ ℝ) |
| 113 | 72 | biimpi 216 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵)) |
| 114 | 113 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × 𝐵)) |
| 115 | 100, 112,
114 | r19.29af 3268 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd
‘𝑧) / 𝑘⦌𝐶 ∈ ℝ) |
| 116 | 115 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → ⦋(2nd
‘𝑧) / 𝑘⦌𝐶 ∈ ℝ) |
| 117 | | xp1st 8046 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) ∈ {𝑥}) |
| 118 | | elsni 4643 |
. . . . . . . . . . 11
⊢
((1st ‘𝑧) ∈ {𝑥} → (1st ‘𝑧) = 𝑥) |
| 119 | 117, 118 | syl 17 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ({𝑥} × 𝐵) → (1st ‘𝑧) = 𝑥) |
| 120 | 119, 101 | jca 511 |
. . . . . . . . 9
⊢ (𝑧 ∈ ({𝑥} × 𝐵) → ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) |
| 121 | | simplll 775 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑) |
| 122 | | simplr 769 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑥 ∈ 𝐴) |
| 123 | | fsumiunle.4 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 0 ≤ 𝐶) |
| 124 | 123 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶) |
| 125 | 121, 122,
124 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑥 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶) |
| 126 | 120, 125 | sylan2 593 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → ∀𝑘 ∈ 𝐵 0 ≤ 𝐶) |
| 127 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘0 |
| 128 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘
≤ |
| 129 | 127, 128,
106 | nfbr 5190 |
. . . . . . . . . 10
⊢
Ⅎ𝑘0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶 |
| 130 | 108 | breq2d 5155 |
. . . . . . . . . 10
⊢ (𝑘 = (2nd ‘𝑧) → (0 ≤ 𝐶 ↔ 0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶)) |
| 131 | 129, 130 | rspc 3610 |
. . . . . . . . 9
⊢
((2nd ‘𝑧) ∈ 𝐵 → (∀𝑘 ∈ 𝐵 0 ≤ 𝐶 → 0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶)) |
| 132 | 131 | imp 406 |
. . . . . . . 8
⊢
(((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 0 ≤ 𝐶) → 0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 133 | 102, 126,
132 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑥} × 𝐵)) → 0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 134 | 100, 133,
114 | r19.29af 3268 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 135 | 134 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) → 0 ≤
⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 136 | 23, 116, 135, 24 | fsumless 15832 |
. . . 4
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ≤ Σ𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 137 | 97, 136 | eqbrtrrd 5167 |
. . 3
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 138 | 12, 13, 14 | cbvsum 15731 |
. . . . . . 7
⊢
Σ𝑘 ∈
𝐵 𝐶 = Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶 |
| 139 | 138 | a1i 11 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶) |
| 140 | 139 | sumeq2sdv 15739 |
. . . . 5
⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶) |
| 141 | | vex 3484 |
. . . . . . . . . 10
⊢ 𝑥 ∈ V |
| 142 | | vex 3484 |
. . . . . . . . . 10
⊢ 𝑦 ∈ V |
| 143 | 141, 142 | op2ndd 8025 |
. . . . . . . . 9
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) |
| 144 | 143 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑦 = (2nd ‘𝑧)) |
| 145 | 144 | csbeq1d 3903 |
. . . . . . 7
⊢ (𝑧 = 〈𝑥, 𝑦〉 → ⦋𝑦 / 𝑘⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌𝐶) |
| 146 | 145 | eqcomd 2743 |
. . . . . 6
⊢ (𝑧 = 〈𝑥, 𝑦〉 → ⦋(2nd
‘𝑧) / 𝑘⦌𝐶 = ⦋𝑦 / 𝑘⦌𝐶) |
| 147 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) |
| 148 | 14 | nfel1 2922 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ |
| 149 | 147, 148 | nfim 1896 |
. . . . . . . 8
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ) |
| 150 | | eleq1w 2824 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) |
| 151 | 150 | anbi2d 630 |
. . . . . . . . 9
⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵))) |
| 152 | 151, 81 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑘 = 𝑦 → ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ))) |
| 153 | 86 | recnd 11289 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
| 154 | 149, 152,
153 | chvarfv 2240 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ) |
| 155 | 154 | anasss 466 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑦 / 𝑘⦌𝐶 ∈ ℂ) |
| 156 | 146, 1, 2, 155 | fsum2d 15807 |
. . . . 5
⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 ⦋𝑦 / 𝑘⦌𝐶 = Σ𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 157 | 140, 156 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 158 | 157 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑧 ∈ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶) |
| 159 | 137, 158 | breqtrrd 5171 |
. 2
⊢ ((𝜑 ∧ ((𝑓:∪ 𝑥 ∈ 𝐴 𝐵–1-1-onto→ran
𝑓 ∧ ∀𝑙 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪
𝑥 ∈ 𝐴 ({𝑥} × 𝐵))) → Σ𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶) |
| 160 | 11, 159 | exlimddv 1935 |
1
⊢ (𝜑 → Σ𝑘 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝐶 ≤ Σ𝑥 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐶) |