Proof of Theorem poimirlem7
| Step | Hyp | Ref
| Expression |
| 1 | | poimirlem9.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
| 2 | | elrabi 3687 |
. . . . . . . . 9
⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 3 | | poimirlem22.s |
. . . . . . . . 9
⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
| 4 | 2, 3 | eleq2s 2859 |
. . . . . . . 8
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 5 | 1, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 6 | | xp1st 8046 |
. . . . . . 7
⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 8 | | xp2nd 8047 |
. . . . . 6
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 9 | 7, 8 | syl 17 |
. . . . 5
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 10 | | fvex 6919 |
. . . . . 6
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
| 11 | | f1oeq1 6836 |
. . . . . 6
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
| 12 | 10, 11 | elab 3679 |
. . . . 5
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 13 | 9, 12 | sylib 218 |
. . . 4
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 14 | | f1of 6848 |
. . . 4
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
| 15 | 13, 14 | syl 17 |
. . 3
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
| 16 | | poimirlem9.2 |
. . . . . . . . 9
⊢ (𝜑 → (2nd
‘𝑇) ∈
(1...(𝑁 −
1))) |
| 17 | | elfznn 13593 |
. . . . . . . . 9
⊢
((2nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2nd
‘𝑇) ∈
ℕ) |
| 18 | 16, 17 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℕ) |
| 19 | 18 | peano2nnd 12283 |
. . . . . . 7
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ∈
ℕ) |
| 20 | 19 | peano2nnd 12283 |
. . . . . 6
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ∈
ℕ) |
| 21 | | nnuz 12921 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
| 22 | 20, 21 | eleqtrdi 2851 |
. . . . 5
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ∈
(ℤ≥‘1)) |
| 23 | | fzss1 13603 |
. . . . 5
⊢
((((2nd ‘𝑇) + 1) + 1) ∈
(ℤ≥‘1) → ((((2nd ‘𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
| 24 | 22, 23 | syl 17 |
. . . 4
⊢ (𝜑 → ((((2nd
‘𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
| 25 | | poimirlem7.3 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) |
| 26 | 24, 25 | sseldd 3984 |
. . 3
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
| 27 | 15, 26 | ffvelcdmd 7105 |
. 2
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) |
| 28 | | xp1st 8046 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
| 29 | 7, 28 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
| 30 | | elmapfn 8905 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 31 | 29, 30 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 32 | 31 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 33 | | 1ex 11257 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
| 34 | | fnconstg 6796 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1)))) |
| 35 | 33, 34 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) |
| 36 | | c0ex 11255 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
| 37 | | fnconstg 6796 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
| 38 | 36, 37 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) |
| 39 | 35, 38 | pm3.2i 470 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
| 40 | | dff1o3 6854 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
| 41 | 40 | simprbi 496 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
| 42 | 13, 41 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
| 43 | | imain 6651 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
| 44 | 42, 43 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
| 45 | 25 | elfzelzd 13565 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 46 | 45 | zred 12722 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 47 | 46 | ltm1d 12200 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) < 𝑀) |
| 48 | | fzdisj 13591 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
| 49 | 47, 48 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
| 50 | 49 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
| 51 | | ima0 6095 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
| 52 | 50, 51 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅) |
| 53 | 44, 52 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅) |
| 54 | | fnun 6682 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
| 55 | 39, 53, 54 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
| 56 | 45 | zcnd 12723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 57 | | npcan1 11688 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
| 58 | 56, 57 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
| 59 | | 1red 11262 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 1 ∈
ℝ) |
| 60 | 20 | nnred 12281 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ∈
ℝ) |
| 61 | 19 | nnred 12281 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ∈
ℝ) |
| 62 | 19 | nnge1d 12314 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 1 ≤ ((2nd
‘𝑇) +
1)) |
| 63 | 61 | ltp1d 12198 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((2nd
‘𝑇) + 1) <
(((2nd ‘𝑇)
+ 1) + 1)) |
| 64 | 59, 61, 60, 62, 63 | lelttrd 11419 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 1 < (((2nd
‘𝑇) + 1) +
1)) |
| 65 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → (((2nd
‘𝑇) + 1) + 1) ≤
𝑀) |
| 66 | 25, 65 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ≤
𝑀) |
| 67 | 59, 60, 46, 64, 66 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 1 < 𝑀) |
| 68 | 59, 46, 67 | ltled 11409 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 1 ≤ 𝑀) |
| 69 | | elnnz1 12643 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤
𝑀)) |
| 70 | 45, 68, 69 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 71 | 70, 21 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘1)) |
| 72 | 58, 71 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ≥‘1)) |
| 73 | | peano2zm 12660 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈
ℤ) |
| 74 | 45, 73 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
| 75 | | uzid 12893 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈ ℤ
→ (𝑀 − 1) ∈
(ℤ≥‘(𝑀 − 1))) |
