Proof of Theorem poimirlem19
Step | Hyp | Ref
| Expression |
1 | | poimirlem22.2 |
. . 3
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | fveq2 6739 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
3 | 2 | breq2d 5082 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
4 | 3 | ifbid 4479 |
. . . . . . . 8
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
5 | | 2fveq3 6744 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
6 | | 2fveq3 6744 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
7 | 6 | imaeq1d 5946 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
8 | 7 | xpeq1d 5598 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
9 | 6 | imaeq1d 5946 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
10 | 9 | xpeq1d 5598 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
11 | 8, 10 | uneq12d 4095 |
. . . . . . . . 9
⊢ (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}))) |
12 | 5, 11 | oveq12d 7253 |
. . . . . . . 8
⊢ (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})))) |
13 | 4, 12 | csbeq12dv 3837 |
. . . . . . 7
⊢ (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})))) |
14 | 13 | mpteq2dv 5168 |
. . . . . 6
⊢ (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}))))) |
15 | 14 | eqeq2d 2750 |
. . . . 5
⊢ (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})))))) |
16 | | poimirlem22.s |
. . . . 5
⊢ 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}))))} |
17 | 15, 16 | elrab2 3620 |
. . . 4
⊢ (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})))))) |
18 | 17 | simprbi 500 |
. . 3
⊢ (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}))))) |
19 | 1, 18 | syl 17 |
. 2
⊢ (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}))))) |
20 | | elrabi 3611 |
. . . . . . . . . . . 12
⊢ (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))) |
21 | 20, 16 | eleq2s 2858 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
22 | 1, 21 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
23 | | xp1st 7815 |
. . . . . . . . . 10
⊢ (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)})) |
24 | 22, 23 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
25 | | xp1st 7815 |
. . . . . . . . 9
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
26 | 24, 25 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
27 | | elmapfn 8570 |
. . . . . . . 8
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
28 | 26, 27 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
29 | 28 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
30 | | 1ex 10859 |
. . . . . . . . . 10
⊢ 1 ∈
V |
31 | | fnconstg 6629 |
. . . . . . . . . 10
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦))) |
32 | 30, 31 | ax-mp 5 |
. . . . . . . . 9
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) |
33 | | c0ex 10857 |
. . . . . . . . . 10
⊢ 0 ∈
V |
34 | | fnconstg 6629 |
. . . . . . . . . 10
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
35 | 33, 34 | ax-mp 5 |
. . . . . . . . 9
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) |
36 | 32, 35 | pm3.2i 474 |
. . . . . . . 8
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
37 | | xp2nd 7816 |
. . . . . . . . . . . . 13
⊢
((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)}) |
38 | 24, 37 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
39 | | fvex 6752 |
. . . . . . . . . . . . 13
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
40 | | f1oeq1 6671 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
41 | 39, 40 | elab 3602 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
42 | 38, 41 | sylib 221 |
. . . . . . . . . . 11
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
43 | | dff1o3 6689 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
44 | 43 | simprbi 500 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
45 | 42, 44 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
46 | | imain 6486 |
. . . . . . . . . 10
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) |
47 | 45, 46 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) |
48 | | elfznn0 13235 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0) |
49 | 48 | nn0red 12181 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
50 | 49 | ltp1d 11792 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1)) |
51 | | fzdisj 13169 |
. . . . . . . . . . . 12
⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅) |
53 | 52 | imaeq2d 5947 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
54 | | ima0 5963 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
55 | 53, 54 | eqtrdi 2796 |
. . . . . . . . 9
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅) |
56 | 47, 55 | sylan9req 2801 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) |
57 | | fnun 6512 |
. . . . . . . 8
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) |
58 | 36, 56, 57 | sylancr 590 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) |
59 | | imaundi 6031 |
. . . . . . . . 9
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
60 | | nn0p1nn 12159 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ0
→ (𝑦 + 1) ∈
ℕ) |
61 | 48, 60 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ) |
62 | | nnuz 12507 |
. . . . . . . . . . . . . 14
⊢ ℕ =
(ℤ≥‘1) |
63 | 61, 62 | eleqtrdi 2850 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈
(ℤ≥‘1)) |
64 | 63 | adantl 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈
(ℤ≥‘1)) |
65 | | poimir.0 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℕ) |
66 | 65 | nncnd 11876 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℂ) |
67 | | npcan1 11287 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
69 | 68 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
70 | | elfzuz3 13139 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ≥‘𝑦)) |
71 | | peano2uz 12527 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
72 | 70, 71 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
73 | 72 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
74 | 69, 73 | eqeltrrd 2841 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘𝑦)) |
75 | | fzsplit2 13167 |
. . . . . . . . . . . 12
⊢ (((𝑦 + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) |
76 | 64, 74, 75 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) |
77 | 76 | imaeq2d 5947 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))) |
78 | | f1ofo 6690 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
79 | | foima 6660 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
80 | 42, 78, 79 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
81 | 80 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
82 | 77, 81 | eqtr3d 2781 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁)) |
83 | 59, 82 | eqtr3id 2794 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁)) |
84 | 83 | fneq2d 6494 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
85 | 58, 84 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
86 | | ovexd 7270 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V) |
87 | | inidm 4150 |
. . . . . 6
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
88 | | eqidd 2740 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st
‘𝑇))‘𝑛)) |
89 | | eqidd 2740 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
90 | 29, 85, 86, 86, 87, 88, 89 | offval 7499 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
91 | | elmapi 8554 |
. . . . . . . . . . . . 13
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
92 | 26, 91 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
93 | 92 | ffvelrnda 6926 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾)) |
94 | | elfzonn0 13317 |
. . . . . . . . . . 11
⊢
(((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈
ℕ0) |
95 | 93, 94 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈
ℕ0) |
96 | 95 | nn0cnd 12182 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ ℂ) |
97 | 96 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ ℂ) |
98 | | ax-1cn 10817 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
99 | | 0cn 10855 |
. . . . . . . . . 10
⊢ 0 ∈
ℂ |
100 | 98, 99 | ifcli 4503 |
. . . . . . . . 9
⊢ if(𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁), 1, 0) ∈ ℂ |
101 | 100 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0) ∈
ℂ) |
102 | | snssi 4738 |
. . . . . . . . . . 11
⊢ (1 ∈
ℂ → {1} ⊆ ℂ) |
103 | 98, 102 | ax-mp 5 |
. . . . . . . . . 10
⊢ {1}
⊆ ℂ |
104 | | snssi 4738 |
. . . . . . . . . . 11
⊢ (0 ∈
ℂ → {0} ⊆ ℂ) |
105 | 99, 104 | ax-mp 5 |
. . . . . . . . . 10
⊢ {0}
⊆ ℂ |
106 | 103, 105 | unssi 4116 |
. . . . . . . . 9
⊢ ({1}
∪ {0}) ⊆ ℂ |
107 | 30 | fconst 6627 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 +
1)))⟶{1} |
108 | 33 | fconst 6627 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0} |
109 | 107, 108 | pm3.2i 474 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}) |
110 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁)) |
111 | 65 | nnzd 12311 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑁 ∈ ℤ) |
112 | | 1z 12237 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 1 ∈
ℤ |
113 | | peano2z 12248 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (1 ∈
ℤ → (1 + 1) ∈ ℤ) |
114 | 112, 113 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (1 + 1)
∈ ℤ |
115 | 111, 114 | jctil 523 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1 + 1) ∈ ℤ
∧ 𝑁 ∈
ℤ)) |
116 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ) |
117 | 116, 112 | jctir 524 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
118 | | fzsubel 13178 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((1 +
1) ∈ ℤ ∧ 𝑁
∈ ℤ) ∧ (𝑛
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1)))) |
119 | 115, 117,
118 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1)))) |
120 | 110, 119 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1))) |
121 | 98, 98 | pncan3oi 11124 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((1 + 1)
− 1) = 1 |
122 | 121 | oveq1i 7245 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((1 + 1)
− 1)...(𝑁 − 1))
= (1...(𝑁 −
1)) |
123 | 120, 122 | eleqtrdi 2850 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1))) |
124 | 123 | ralrimiva 3108 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1))) |
125 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1))) |
126 | | peano2zm 12250 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
127 | 111, 126 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
128 | 127, 112 | jctil 523 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1 ∈ ℤ ∧
(𝑁 − 1) ∈
ℤ)) |
129 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
130 | 129, 112 | jctir 524 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈
ℤ)) |
131 | | fzaddel 13176 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((1
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
→ (𝑦 ∈
(1...(𝑁 − 1)) ↔
(𝑦 + 1) ∈ ((1 +
1)...((𝑁 − 1) +
1)))) |
132 | 128, 130,
131 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))) |
133 | 125, 132 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))) |
134 | 68 | oveq2d 7251 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 +
1)...𝑁)) |
135 | 134 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 +
1)...𝑁)) |
136 | 133, 135 | eleqtrd 2842 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁)) |
137 | 116 | zcnd 12313 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ) |
138 | 129 | zcnd 12313 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ) |
139 | | subadd2 11112 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝑦 ∈
ℂ) → ((𝑛 −
1) = 𝑦 ↔ (𝑦 + 1) = 𝑛)) |
140 | 98, 139 | mp3an2 1451 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛)) |
141 | | eqcom 2746 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦) |
142 | | eqcom 2746 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛) |
143 | 140, 141,
142 | 3bitr4g 317 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) |
144 | 137, 138,
143 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) |
145 | 144 | ralrimiva 3108 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) |
146 | 145 | adantl 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) |
147 | | reu6i 3658 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)) |
148 | 136, 146,
147 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)) |
149 | 148 | ralrimiva 3108 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)) |
150 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) |
151 | 150 | f1ompt 6950 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ↔ (∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))) |
152 | 124, 149,
151 | sylanbrc 586 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1))) |
153 | | f1osng 6723 |
. . . . . . . . . . . . . . . . . 18
⊢ ((1
∈ V ∧ 𝑁 ∈
ℕ) → {〈1, 𝑁〉}:{1}–1-1-onto→{𝑁}) |
154 | 30, 65, 153 | sylancr 590 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → {〈1, 𝑁〉}:{1}–1-1-onto→{𝑁}) |
155 | 65 | nnred 11875 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ ℝ) |
156 | 155 | ltm1d 11794 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
157 | 127 | zred 12312 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
158 | 157, 155 | ltnled 11009 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1))) |
159 | 156, 158 | mpbid 235 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1)) |
160 | | elfzle2 13146 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
161 | 159, 160 | nsyl 142 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1))) |
162 | | disjsn 4644 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(1...(𝑁 −
1))) |
163 | 161, 162 | sylibr 237 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
164 | | 1re 10863 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 1 ∈
ℝ |
165 | 164 | ltp1i 11766 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 < (1
+ 1) |
166 | 114 | zrei 12212 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (1 + 1)
∈ ℝ |
167 | 164, 166 | ltnlei 10983 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (1 <
(1 + 1) ↔ ¬ (1 + 1) ≤ 1) |
168 | 165, 167 | mpbi 233 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ¬ (1
+ 1) ≤ 1 |
169 | | elfzle1 13145 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (1 ∈
((1 + 1)...𝑁) → (1 +
1) ≤ 1) |
170 | 168, 169 | mto 200 |
. . . . . . . . . . . . . . . . . . 19
⊢ ¬ 1
∈ ((1 + 1)...𝑁) |
171 | | disjsn 4644 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((1 +
1)...𝑁) ∩ {1}) =
∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁)) |
172 | 170, 171 | mpbir 234 |
. . . . . . . . . . . . . . . . . 18
⊢ (((1 +
1)...𝑁) ∩ {1}) =
∅ |
173 | | f1oun 6702 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {〈1, 𝑁〉}:{1}–1-1-onto→{𝑁}) ∧ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {〈1, 𝑁〉}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁})) |
174 | 172, 173 | mpanr1 703 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {〈1, 𝑁〉}:{1}–1-1-onto→{𝑁}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {〈1, 𝑁〉}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁})) |
175 | 152, 154,
163, 174 | syl21anc 838 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {〈1, 𝑁〉}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁})) |
176 | | eleq1 2827 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁))) |
177 | 170, 176 | mtbiri 330 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁)) |
178 | 177 | necon2ai 2973 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1) |
179 | | ifnefalse 4468 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1)) |
180 | 178, 179 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1)) |
181 | 180 | mpteq2ia 5163 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) |
182 | 181 | uneq1i 4090 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {〈1, 𝑁〉}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {〈1, 𝑁〉}) |
183 | 30 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 1 ∈
V) |
184 | 65, 62 | eleqtrdi 2850 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
185 | | fzpred 13190 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁))) |
186 | 184, 185 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁))) |
187 | | uncom 4084 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({1}
∪ ((1 + 1)...𝑁)) = (((1
+ 1)...𝑁) ∪
{1}) |
188 | 186, 187 | eqtr2di 2797 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁)) |
189 | | iftrue 4462 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁) |
190 | 189 | adantl 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁) |
191 | 183, 65, 188, 190 | fmptapd 7008 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {〈1, 𝑁〉}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) |
192 | 182, 191 | eqtr3id 2794 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {〈1, 𝑁〉}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) |
193 | 68, 184 | eqeltrd 2840 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘1)) |
194 | | uzid 12483 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
195 | | peano2uz 12527 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
196 | 127, 194,
195 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
197 | 68, 196 | eqeltrrd 2841 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
198 | | fzsplit2 13167 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
199 | 193, 197,
198 | syl2anc 587 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
200 | 68 | oveq1d 7250 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁)) |
201 | | fzsn 13184 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) |
202 | 111, 201 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑁...𝑁) = {𝑁}) |
203 | 200, 202 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁}) |
204 | 203 | uneq2d 4094 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
205 | 199, 204 | eqtr2d 2780 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁)) |
206 | 192, 188,
205 | f1oeq123d 6677 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {〈1, 𝑁〉}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))) |
207 | 175, 206 | mpbid 235 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) |
208 | | f1oco 6705 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁)) |
209 | 42, 207, 208 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁)) |
210 | | dff1o3 6689 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))) |
211 | 210 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))) |
212 | | imain 6486 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))) |
213 | 209, 211,
212 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))) |
214 | 61 | nnred 11875 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ) |
215 | 214 | ltp1d 11792 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1)) |
216 | | fzdisj 13169 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
217 | 215, 216 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
218 | 217 | imaeq2d 5947 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “
∅)) |
219 | | ima0 5963 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅) =
∅ |
220 | 218, 219 | eqtrdi 2796 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
221 | 213, 220 | sylan9req 2801 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
222 | | fun 6603 |
. . . . . . . . . . . 12
⊢
(((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1}):(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}) ∧ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0})) |
223 | 109, 221,
222 | sylancr 590 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0})) |
224 | | imaundi 6031 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) |
225 | 61 | peano2nnd 11877 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ) |
226 | 225, 62 | eleqtrdi 2850 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
227 | 226 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
228 | | eluzp1p1 12496 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
229 | 70, 228 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
230 | 229 | adantl 485 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
231 | 69, 230 | eqeltrrd 2841 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) |
232 | | fzsplit2 13167 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑦 + 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
233 | 227, 231,
232 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
234 | 233 | imaeq2d 5947 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))) |
235 | | f1ofo 6690 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁)) |
236 | | foima 6660 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁)) |
237 | 209, 235,
236 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁)) |
238 | 237 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁)) |
239 | 234, 238 | eqtr3d 2781 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁)) |
240 | 224, 239 | eqtr3id 2794 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁)) |
241 | 240 | feq2d 6553 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}) ↔
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))) |
242 | 223, 241 | mpbid 235 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})) |
243 | 242 | ffvelrnda 6926 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0})) |
244 | 106, 243 | sseldi 3916 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ) |
245 | 97, 101, 244 | subadd23d 11241 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st
‘𝑇))‘𝑛) + (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)))) |
246 | | oveq2 7243 |
. . . . . . . . . 10
⊢ (1 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁), 1, 0) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) |
247 | 246 | eqeq1d 2741 |
. . . . . . . . 9
⊢ (1 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁), 1, 0) → ((((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
248 | | oveq2 7243 |
. . . . . . . . . 10
⊢ (0 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁), 1, 0) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) |
249 | 248 | eqeq1d 2741 |
. . . . . . . . 9
⊢ (0 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁), 1, 0) → ((((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
250 | | 1m1e0 11932 |
. . . . . . . . . . . . 13
⊢ (1
− 1) = 0 |
251 | | f1ofn 6684 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
252 | 42, 251 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
253 | 252 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
254 | | imassrn 5958 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) |
255 | | f1of 6683 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁)) |
256 | 207, 255 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁)) |
257 | 256 | frnd 6575 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁)) |
258 | 254, 257 | sstrid 3929 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁)) |
259 | 258 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁)) |
260 | | eqidd 2740 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) |
261 | | eluzfz1 13149 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ≥‘1) → 1 ∈ (1...𝑁)) |
262 | 184, 261 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 1 ∈ (1...𝑁)) |
263 | 260, 190,
262, 65 | fvmptd 6847 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁) |
264 | 263 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁) |
265 | | f1ofn 6684 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁)) |
266 | 207, 265 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁)) |
267 | 266 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁)) |
268 | | fzss2 13182 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
269 | 231, 268 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
270 | | eluzfz1 13149 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → 1 ∈ (1...(𝑦 + 1))) |
271 | 63, 270 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1))) |
272 | 271 | adantl 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1))) |
273 | | fnfvima 7071 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) |
274 | 267, 269,
272, 273 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) |
275 | 264, 274 | eqeltrrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) |
276 | | fnfvima 7071 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) → ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))) |
277 | 253, 259,
275, 276 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))) |
278 | | imaco 6133 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) |
279 | 277, 278 | eleqtrrdi 2851 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))) |
280 | | fnconstg 6629 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))) |
281 | 30, 280 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) |
282 | | fnconstg 6629 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) |
283 | 33, 282 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) |
284 | | fvun1 6824 |
. . . . . . . . . . . . . . . . 17
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∧ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) ∧ (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))) →
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1})‘((2nd ‘(1st ‘𝑇))‘𝑁))) |
285 | 281, 283,
284 | mp3an12 1453 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))) →
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1})‘((2nd ‘(1st ‘𝑇))‘𝑁))) |
286 | 221, 279,
285 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1})‘((2nd ‘(1st ‘𝑇))‘𝑁))) |
287 | 30 | fvconst2 7041 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇))‘𝑁) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) →
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 1) |
288 | 279, 287 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1})‘((2nd ‘(1st ‘𝑇))‘𝑁)) = 1) |
289 | 286, 288 | eqtrd 2779 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = 1) |
290 | 289 | oveq1d 7250 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) − 1) = (1 −
1)) |
291 | | fzss1 13181 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
292 | 63, 291 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
293 | 292 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
294 | | eluzfz2 13150 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁)) |
295 | 231, 294 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁)) |
296 | | fnfvima 7071 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
297 | 253, 293,
295, 296 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
298 | | fvun2 6825 |
. . . . . . . . . . . . . . . 16
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑁))) |
299 | 32, 35, 298 | mp3an12 1453 |
. . . . . . . . . . . . . . 15
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑁))) |
300 | 56, 297, 299 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑁))) |
301 | 33 | fvconst2 7041 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘(1st ‘𝑇))‘𝑁) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑁)) = 0) |
302 | 297, 301 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑁)) = 0) |
303 | 300, 302 | eqtrd 2779 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) = 0) |
304 | 250, 290,
303 | 3eqtr4a 2806 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁))) |
305 | | fveq2 6739 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁))) |
306 | 305 | oveq1d 7250 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) − 1)) |
307 | | fveq2 6739 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁))) |
308 | 306, 307 | eqeq12d 2755 |
. . . . . . . . . . . 12
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑁) → ((((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑁)))) |
309 | 304, 308 | syl5ibrcom 250 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
310 | 309 | imp 410 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
311 | 310 | adantlr 715 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
312 | 244 | subid1d 11208 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
313 | 312 | adantr 484 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
314 | | eldifsn 4717 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑁))) |
315 | | df-ne 2944 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ≠ ((2nd
‘(1st ‘𝑇))‘𝑁) ↔ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) |
316 | 315 | anbi2i 626 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑁)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁))) |
317 | 314, 316 | bitri 278 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁))) |
318 | | fnconstg 6629 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn
((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) |
319 | 33, 318 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn
((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) |
