Proof of Theorem poimirlem16
Step | Hyp | Ref
| Expression |
1 | | poimirlem22.2 |
. . 3
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
3 | 2 | breq2d 4595 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
4 | 3 | ifbid 4058 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
5 | 4 | csbeq1d 3506 |
. . . . . . . 8
⊢ (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})))) |
6 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇)) |
7 | 6 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
8 | 6 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
9 | 8 | imaeq1d 5384 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
10 | 9 | xpeq1d 5062 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
11 | 8 | imaeq1d 5384 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
12 | 11 | xpeq1d 5062 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
13 | 10, 12 | uneq12d 3730 |
. . . . . . . . . 10
⊢ (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}))) |
14 | 7, 13 | oveq12d 6567 |
. . . . . . . . 9
⊢ (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})))) |
15 | 14 | csbeq2dv 3944 |
. . . . . . . 8
⊢ (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})))) |
16 | 5, 15 | eqtrd 2644 |
. . . . . . 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})))) |
17 | 16 | mpteq2dv 4673 |
. . . . . 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}))))) |
18 | 17 | eqeq2d 2620 |
. . . . 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})))))) |
19 | | 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}))))} |
20 | 18, 19 | elrab2 3333 |
. . . 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})))))) |
21 | 20 | simprbi 479 |
. . 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}))))) |
22 | 1, 21 | 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}))))) |
23 | | elrabi 3328 |
. . . . . . . . . . . 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))) |
24 | 23, 19 | eleq2s 2706 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
25 | 1, 24 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
26 | | xp1st 7089 |
. . . . . . . . . 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)})) |
27 | 25, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
28 | | xp1st 7089 |
. . . . . . . . 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))) |
29 | 27, 28 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
30 | | elmapfn 7766 |
. . . . . . . 8
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
32 | 31 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
33 | | 1ex 9914 |
. . . . . . . . . 10
⊢ 1 ∈
V |
34 | | fnconstg 6006 |
. . . . . . . . . 10
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) |
35 | 33, 34 | ax-mp 5 |
. . . . . . . . 9
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) |
36 | | c0ex 9913 |
. . . . . . . . . 10
⊢ 0 ∈
V |
37 | | fnconstg 6006 |
. . . . . . . . . 10
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
38 | 36, 37 | ax-mp 5 |
. . . . . . . . 9
⊢
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) |
39 | 35, 38 | pm3.2i 470 |
. . . . . . . 8
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
40 | | xp2nd 7090 |
. . . . . . . . . . . . 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)}) |
41 | 27, 40 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
42 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
43 | | f1oeq1 6040 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
44 | 42, 43 | elab 3319 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
45 | 41, 44 | sylib 207 |
. . . . . . . . . . 11
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
46 | | dff1o3 6056 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
47 | 46 | simprbi 479 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
48 | 45, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
49 | | imain 5888 |
. . . . . . . . . 10
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
50 | 48, 49 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
51 | | elfznn0 12302 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0) |
52 | | nn0p1nn 11209 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ0
→ (𝑦 + 1) ∈
ℕ) |
53 | 51, 52 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ) |
54 | 53 | nnred 10912 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ) |
55 | 54 | ltp1d 10833 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1)) |
56 | | fzdisj 12239 |
. . . . . . . . . . . 12
⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
57 | 55, 56 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
58 | 57 | imaeq2d 5385 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
59 | | ima0 5400 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
60 | 58, 59 | syl6eq 2660 |
. . . . . . . . 9
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
61 | 50, 60 | sylan9req 2665 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
62 | | fnun 5911 |
. . . . . . . 8
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
63 | 39, 61, 62 | sylancr 694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
64 | | imaundi 5464 |
. . . . . . . . 9
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
65 | | nnuz 11599 |
. . . . . . . . . . . . . . 15
⊢ ℕ =
(ℤ≥‘1) |
66 | 53, 65 | syl6eleq 2698 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈
(ℤ≥‘1)) |
67 | | peano2uz 11617 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
69 | 68 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
70 | | poimir.0 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℕ) |
71 | 70 | nncnd 10913 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℂ) |
72 | | npcan1 10334 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
73 | 71, 72 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
74 | 73 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
75 | | elfzuz3 12210 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ≥‘𝑦)) |