| 76 | | peano2uz 12943 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈
(ℤ≥‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
| 77 | 74, 75, 76 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
| 78 | 58, 77 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
| 79 | | uzss 12901 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈
(ℤ≥‘(𝑀 − 1)) →
(ℤ≥‘𝑀) ⊆
(ℤ≥‘(𝑀 − 1))) |
| 80 | 78, 79 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
(ℤ≥‘𝑀) ⊆
(ℤ≥‘(𝑀 − 1))) |
| 81 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 82 | 25, 81 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 83 | 80, 82 | sseldd 3984 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) |
| 84 | | fzsplit2 13589 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
| 85 | 72, 83, 84 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
| 86 | 58 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁)) |
| 87 | 86 | uneq2d 4168 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
| 88 | 85, 87 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
| 89 | 88 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...(𝑀 − 1)) ∪
(𝑀...𝑁)))) |
| 90 | | imaundi 6169 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
| 91 | 89, 90 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
| 92 | | f1ofo 6855 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
| 93 | 13, 92 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
| 94 | | foima 6825 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
| 95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
| 96 | 91, 95 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = (1...𝑁)) |
| 97 | 96 | fneq2d 6662 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))) |
| 98 | 55, 97 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
| 99 | 98 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
| 100 | | ovexd 7466 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → (1...𝑁) ∈ V) |
| 101 | | inidm 4227 |
. . . . . . . . . 10
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
| 102 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st
‘𝑇))‘𝑛)) |
| 103 | | imaundi 6169 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ {𝑀}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 104 | | fzpred 13612 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
| 105 | 82, 104 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
| 106 | 105 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁)))) |
| 107 | | f1ofn 6849 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 108 | 13, 107 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 109 | | fnsnfv 6988 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑇))‘𝑀)} = ((2nd ‘(1st
‘𝑇)) “ {𝑀})) |
| 110 | 108, 26, 109 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘𝑀)} = ((2nd ‘(1st
‘𝑇)) “ {𝑀})) |
| 111 | 110 | uneq1d 4167 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ({((2nd
‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ {𝑀}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 112 | 103, 106,
111 | 3eqtr4a 2803 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) = ({((2nd
‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 113 | 112 | xpeq1d 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0})) |
| 114 | | xpundir 5755 |
. . . . . . . . . . . . . . . 16
⊢
(({((2nd ‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
| 115 | 113, 114 | eqtrdi 2793 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 116 | 115 | uneq2d 4168 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 117 | | un12 4173 |
. . . . . . . . . . . . . 14
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 118 | 116, 117 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 119 | 118 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 120 | 119 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 121 | | fnconstg 6796 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 122 | 36, 121 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
| 123 | 35, 122 | pm3.2i 470 |
. . . . . . . . . . . . . . 15
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 124 | | imain 6651 |
. . . . . . . . . . . . . . . . 17
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 125 | 42, 124 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 126 | 74 | zred 12722 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
| 127 | | peano2re 11434 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
| 128 | 46, 127 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 + 1) ∈ ℝ) |
| 129 | 46 | ltp1d 12198 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
| 130 | 126, 46, 128, 47, 129 | lttrd 11422 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀 − 1) < (𝑀 + 1)) |
| 131 | | fzdisj 13591 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
| 132 | 130, 131 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
| 133 | 132 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
| 134 | 133, 51 | eqtrdi 2793 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
| 135 | 125, 134 | eqtr3d 2779 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
| 136 | | fnun 6682 |
. . . . . . . . . . . . . . 15
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 137 | 123, 135,
136 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 138 | | imaundi 6169 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 139 | | imadif 6650 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑀}))) |
| 140 | 42, 139 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑀}))) |
| 141 | | fzsplit 13590 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
| 142 | 26, 141 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
| 143 | 142 | difeq1d 4125 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀})) |
| 144 | | difundir 4291 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑀) ∪
((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) |
| 145 | | fzsplit2 13589 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀))) |
| 146 | 72, 78, 145 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀))) |
| 147 | 58 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀)) |
| 148 | | fzsn 13606 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