320 | 32, 319 | pm3.2i 474 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn
((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) |
321 | | imain 6486 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))) |
322 | 45, 321 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))) |
323 | | fzdisj 13169 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅) |
324 | 50, 323 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅) |
325 | 324 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
326 | 325, 54 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ∅) |
327 | 322, 326 | sylan9req 2801 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅) |
328 | | fnun 6512 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn
((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅) →
((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn
(((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))) |
329 | 320, 327,
328 | sylancr 590 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn
(((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))) |
330 | | imaundi 6031 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) |
331 | 199, 204 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
332 | 331 | difeq1d 4053 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁})) |
333 | | difun2 4412 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑁 −
1)) ∪ {𝑁}) ∖
{𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁}) |
334 | 332, 333 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})) |
335 | | difsn 4728 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1))) |
336 | 161, 335 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1))) |
337 | 334, 336 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1))) |
338 | 337 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1))) |
339 | 70 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈
(ℤ≥‘𝑦)) |
340 | | fzsplit2 13167 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑦 + 1) ∈
(ℤ≥‘1) ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑦)) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) |
341 | 64, 339, 340 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) |
342 | 338, 341 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) |
343 | 342 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))) |
344 | | imadif 6485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑁}))) |
345 | 45, 344 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑁}))) |
346 | | elfz1end 13172 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) |
347 | 65, 346 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑁 ∈ (1...𝑁)) |
348 | | fnsnfv 6812 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st
‘𝑇)) “ {𝑁})) |
349 | 252, 347,
348 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st
‘𝑇)) “ {𝑁})) |
350 | 349 | eqcomd 2745 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {𝑁}) = {((2nd
‘(1st ‘𝑇))‘𝑁)}) |
351 | 80, 350 | difeq12d 4055 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑁})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
352 | 345, 351 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
353 | 352 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
354 | 343, 353 | eqtr3d 2781 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
355 | 330, 354 | eqtr3id 2794 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
356 | 355 | fneq2d 6494 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn
(((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}))) |
357 | 329, 356 | mpbid 235 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
358 | | disjdifr 4404 |
. . . . . . . . . . . . . . 15
⊢
(((1...𝑁) ∖
{((2nd ‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = ∅ |
359 | | fnconstg 6629 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑁)}) |
360 | 30, 359 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑁)} |
361 | | fvun1 6824 |
. . . . . . . . . . . . . . . . 17
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∧ ({((2nd
‘(1st ‘𝑇))‘𝑁)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛)) |
362 | 360, 361 | mp3an2 1451 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛)) |
363 | | fnconstg 6629 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
V → ({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑁)}) |
364 | 33, 363 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑁)} |
365 | | fvun1 6824 |
. . . . . . . . . . . . . . . . 17
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∧ ({((2nd
‘(1st ‘𝑇))‘𝑁)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛)) |
366 | 364, 365 | mp3an2 1451 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛)) |
367 | 362, 366 | eqtr4d 2782 |
. . . . . . . . . . . . . . 15
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∩ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}))) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
368 | 358, 367 | mpanr1 703 |
. . . . . . . . . . . . . 14
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
369 | 357, 368 | sylan 583 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑁)})) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
370 | 317, 369 | sylan2br 598 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁))) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
371 | 370 | anassrs 471 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
372 | | imaundi 6031 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) =
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1})) |
373 | | imaco 6133 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) |
374 | | imaco 6133 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}) =
((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})) |
375 | 373, 374 | uneq12i 4092 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1})) =
(((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))) |
376 | 372, 375 | eqtri 2767 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) =
(((2nd ‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))) |
377 | | fzpred 13190 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1)))) |
378 | 63, 377 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1)))) |
379 | | uncom 4084 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ({1}
∪ ((1 + 1)...(𝑦 + 1)))
= (((1 + 1)...(𝑦 + 1))
∪ {1}) |
380 | 378, 379 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = (((1 + 1)...(𝑦 + 1)) ∪ {1})) |
381 | 380 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪
{1}))) |
382 | 381 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪
{1}))) |
383 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
384 | 121 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℤ → ((1 + 1)
− 1) = 1) |
385 | | zcn 12211 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ℤ → 𝑦 ∈
ℂ) |
386 | | pncan1 11286 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦) |
387 | 385, 386 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℤ → ((𝑦 + 1) − 1) = 𝑦) |
388 | 384, 387 | oveq12d 7253 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ℤ → (((1 + 1)
− 1)...((𝑦 + 1)
− 1)) = (1...𝑦)) |
389 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ 𝑗 ∈