76 | | eluzp1p1 11589 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
77 | 75, 76 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
78 | 77 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
79 | 74, 78 | eqeltrrd 2689 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) |
80 | | fzsplit2 12237 |
. . . . . . . . . . . 12
⊢ ((((𝑦 + 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
81 | 69, 79, 80 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
82 | 81 | imaeq2d 5385 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...(𝑦 + 1)) ∪
(((𝑦 + 1) + 1)...𝑁)))) |
83 | | f1ofo 6057 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
84 | | foima 6033 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
85 | 45, 83, 84 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
86 | 85 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
87 | 82, 86 | eqtr3d 2646 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁)) |
88 | 64, 87 | syl5eqr 2658 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁)) |
89 | 88 | fneq2d 5896 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
90 | 63, 89 | mpbid 221 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
91 | | ovex 6577 |
. . . . . . 7
⊢
(1...𝑁) ∈
V |
92 | 91 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V) |
93 | | inidm 3784 |
. . . . . 6
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
94 | | eqidd 2611 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st
‘𝑇))‘𝑛)) |
95 | | eqidd 2611 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
96 | 32, 90, 92, 92, 93, 94, 95 | offval 6802 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))) |
97 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (1 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘1), 1, 0) → (1 +
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
98 | 97 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (1 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘1), 1, 0) →
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
99 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (0 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘1), 1, 0) → (0 +
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
100 | 99 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (0 =
if(𝑛 = ((2nd
‘(1st ‘𝑇))‘1), 1, 0) →
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (0 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
101 | | 1p0e1 11010 |
. . . . . . . . . . . . . 14
⊢ (1 + 0) =
1 |
102 | 101 | eqcomi 2619 |
. . . . . . . . . . . . 13
⊢ 1 = (1 +
0) |
103 | | f1ofn 6051 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
104 | 45, 103 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
105 | 104 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
106 | | fzss2 12252 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
107 | 79, 106 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
108 | | eluzfz1 12219 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → 1 ∈ (1...(𝑦 + 1))) |
109 | 66, 108 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1))) |
110 | 109 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1))) |
111 | | fnfvima 6400 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) |
112 | 105, 107,
110, 111 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) |
113 | | fvun1 6179 |
. . . . . . . . . . . . . . . 16
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1))) |
114 | 35, 38, 113 | mp3an12 1406 |
. . . . . . . . . . . . . . 15
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1))) |
115 | 61, 112, 114 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1))) |
116 | 33 | fvconst2 6374 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1)) = 1) |
117 | 112, 116 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1)) = 1) |
118 | 115, 117 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = 1) |
119 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1))) |
120 | 70 | nnzd 11357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝑁 ∈ ℤ) |
121 | | peano2zm 11297 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
122 | 120, 121 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
123 | | 1z 11284 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 1 ∈
ℤ |
124 | 122, 123 | jctil 558 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (1 ∈ ℤ ∧
(𝑁 − 1) ∈
ℤ)) |
125 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ) |
126 | 125, 123 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
127 | | fzaddel 12246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((1
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
→ (𝑛 ∈
(1...(𝑁 − 1)) ↔
(𝑛 + 1) ∈ ((1 +
1)...((𝑁 − 1) +
1)))) |
128 | 124, 126,
127 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))) |
129 | 119, 128 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))) |
130 | 73 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 +
1)...𝑁)) |
131 | 130 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 +
1)...𝑁)) |
132 | 129, 131 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁)) |
133 | 132 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁)) |
134 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁)) |
135 | | peano2z 11295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (1 ∈
ℤ → (1 + 1) ∈ ℤ) |
136 | 123, 135 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (1 + 1)
∈ ℤ |
137 | 120, 136 | jctil 558 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → ((1 + 1) ∈ ℤ
∧ 𝑁 ∈
ℤ)) |
138 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ) |
139 | 138, 123 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈
ℤ)) |
140 | | fzsubel 12248 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((1 +
1) ∈ ℤ ∧ 𝑁
∈ ℤ) ∧ (𝑦
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1)))) |
141 | 137, 139,
140 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1)))) |