| 149 | 45, 148 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
| 150 | 147, 149 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀}) |
| 151 | 150 | uneq2d 4168 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀})) |
| 152 | 146, 151 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀})) |
| 153 | 152 | difeq1d 4125 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀})) |
| 154 | | difun2 4481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑀 −
1)) ∪ {𝑀}) ∖
{𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀}) |
| 155 | 126, 46 | ltnled 11408 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1))) |
| 156 | 47, 155 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1)) |
| 157 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1)) |
| 158 | 156, 157 | nsyl 140 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1))) |
| 159 | | difsn 4798 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1))) |
| 160 | 158, 159 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1))) |
| 161 | 154, 160 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1))) |
| 162 | 153, 161 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1))) |
| 163 | 46, 128 | ltnled 11408 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
| 164 | 129, 163 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀) |
| 165 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀) |
| 166 | 164, 165 | nsyl 140 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) |
| 167 | | difsn 4798 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁)) |
| 168 | 166, 167 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁)) |
| 169 | 162, 168 | uneq12d 4169 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
| 170 | 144, 169 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
| 171 | 143, 170 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
| 172 | 171 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))) |
| 173 | 110 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {𝑀}) = {((2nd
‘(1st ‘𝑇))‘𝑀)}) |
| 174 | 95, 173 | difeq12d 4127 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 175 | 140, 172,
174 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 176 | 138, 175 | eqtr3id 2791 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 177 | 176 | fneq2d 6662 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) |
| 178 | 137, 177 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 179 | | eldifsn 4786 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) |
| 180 | 179 | biimpri 228 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 181 | 180 | ancoms 458 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ≠ ((2nd
‘(1st ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 182 | | disjdif 4472 |
. . . . . . . . . . . . . 14
⊢
({((2nd ‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ |
| 183 | | fnconstg 6796 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
V → ({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)}) |
| 184 | 36, 183 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} |
| 185 | | fvun2 7001 |
. . . . . . . . . . . . . . 15
⊢
((({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 186 | 184, 185 | mp3an1 1450 |
. . . . . . . . . . . . . 14
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 187 | 182, 186 | mpanr1 703 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 188 | 178, 181,
187 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 189 | 188 | anassrs 467 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 190 | 120, 189 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 191 | 32, 99, 100, 100, 101, 102, 190 | ofval 7708 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))) |
| 192 | | fnconstg 6796 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
| 193 | 33, 192 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) |
| 194 | 193, 122 | pm3.2i 470 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 195 | | imain 6651 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 196 | 42, 195 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 197 | | fzdisj 13591 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
| 198 | 129, 197 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
| 199 | 198 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
| 200 | 199, 51 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
| 201 | 196, 200 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
| 202 | | fnun 6682 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 203 | 194, 201,
202 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 204 | 142 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
| 205 | | imaundi 6169 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 206 | 204, 205 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 207 | 206, 95 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
| 208 | 207 | fneq2d 6662 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
| 209 | 203, 208 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
| 210 | 209 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
| 211 | | imaundi 6169 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ {𝑀})) |
| 212 | 152 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) = ((2nd ‘(1st
‘𝑇)) “
((1...(𝑀 − 1)) ∪
{𝑀}))) |
| 213 | 110 | uneq2d 4168 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ {𝑀}))) |
| 214 | 211, 212,
213 | 3eqtr4a 2803 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
| 215 | 214 | xpeq1d 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)}) × {1})) |
| 216 | | xpundir 5755 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)}) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) |
| 217 | 215, 216 | eqtrdi 2793 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}))) |
| 218 | 217 | uneq1d 4167 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 219 | | un23 4174 |
. . . . . . . . . . . . . . 15
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1})) |
| 220 | 219 | equncomi 4160 |
. . . . . . . . . . . . . 14