ℤ) |
390 | 389 | zcnd 12313 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ 𝑗 ∈
ℂ) |
391 | | pncan1 11286 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗) |
392 | 390, 391 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ ((𝑗 + 1) − 1)
= 𝑗) |
393 | 392 | eleq1d 2824 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ (((𝑗 + 1) − 1)
∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) −
1)))) |
394 | 393 | ibir 271 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ ((𝑗 + 1) − 1)
∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) |
395 | 394 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1)))
→ ((𝑗 + 1) − 1)
∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) |
396 | | peano2z 12248 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈
ℤ) |
397 | 396, 114 | jctil 523 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 ∈ ℤ → ((1 + 1)
∈ ℤ ∧ (𝑦 +
1) ∈ ℤ)) |
398 | 389 | peano2zd 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ (𝑗 + 1) ∈
ℤ) |
399 | 398, 112 | jctir 524 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ ((𝑗 + 1) ∈
ℤ ∧ 1 ∈ ℤ)) |
400 | | fzsubel 13178 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((1 +
1) ∈ ℤ ∧ (𝑦
+ 1) ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
→ ((𝑗 + 1) ∈ ((1
+ 1)...(𝑦 + 1)) ↔
((𝑗 + 1) − 1) ∈
(((1 + 1) − 1)...((𝑦
+ 1) − 1)))) |
401 | 397, 399,
400 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1)))
→ ((𝑗 + 1) ∈ ((1
+ 1)...(𝑦 + 1)) ↔
((𝑗 + 1) − 1) ∈
(((1 + 1) − 1)...((𝑦
+ 1) − 1)))) |
402 | 395, 401 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1)))
→ (𝑗 + 1) ∈ ((1 +
1)...(𝑦 +
1))) |
403 | 392 | eqcomd 2745 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ 𝑗 = ((𝑗 + 1) −
1)) |
404 | 403 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1)))
→ 𝑗 = ((𝑗 + 1) −
1)) |
405 | | oveq1 7242 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = (𝑗 + 1) → (𝑛 − 1) = ((𝑗 + 1) − 1)) |
406 | 405 | rspceeqv 3567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)) |
407 | 402, 404,
406 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1)))
→ ∃𝑛 ∈ ((1
+ 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)) |
408 | 407 | ex 416 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
→ ∃𝑛 ∈ ((1
+ 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))) |
409 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) |
410 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ∈ ℤ) |
411 | 410, 112 | jctir 524 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
412 | | fzsubel 13178 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((1 +
1) ∈ ℤ ∧ (𝑦
+ 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
→ (𝑛 ∈ ((1 +
1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1)
− 1)...((𝑦 + 1)
− 1)))) |
413 | 397, 411,
412 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1)
− 1)...((𝑦 + 1)
− 1)))) |
414 | 409, 413 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 − 1) ∈ (((1 + 1)
− 1)...((𝑦 + 1)
− 1))) |
415 | | eleq1 2827 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = (𝑛 − 1) → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ (𝑛 − 1) ∈ (((1 + 1)
− 1)...((𝑦 + 1)
− 1)))) |
416 | 414, 415 | syl5ibrcom 250 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) −
1)))) |
417 | 416 | rexlimdva 3213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ℤ →
(∃𝑛 ∈ ((1 +
1)...(𝑦 + 1))𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) −
1)))) |
418 | 408, 417 | impbid 215 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
↔ ∃𝑛 ∈ ((1
+ 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))) |
419 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) |
420 | 419 | elrnmpt 5843 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))) |
421 | 420 | elv 3429 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)) |
422 | 418, 421 | bitr4di 292 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) −
1)...((𝑦 + 1) − 1))
↔ 𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))) |
423 | 422 | eqrdv 2737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ ℤ → (((1 + 1)
− 1)...((𝑦 + 1)
− 1)) = ran (𝑛 ∈
((1 + 1)...(𝑦 + 1)) ↦
(𝑛 −
1))) |
424 | 388, 423 | eqtr3d 2781 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ℤ →
(1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))) |
425 | 383, 424 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))) |
426 | 425 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))) |
427 | | df-ima 5582 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) |
428 | | uzid 12483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (1 ∈
ℤ → 1 ∈ (ℤ≥‘1)) |
429 | | peano2uz 12527 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (1 ∈
(ℤ≥‘1) → (1 + 1) ∈
(ℤ≥‘1)) |
430 | 112, 428,
429 | mp2b 10 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (1 + 1)
∈ (ℤ≥‘1) |
431 | | fzss1 13181 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((1 + 1)
∈ (ℤ≥‘1) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1))) |
432 | 430, 431 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((1 +
1)...(𝑦 + 1)) ⊆
(1...(𝑦 +
1)) |
433 | 432, 269 | sstrid 3929 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...𝑁)) |
434 | 433 | resmptd 5926 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) |
435 | | elfzle1 13145 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (1 ∈
((1 + 1)...(𝑦 + 1)) →
(1 + 1) ≤ 1) |
436 | 168, 435 | mto 200 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ¬ 1
∈ ((1 + 1)...(𝑦 +
1)) |
437 | | eleq1 2827 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ 1 ∈ ((1 + 1)...(𝑦 + 1)))) |
438 | 436, 437 | mtbiri 330 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) |
439 | 438 | necon2ai 2973 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ≠ 1) |
440 | 439, 179 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1)) |
441 | 440 | mpteq2ia 5163 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) |
442 | 434, 441 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))) |
443 | 442 | rneqd 5825 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))) |
444 | 427, 443 | syl5eq 2792 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))) |
445 | 426, 444 | eqtr4d 2782 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) |
446 | 445 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))) |
447 | 263 | sneqd 4570 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = {𝑁}) |
448 | | fnsnfv 6812 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})) |
449 | 266, 262,
448 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})) |
450 | 447, 449 | eqtr3d 2781 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {𝑁} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})) |
451 | 450 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {𝑁}) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))) |
452 | 349, 451 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))) |
453 | 452 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd
‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))) |
454 | 446, 453 | uneq12d 4095 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “
{1})))) |
455 | 376, 382,
454 | 3eqtr4a 2806 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