142 | 134, 141 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1))) |
143 | | ax-1cn 9873 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 1 ∈
ℂ |
144 | 143, 143 | pncan3oi 10176 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((1 + 1)
− 1) = 1 |
145 | 144 | oveq1i 6559 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((1 + 1)
− 1)...(𝑁 − 1))
= (1...(𝑁 −
1)) |
146 | 142, 145 | syl6eleq 2698 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1))) |
147 | 138 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ) |
148 | | elfznn 12241 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ) |
149 | 148 | nncnd 10913 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ) |
150 | | subadd2 10164 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑦 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝑛 ∈
ℂ) → ((𝑦 −
1) = 𝑛 ↔ (𝑛 + 1) = 𝑦)) |
151 | 143, 150 | mp3an2 1404 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦)) |
152 | 151 | bicomd 212 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛)) |
153 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑛 + 1) = 𝑦 ↔ 𝑦 = (𝑛 + 1)) |
154 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 − 1) = 𝑛 ↔ 𝑛 = (𝑦 − 1)) |
155 | 152, 153,
154 | 3bitr3g 301 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
156 | 147, 149,
155 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
157 | 156 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
158 | 157 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
159 | | reu6i 3364 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)) |
160 | 146, 158,
159 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)) |
161 | 160 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)) |
162 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) |
163 | 162 | f1ompt 6290 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 +
1)...𝑁) ↔
(∀𝑛 ∈
(1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))) |
164 | 133, 161,
163 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 +
1)...𝑁)) |
165 | | f1osng 6089 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑁 ∈ ℕ ∧ 1 ∈
V) → {〈𝑁,
1〉}:{𝑁}–1-1-onto→{1}) |
166 | 70, 33, 165 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {〈𝑁, 1〉}:{𝑁}–1-1-onto→{1}) |
167 | 70 | nnred 10912 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ ℝ) |
168 | 167 | ltm1d 10835 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
169 | 122 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
170 | 169, 167 | ltnled 10063 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1))) |
171 | 168, 170 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1)) |
172 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
173 | 171, 172 | nsyl 134 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1))) |
174 | | disjsn 4192 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(1...(𝑁 −
1))) |
175 | 173, 174 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
176 | | 1re 9918 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 1 ∈
ℝ |
177 | 176 | ltp1i 10806 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 1 < (1
+ 1) |
178 | 176, 176 | readdcli 9932 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (1 + 1)
∈ ℝ |
179 | 176, 178 | ltnlei 10037 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (1 <
(1 + 1) ↔ ¬ (1 + 1) ≤ 1) |
180 | 177, 179 | mpbi 219 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ¬ (1
+ 1) ≤ 1 |
181 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1 ∈
((1 + 1)...𝑁) → (1 +
1) ≤ 1) |
182 | 180, 181 | mto 187 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ¬ 1
∈ ((1 + 1)...𝑁) |
183 | | disjsn 4192 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((1 +
1)...𝑁) ∩ {1}) =
∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁)) |
184 | 182, 183 | mpbir 220 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((1 +
1)...𝑁) ∩ {1}) =
∅ |
185 | | f1oun 6069 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 +
1)...𝑁) ∧ {〈𝑁, 1〉}:{𝑁}–1-1-onto→{1})
∧ (((1...(𝑁 − 1))
∩ {𝑁}) = ∅ ∧
(((1 + 1)...𝑁) ∩ {1}) =
∅)) → ((𝑛 ∈
(1...(𝑁 − 1)) ↦
(𝑛 + 1)) ∪ {〈𝑁, 1〉}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1
+ 1)...𝑁) ∪
{1})) |
186 | 184, 185 | mpanr2 716 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 +
1)...𝑁) ∧ {〈𝑁, 1〉}:{𝑁}–1-1-onto→{1})
∧ ((1...(𝑁 − 1))
∩ {𝑁}) = ∅)
→ ((𝑛 ∈
(1...(𝑁 − 1)) ↦
(𝑛 + 1)) ∪ {〈𝑁, 1〉}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1
+ 1)...𝑁) ∪
{1})) |
187 | 164, 166,
175, 186 | syl21anc 1317 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1
+ 1)...𝑁) ∪
{1})) |
188 | | ssv 3588 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ℕ
⊆ V |
189 | 188, 70 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ V) |
190 | 33 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 1 ∈
V) |
191 | 70, 65 | syl6eleq 2698 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
192 | 73, 191 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘1)) |
193 | | uzid 11578 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
194 | | peano2uz 11617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
195 | 122, 193,
194 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
196 | 73, 195 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
197 | | fzsplit2 12237 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
198 | 192, 196,
197 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
199 | 73 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁)) |
200 | | fzsn 12254 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) |
201 | 120, 200 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝑁...𝑁) = {𝑁}) |
202 | 199, 201 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁}) |
203 | 202 | uneq2d 3729 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
204 | 198, 203 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁)) |