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 221 | 218, 220 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 222 | 221 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 223 | 222 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 224 | | fnconstg 6796 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
V → ({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)}) |
| 225 | 33, 224 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} |
| 226 | | fvun2 7001 |
. . . . . . . . . . . . . . 15
⊢
((({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 227 | 225, 226 | mp3an1 1450 |
. . . . . . . . . . . . . 14
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 228 | 182, 227 | mpanr1 703 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 229 | 178, 181,
228 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 230 | 229 | anassrs 467 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 231 | 223, 230 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
| 232 | 32, 210, 100, 100, 101, 102, 231 | ofval 7708 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))) |
| 233 | 191, 232 | eqtr4d 2780 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 234 | 233 | an32s 652 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 235 | 234 | anasss 466 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 236 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
| 237 | 236 | breq2d 5155 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
| 238 | 237 | ifbid 4549 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
| 239 | 238 | csbeq1d 3903 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 240 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
| 241 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
| 242 | 241 | imaeq1d 6077 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
| 243 | 242 | xpeq1d 5714 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
| 244 | 241 | imaeq1d 6077 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
| 245 | 244 | xpeq1d 5714 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
| 246 | 243, 245 | uneq12d 4169 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
| 247 | 240, 246 | oveq12d 7449 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 248 | 247 | csbeq2dv 3906 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 249 | 239, 248 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 250 | 249 | mpteq2dv 5244 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 251 | 250 | eqeq2d 2748 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
| 252 | 251, 3 | elrab2 3695 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
| 253 | 252 | simprbi 496 |
. . . . . . . . . 10
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 254 | 1, 253 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 255 | | breq1 5146 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 2) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 2) < (2nd
‘𝑇))) |
| 256 | 255 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 2) < (2nd
‘𝑇))) |
| 257 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 2) → (𝑦 + 1) = ((𝑀 − 2) + 1)) |
| 258 | | sub1m1 12518 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) − 1) = (𝑀 − 2)) |
| 259 | 56, 258 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑀 − 1) − 1) = (𝑀 − 2)) |
| 260 | 259 | oveq1d 7446 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑀 − 1) − 1) + 1) = ((𝑀 − 2) +
1)) |
| 261 | 74 | zcnd 12723 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 − 1) ∈ ℂ) |
| 262 | | npcan1 11688 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 − 1) ∈ ℂ
→ (((𝑀 − 1)
− 1) + 1) = (𝑀
− 1)) |
| 263 | 261, 262 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1)) |
| 264 | 260, 263 | eqtr3d 2779 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑀 − 2) + 1) = (𝑀 − 1)) |
| 265 | 257, 264 | sylan9eqr 2799 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → (𝑦 + 1) = (𝑀 − 1)) |
| 266 | 256, 265 | ifbieq2d 4552 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 2) < (2nd
‘𝑇), 𝑦, (𝑀 − 1))) |
| 267 | 18 | nncnd 12282 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℂ) |
| 268 | | add1p1 12517 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑇) ∈ ℂ → (((2nd
‘𝑇) + 1) + 1) =
((2nd ‘𝑇)
+ 2)) |
| 269 | 267, 268 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) =
((2nd ‘𝑇)
+ 2)) |
| 270 | 269, 66 | eqbrtrrd 5167 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘𝑇) + 2) ≤ 𝑀) |
| 271 | 18 | nnred 12281 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℝ) |
| 272 | | 2re 12340 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℝ |
| 273 | | leaddsub 11739 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 2 ∈ ℝ
∧ 𝑀 ∈ ℝ)
→ (((2nd ‘𝑇) + 2) ≤ 𝑀 ↔ (2nd ‘𝑇) ≤ (𝑀 − 2))) |
| 274 | 272, 273 | mp3an2 1451 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd
‘𝑇) + 2) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 2))) |
| 275 | 271, 46, 274 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘𝑇) + 2) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 2))) |
| 276 | 59, 46 | posdifd 11850 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1 < 𝑀 ↔ 0 < (𝑀 − 1))) |
| 277 | 67, 276 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 < (𝑀 − 1)) |
| 278 | | elnnz 12623 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑀 − 1) ∈ ℕ
↔ ((𝑀 − 1)
∈ ℤ ∧ 0 < (𝑀 − 1))) |
| 279 | 74, 277, 278 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 − 1) ∈ ℕ) |
| 280 | | nnm1nn0 12567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 − 1) ∈ ℕ
→ ((𝑀 − 1)
− 1) ∈ ℕ0) |
| 281 | 279, 280 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑀 − 1) − 1) ∈
ℕ0) |
| 282 | 259, 281 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 − 2) ∈
ℕ0) |
| 283 | 282 | nn0red 12588 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 2) ∈ ℝ) |
| 284 | 271, 283 | lenltd 11407 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑇) ≤ (𝑀 − 2) ↔ ¬ (𝑀 − 2) < (2nd
‘𝑇))) |
| 285 | 275, 284 | bitrd 279 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘𝑇) + 2) ≤ 𝑀 ↔ ¬ (𝑀 − 2) < (2nd
‘𝑇))) |
| 286 | 270, 285 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ (𝑀 − 2) < (2nd
‘𝑇)) |
| 287 | 286 | iffalsed 4536 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if((𝑀 − 2) < (2nd
‘𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1)) |