456 | 455 | xpeq1d 5598 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) =
((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)}) × {1})) |
457 | | xpundir 5636 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)}) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘𝑁)} × {1})) |
458 | 456, 457 | eqtrdi 2796 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) =
((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘𝑁)} × {1}))) |
459 | | imaco 6133 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁))) |
460 | | df-ima 5582 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) |
461 | | fzss1 13181 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑦 + 1) + 1) ∈
(ℤ≥‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
462 | 227, 461 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
463 | 462 | resmptd 5926 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) |
464 | | 1red 10864 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈
ℝ) |
465 | 61 | nnzd 12311 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ) |
466 | 465 | peano2zd 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℤ) |
467 | 466 | zred 12312 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ) |
468 | 61 | nnge1d 11908 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ≤ (𝑦 + 1)) |
469 | 464, 214,
467, 468, 215 | lelttrd 11020 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 < ((𝑦 + 1) + 1)) |
470 | 464, 467 | ltnled 11009 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1 < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤
1)) |
471 | 469, 470 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ ((𝑦 + 1) + 1) ≤
1) |
472 | | elfzle1 13145 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (1 ∈
(((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ 1) |
473 | 471, 472 | nsyl 142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁)) |
474 | | eleq1 2827 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 = 1 → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ 1 ∈ (((𝑦 + 1) + 1)...𝑁))) |
475 | 474 | notbid 321 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 = 1 → (¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁))) |
476 | 473, 475 | syl5ibrcom 250 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 = 1 → ¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁))) |
477 | 476 | necon2ad 2958 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ≠ 1)) |
478 | 477 | imp 410 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ≠ 1) |
479 | 478, 179 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1)) |
480 | 479 | mpteq2dva 5167 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))) |
481 | 480 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))) |
482 | 463, 481 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))) |
483 | 482 | rneqd 5825 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))) |
484 | 460, 483 | syl5eq 2792 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))) |
485 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) |
486 | 485 | elrnmpt 5843 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))) |
487 | 486 | elv 3429 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)) |
488 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) |
489 | 111, 466 | anim12ci 617 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈
ℤ)) |
490 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ∈ ℤ) |
491 | 490, 112 | jctir 524 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
492 | | fzsubel 13178 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝑦 + 1) + 1)
∈ ℤ ∧ 𝑁
∈ ℤ) ∧ (𝑛
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
493 | 489, 491,
492 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
494 | 488, 493 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) |
495 | | eleq1 2827 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = (𝑛 − 1) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
496 | 494, 495 | syl5ibrcom 250 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
497 | 496 | rexlimdva 3213 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
498 | | elfzelz 13142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℤ) |
499 | 498 | zcnd 12313 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℂ) |
500 | 499, 391 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) = 𝑗) |
501 | 500 | eleq1d 2824 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
502 | 501 | ibir 271 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) |
503 | 502 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) |
504 | 498 | peano2zd 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (𝑗 + 1) ∈ ℤ) |
505 | 504, 112 | jctir 524 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈
ℤ)) |
506 | | fzsubel 13178 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝑦 + 1) + 1)
∈ ℤ ∧ 𝑁
∈ ℤ) ∧ ((𝑗 +
1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
507 | 489, 505,
506 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
508 | 503, 507 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → (𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁)) |
509 | 500 | eqcomd 2745 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 = ((𝑗 + 1) − 1)) |
510 | 509 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → 𝑗 = ((𝑗 + 1) − 1)) |
511 | 405 | rspceeqv 3567 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)) |
512 | 508, 510,
511 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)) |
513 | 512 | ex 416 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))) |
514 | 497, 513 | impbid 215 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
515 | 487, 514 | syl5bb 286 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))) |
516 | 515 | eqrdv 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) |
517 | 61 | nncnd 11876 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℂ) |
518 | | pncan1 11286 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 + 1) ∈ ℂ →
(((𝑦 + 1) + 1) − 1) =
(𝑦 + 1)) |
519 | 517, 518 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1)) |
520 | 519 | oveq1d 7250 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1))) |
521 | 520 | adantl 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1))) |
522 | 484, 516,
521 | 3eqtrd 2783 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ((𝑦 + 1)...(𝑁 − 1))) |
523 | 522 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) |
524 | 459, 523 | syl5eq 2792 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) |
525 | 524 | xpeq1d 5598 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) |
526 | 458, 525 | uneq12d 4095 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘𝑁)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ×
{0}))) |
527 | | un23 4099 |
. . . . . . . . . . . . . 14
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘𝑁)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) =
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1})) |
528 | 526, 527 | eqtrdi 2796 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))) |
529 | 528 | fveq1d 6741 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛)) |