205 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1) |
206 | 205 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1) |
207 | 189, 190,
204, 206 | fmptapd 6342 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {〈𝑁, 1〉}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) |
208 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1)))) |
209 | 208 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))) |
210 | 173, 209 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1)))) |
211 | 210 | necon2ad 2797 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁)) |
212 | 211 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁) |
213 | | ifnefalse 4048 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 ≠ 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
214 | 212, 213 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
215 | 214 | mpteq2dva 4672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))) |
216 | 215 | uneq1d 3728 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {〈𝑁, 1〉}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉})) |
217 | 207, 216 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉})) |
218 | 198, 203 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
219 | | uzid 11578 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1 ∈
ℤ → 1 ∈ (ℤ≥‘1)) |
220 | | peano2uz 11617 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1 ∈
(ℤ≥‘1) → (1 + 1) ∈
(ℤ≥‘1)) |
221 | 123, 219,
220 | mp2b 10 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (1 + 1)
∈ (ℤ≥‘1) |
222 | | fzsplit2 12237 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((1 + 1)
∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘1))
→ (1...𝑁) = ((1...1)
∪ ((1 + 1)...𝑁))) |
223 | 221, 191,
222 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁))) |
224 | | fzsn 12254 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (1 ∈
ℤ → (1...1) = {1}) |
225 | 123, 224 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1...1) =
{1} |
226 | 225 | uneq1i 3725 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((1...1)
∪ ((1 + 1)...𝑁)) = ({1}
∪ ((1 + 1)...𝑁)) |
227 | 226 | equncomi 3721 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1...1)
∪ ((1 + 1)...𝑁)) = (((1
+ 1)...𝑁) ∪
{1}) |
228 | 223, 227 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1})) |
229 | 217, 218,
228 | f1oeq123d 6046 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1
+ 1)...𝑁) ∪
{1}))) |
230 | 187, 229 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) |
231 | | f1oco 6072 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁)) |
232 | 45, 230, 231 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁)) |
233 | | dff1o3 6056 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))) |
234 | 233 | simprbi 479 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))) |
235 | 232, 234 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → Fun ◡((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))) |
236 | | imain 5888 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))) |
237 | 235, 236 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))) |
238 | 51 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
239 | 238 | ltp1d 10833 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1)) |
240 | | fzdisj 12239 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅) |
241 | 239, 240 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅) |
242 | 241 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅)) |
243 | | ima0 5400 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅) =
∅ |
244 | 242, 243 | syl6eq 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅) |
245 | 237, 244 | sylan9req 2665 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅) |
246 | | imassrn 5396 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) |
247 | | f1of 6050 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟶(1...𝑁)) |
248 | | frn 5966 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟶(1...𝑁) → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁)) |
249 | 230, 247,
248 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁)) |
250 | 246, 249 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁)) |
251 | 250 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁)) |
252 | | elfz1end 12242 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) |
253 | 70, 252 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ (1...𝑁)) |
254 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) |
255 | 205, 254,
33 | fvmpt 6191 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1) |
256 | 253, 255 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1) |
257 | 256 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1) |
258 | | f1ofn 6051 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁)) |
259 | 230, 258 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁)) |
260 | 259 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁)) |
261 | | fzss1 12251 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
262 | 66, 261 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
263 | 262 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
264 | | eluzfz2 12220 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁)) |
265 | 79, 264 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁)) |
266 | | fnfvima 6400 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) |
267 | 260, 263,
265, 266 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) |
268 | 257, 267 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) |
269 | | fnfvima 6400 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁) ∧ 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) → ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))) |
270 | 105, 251,
268, 269 | syl3anc 1318 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))) |
271 | | imaco 5557 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) |