| 288 | 287 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if((𝑀 − 2) < (2nd
‘𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1)) |
| 289 | 266, 288 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1)) |
| 290 | 289 | csbeq1d 3903 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 − 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 291 | | oveq2 7439 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1))) |
| 292 | 291 | imaeq2d 6078 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑀 − 1) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...(𝑀 −
1)))) |
| 293 | 292 | xpeq1d 5714 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑀 − 1) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1})) |
| 294 | 293 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1})) |
| 295 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1)) |
| 296 | 295, 58 | sylan9eqr 2799 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀) |
| 297 | 296 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁)) |
| 298 | 297 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ (𝑀...𝑁))) |
| 299 | 298 | xpeq1d 5714 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) |
| 300 | 294, 299 | uneq12d 4169 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))) |
| 301 | 300 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
| 302 | 74, 301 | csbied 3935 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋(𝑀 − 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
| 303 | 302 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋(𝑀 − 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
| 304 | 290, 303 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
| 305 | | poimir.0 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 306 | | nnm1nn0 12567 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
| 307 | 305, 306 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ0) |
| 308 | 305 | nnred 12281 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 309 | 46 | lem1d 12201 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀 − 1) ≤ 𝑀) |
| 310 | | elfzle2 13568 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → 𝑀 ≤ 𝑁) |
| 311 | 25, 310 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
| 312 | 126, 46, 308, 309, 311 | letrd 11418 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑀 − 1) ≤ 𝑁) |
| 313 | 126, 308,
59, 312 | lesub1dd 11879 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑀 − 1) − 1) ≤ (𝑁 − 1)) |
| 314 | 259, 313 | eqbrtrrd 5167 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 2) ≤ (𝑁 − 1)) |
| 315 | | elfz2nn0 13658 |
. . . . . . . . . 10
⊢ ((𝑀 − 2) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 2) ∈
ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧
(𝑀 − 2) ≤ (𝑁 − 1))) |
| 316 | 282, 307,
314, 315 | syl3anbrc 1344 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 − 2) ∈ (0...(𝑁 − 1))) |
| 317 | | ovexd 7466 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V) |
| 318 | 254, 304,
316, 317 | fvmptd 7023 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑀 − 2)) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
| 319 | 318 | fveq1d 6908 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛)) |
| 320 | 319 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛)) |
| 321 | | breq1 5146 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 1) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < (2nd
‘𝑇))) |
| 322 | 321 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < (2nd
‘𝑇))) |
| 323 | | oveq1 7438 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1)) |
| 324 | 323, 58 | sylan9eqr 2799 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀) |
| 325 | 322, 324 | ifbieq2d 4552 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2nd
‘𝑇), 𝑦, 𝑀)) |
| 326 | 61 | lep1d 12199 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ≤
(((2nd ‘𝑇)
+ 1) + 1)) |
| 327 | 61, 60, 46, 326, 66 | letrd 11418 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ≤ 𝑀) |
| 328 | | 1re 11261 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℝ |
| 329 | | leaddsub 11739 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 1 ∈ ℝ
∧ 𝑀 ∈ ℝ)
→ (((2nd ‘𝑇) + 1) ≤ 𝑀 ↔ (2nd ‘𝑇) ≤ (𝑀 − 1))) |
| 330 | 328, 329 | mp3an2 1451 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd
‘𝑇) + 1) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 1))) |
| 331 | 271, 46, 330 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘𝑇) + 1) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 1))) |
| 332 | 271, 126 | lenltd 11407 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑇) ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < (2nd
‘𝑇))) |
| 333 | 331, 332 | bitrd 279 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘𝑇) + 1) ≤ 𝑀 ↔ ¬ (𝑀 − 1) < (2nd
‘𝑇))) |
| 334 | 327, 333 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ (𝑀 − 1) < (2nd
‘𝑇)) |
| 335 | 334 | iffalsed 4536 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if((𝑀 − 1) < (2nd
‘𝑇), 𝑦, 𝑀) = 𝑀) |
| 336 | 335 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < (2nd
‘𝑇), 𝑦, 𝑀) = 𝑀) |
| 337 | 325, 336 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
| 338 | 337 | csbeq1d 3903 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 339 | | oveq2 7439 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
| 340 | 339 | imaeq2d 6078 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑀))) |
| 341 | 340 | xpeq1d 5714 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})) |
| 342 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
| 343 | 342 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
| 344 | 343 | imaeq2d 6078 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 345 | 344 | xpeq1d 5714 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
| 346 | 341, 345 | uneq12d 4169 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 347 | 346 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑀 → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 348 | 347 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 349 | 25, 348 | csbied 3935 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 350 | 349 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 351 | 338, 350 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 352 | 279 | nnnn0d 12587 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ0) |