530 | 529 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {1}))‘𝑛)) |
531 | | imaundi 6031 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) = (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ {𝑁})) |
532 | | fzsplit2 13167 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
533 | 229, 197,
532 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
534 | 203 | uneq2d 4094 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
535 | 534 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
536 | 533, 535 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
537 | 536 | imaeq2d 5947 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))) |
538 | 349 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd
‘(1st ‘𝑇))‘𝑁)} = ((2nd ‘(1st
‘𝑇)) “ {𝑁})) |
539 | 538 | uneq2d 4094 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ {𝑁}))) |
540 | 531, 537,
539 | 3eqtr4a 2806 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) = (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)})) |
541 | 540 | xpeq1d 5598 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)}) × {0})) |
542 | | xpundir 5636 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑁)}) × {0}) = ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})) |
543 | 541, 542 | eqtrdi 2796 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))) |
544 | 543 | uneq2d 4094 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})))) |
545 | | unass 4097 |
. . . . . . . . . . . . . 14
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))) |
546 | 544, 545 | eqtr4di 2798 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))) |
547 | 546 | fveq1d 6741 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
548 | 547 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪
({((2nd ‘(1st ‘𝑇))‘𝑁)} × {0}))‘𝑛)) |
549 | 371, 530,
548 | 3eqtr4d 2789 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
550 | 313, 549 | eqtrd 2779 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁)) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
551 | 247, 249,
311, 550 | ifbothda 4494 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) = (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
552 | 551 | oveq2d 7251 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) + (((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) = (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
553 | 245, 552 | eqtr2d 2780 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))) |
554 | 553 | mpteq2dva 5167 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))) |
555 | 90, 554 | eqtrd 2779 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))) |
556 | 49 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ) |
557 | 157 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ) |
558 | 155 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ) |
559 | | elfzle2 13146 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1)) |
560 | 559 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1)) |
561 | 156 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁) |
562 | 556, 557,
558, 560, 561 | lelttrd 11020 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁) |
563 | | poimirlem21.4 |
. . . . . . . . 9
⊢ (𝜑 → (2nd
‘𝑇) = 𝑁) |
564 | 563 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑇) = 𝑁) |
565 | 562, 564 | breqtrrd 5098 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd ‘𝑇)) |
566 | 565 | iftrued 4464 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦) |
567 | 566 | csbeq1d 3832 |
. . . . 5
⊢ ((𝜑 ∧ 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 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
568 | | vex 3427 |
. . . . . 6
⊢ 𝑦 ∈ V |
569 | | oveq2 7243 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦)) |
570 | 569 | imaeq2d 5947 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑦))) |
571 | 570 | xpeq1d 5598 |
. . . . . . . 8
⊢ (𝑗 = 𝑦 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1})) |
572 | | oveq1 7242 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1)) |
573 | 572 | oveq1d 7250 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁)) |
574 | 573 | imaeq2d 5947 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
575 | 574 | xpeq1d 5598 |
. . . . . . . 8
⊢ (𝑗 = 𝑦 → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) |
576 | 571, 575 | uneq12d 4095 |
. . . . . . 7
⊢ (𝑗 = 𝑦 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
577 | 576 | oveq2d 7251 |
. . . . . 6
⊢ (𝑗 = 𝑦 → ((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...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
578 | 568, 577 | csbie 3864 |
. . . . 5
⊢
⦋𝑦 /
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...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
579 | 567, 578 | eqtrdi 2796 |
. . . 4
⊢ ((𝜑 ∧ 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}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
580 | | ovexd 7270 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) ∈
V) |
581 | | fvexd 6754 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ V) |
582 | | eqidd 2740 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)))) |
583 | 242 | ffnd 6568 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
584 | | nfcv 2907 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(2nd ‘(1st
‘𝑇)) |
585 | | nfmpt1 5170 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) |
586 | 584, 585 | nfco 5752 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) |
587 | | nfcv 2907 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(1...(𝑦 + 1)) |
588 | 586, 587 | nfima 5955 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) |
589 | | nfcv 2907 |
. . . . . . . . 9
⊢
Ⅎ𝑛{1} |
590 | 588, 589 | nfxp 5602 |
. . . . . . . 8
⊢
Ⅎ𝑛((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ×
{1}) |
591 | | nfcv 2907 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(((𝑦 + 1) + 1)...𝑁) |
592 | 586, 591 | nfima 5955 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) |
593 | | nfcv 2907 |
. . . . . . . . 9
⊢
Ⅎ𝑛{0} |
594 | 592, 593 | nfxp 5602 |
. . . . . . . 8
⊢
Ⅎ𝑛((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) |
595 | 590, 594 | nfun 4096 |
. . . . . . 7
⊢
Ⅎ𝑛(((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
596 | 595 | dffn5f 6805 |
. . . . . 6
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))) |
597 | 583, 596 | sylib 221 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))) |
598 | 86, 580, 581, 582, 597 | offval2 7510 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) ∘f +
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0)) + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))) |
599 | 555, 579,
598 | 3eqtr4rd 2790 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) ∘f +
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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})))) |
600 | 599 | mpteq2dva 5167 |
. 2
⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) ∘f +
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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}))))) |
601 | 19, 600 | eqtr4d 2782 |
1
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑁), 1, 0))) ∘f +
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))) |