| 353 | 46, 308, 59, 311 | lesub1dd 11879 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1)) |
| 354 | | elfz2nn0 13658 |
. . . . . . . . . 10
⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈
ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧
(𝑀 − 1) ≤ (𝑁 − 1))) |
| 355 | 352, 307,
353, 354 | syl3anbrc 1344 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
| 356 | | ovexd 7466 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
| 357 | 254, 351,
355, 356 | fvmptd 7023 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 358 | 357 | fveq1d 6908 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 359 | 358 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
| 360 | 235, 320,
359 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛)) |
| 361 | 360 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛))) |
| 362 | 361 | necon1d 2962 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) → 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀))) |
| 363 | | elmapi 8889 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
| 364 | 29, 363 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
| 365 | 364, 27 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ (0..^𝐾)) |
| 366 | | elfzonn0 13747 |
. . . . . . . . 9
⊢
(((1st ‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈
ℕ0) |
| 367 | 365, 366 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈
ℕ0) |
| 368 | 367 | nn0red 12588 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ ℝ) |
| 369 | 368 | ltp1d 12198 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) < (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
| 370 | 368, 369 | ltned 11397 |
. . . . . 6
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ≠ (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
| 371 | 318 | fveq1d 6908 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 372 | | ovexd 7466 |
. . . . . . . . 9
⊢ (𝜑 → (1...𝑁) ∈ V) |
| 373 | | eqidd 2738 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((1st ‘(1st
‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 374 | | fzss1 13603 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈
(ℤ≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) |
| 375 | 71, 374 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁)) |
| 376 | | eluzfz1 13571 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
| 377 | 82, 376 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
| 378 | | fnfvima 7253 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
| 379 | 108, 375,
377, 378 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
| 380 | | fvun2 7001 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 381 | 35, 38, 380 | mp3an12 1453 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 382 | 53, 379, 381 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 383 | 36 | fvconst2 7224 |
. . . . . . . . . . . 12
⊢
(((2nd ‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) → ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
| 384 | 379, 383 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
| 385 | 382, 384 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
| 386 | 385 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
| 387 | 31, 98, 372, 372, 101, 373, 386 | ofval 7708 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 0)) |
| 388 | 27, 387 | mpdan 687 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 0)) |
| 389 | 367 | nn0cnd 12589 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ ℂ) |
| 390 | 389 | addridd 11461 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 0) = ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 391 | 371, 388,
390 | 3eqtrd 2781 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((1st ‘(1st
‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 392 | 357 | fveq1d 6908 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 393 | | fzss2 13604 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (1...𝑀) ⊆ (1...𝑁)) |
| 394 | 82, 393 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑀) ⊆ (1...𝑁)) |
| 395 | | elfz1end 13594 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀)) |
| 396 | 70, 395 | sylib 218 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
| 397 | | fnfvima 7253 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
| 398 | 108, 394,
396, 397 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
| 399 | | fvun1 7000 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 400 | 193, 122,
399 | mp3an12 1453 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 401 | 201, 398,
400 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 402 | 33 | fvconst2 7224 |
. . . . . . . . . . . 12
⊢
(((2nd ‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
| 403 | 398, 402 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
| 404 | 401, 403 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
| 405 | 404 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
| 406 | 31, 209, 372, 372, 101, 373, 405 | ofval 7708 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
| 407 | 27, 406 | mpdan 687 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
| 408 | 392, 407 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
| 409 | 370, 391,
408 | 3netr4d 3018 |
. . . . 5
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 410 | | fveq2 6906 |
. . . . . 6
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 411 | | fveq2 6906 |
. . . . . 6
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
| 412 | 410, 411 | neeq12d 3002 |
. . . . 5
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑀) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀)))) |
| 413 | 409, 412 | syl5ibrcom 247 |
. . . 4
⊢ (𝜑 → (𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛))) |
| 414 | 413 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛))) |
| 415 | 362, 414 | impbid 212 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀))) |
| 416 | 27, 415 | riota5 7417 |
1
⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st
‘𝑇))‘𝑀)) |