272 | 270, 271 | syl6eleqr 2699 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘1) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) |
273 | | fnconstg 6006 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))) |
274 | 33, 273 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) |
275 | | fnconstg 6006 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
V → ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) |
276 | 36, 275 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) |
277 | | fvun2 6180 |
. . . . . . . . . . . . . . . . 17
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∧ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) ∧ (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘1) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘1))) |
278 | 274, 276,
277 | mp3an12 1406 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘1) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘1))) |
279 | 245, 272,
278 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘1))) |
280 | 36 | fvconst2 6374 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇))‘1) ∈ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘1)) = 0) |
281 | 272, 280 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘1)) = 0) |
282 | 279, 281 | eqtrd 2644 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = 0) |
283 | 282 | oveq2d 6565 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1 +
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1))) = (1 + 0)) |
284 | 102, 118,
283 | 3eqtr4a 2670 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)))) |
285 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘1) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1))) |
286 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘1) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1))) |
287 | 286 | oveq2d 6565 |
. . . . . . . . . . . . 13
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘1) → (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)))) |
288 | 285, 287 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘1) → ((((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1)) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘1))))) |
289 | 284, 288 | syl5ibrcom 236 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st
‘𝑇))‘1) →
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
290 | 289 | imp 444 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
291 | 290 | adantlr 747 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
292 | | eldifsn 4260 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘1))) |
293 | | df-ne 2782 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ≠ ((2nd
‘(1st ‘𝑇))‘1) ↔ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) |
294 | 293 | anbi2i 726 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘1)) ↔
(𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1))) |
295 | 292, 294 | bitri 263 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1))) |
296 | | fnconstg 6006 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) |
297 | 33, 296 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) |
298 | 297, 38 | pm3.2i 470 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
299 | | imain 5888 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
300 | 48, 299 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
301 | | fzdisj 12239 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
302 | 55, 301 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
303 | 302 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
304 | 303, 59 | syl6eq 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
305 | 300, 304 | sylan9req 2665 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
306 | | fnun 5911 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
307 | 298, 305,
306 | sylancr 694 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
308 | | imaundi 5464 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
309 | | fzpred 12259 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈
(ℤ≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁))) |
310 | 191, 309 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁))) |
311 | | uncom 3719 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({1}
∪ ((1 + 1)...𝑁)) = (((1
+ 1)...𝑁) ∪
{1}) |
312 | 310, 311 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1})) |
313 | 312 | difeq1d 3689 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((((1 + 1)...𝑁) ∪ {1}) ∖
{1})) |
314 | | difun2 4000 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((1 +
1)...𝑁) ∪ {1}) ∖
{1}) = (((1 + 1)...𝑁)
∖ {1}) |
315 | | disj3 3973 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((1 +
1)...𝑁) ∩ {1}) =
∅ ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1})) |
316 | 184, 315 | mpbi 219 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1 +
1)...𝑁) = (((1 + 1)...𝑁) ∖ {1}) |
317 | 314, 316 | eqtr4i 2635 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((1 +
1)...𝑁) ∪ {1}) ∖
{1}) = ((1 + 1)...𝑁) |
318 | 313, 317 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁)) |
319 | 318 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁)) |
320 | | eluzp1p1 11589 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1))) |
321 | 66, 320 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1))) |
322 | 321 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1))) |
323 | | fzsplit2 12237 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
324 | 322, 79, 323 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
325 | 319, 324 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
326 | 325 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))) |
327 | | imadif 5887 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {1}))) |
328 | 48, 327 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {1}))) |
329 | | eluzfz1 12219 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈
(ℤ≥‘1) → 1 ∈ (1...𝑁)) |
330 | 191, 329 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 1 ∈ (1...𝑁)) |
331 | | fnsnfv 6168 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ {1})) |
332 | 104, 330,
331 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ {1})) |
333 | 332 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {1}) = {((2nd
‘(1st ‘𝑇))‘1)}) |
334 | 85, 333 | difeq12d 3691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {1})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
335 | 328, 334 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
336 | 335 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
337 | 326, 336 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
338 | 308, 337 | syl5eqr 2658 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
339 | 338 | fneq2d 5896 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}))) |
340 | 307, 339 | mpbid 221 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
341 | | incom 3767 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ∖
{((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ({((2nd
‘(1st ‘𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
342 | | disjdif 3992 |
. . . . . . . . . . . . . . . 16
⊢
({((2nd ‘(1st ‘𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) = ∅ |
343 | 341, 342 | eqtri 2632 |
. . . . . . . . . . . . . . 15
⊢
(((1...𝑁) ∖
{((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ |
344 | | fnconstg 6006 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → ({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn
{((2nd ‘(1st ‘𝑇))‘1)}) |
345 | 33, 344 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn
{((2nd ‘(1st ‘𝑇))‘1)} |
346 | | fvun1 6179 |
. . . . . . . . . . . . . . . . 17
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∧ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}) Fn
{((2nd ‘(1st ‘𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}))) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
347 | 345, 346 | mp3an2 1404 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}))) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
348 | | fnconstg 6006 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
V → ({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn
{((2nd ‘(1st ‘𝑇))‘1)}) |
349 | 36, 348 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn
{((2nd ‘(1st ‘𝑇))‘1)} |
350 | | fvun1 6179 |
. . . . . . . . . . . . . . . . 17
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∧ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}) Fn
{((2nd ‘(1st ‘𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}))) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
351 | 349, 350 | mp3an2 1404 |
. . . . . . . . . . . . . . . 16
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}))) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) |
352 | 347, 351 | eqtr4d 2647 |
. . . . . . . . . . . . . . 15
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}))) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
353 | 343, 352 | mpanr1 715 |
. . . . . . . . . . . . . 14
⊢
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
354 | 340, 353 | sylan 487 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) → ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
355 | 295, 354 | sylan2br 492 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)))
→ ((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
356 | 355 | anassrs 678 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
((((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
357 | | fzpred 12259 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1)))) |
358 | 66, 357 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1)))) |
359 | 358 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
360 | 359 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
361 | 332 | uneq1d 3728 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) = (((2nd
‘(1st ‘𝑇)) “ {1}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))) |
362 | | uncom 3719 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) = ({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) |
363 | | imaundi 5464 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))) = (((2nd
‘(1st ‘𝑇)) “ {1}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) |
364 | 361, 362,
363 | 3eqtr4g 2669 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
365 | 364 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
366 | 360, 365 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)})) |
367 | 366 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) × {1})) |
368 | | xpundir 5095 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) × {1}) =
((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1})) |
369 | 367, 368 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))) |
370 | 369 | uneq1d 3728 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1})) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
371 | | un23 3734 |
. . . . . . . . . . . . . 14
⊢
(((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1})) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1})) |
372 | 370, 371 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))) |
373 | 372 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛)) |
374 | 373 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {1}))‘𝑛)) |
375 | | imaco 5557 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦))) |
376 | | df-ima 5051 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) |
377 | | peano2uz 11617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
378 | 75, 377 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
379 | 378 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
380 | 74, 379 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘𝑦)) |
381 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈
(ℤ≥‘𝑦) → (1...𝑦) ⊆ (1...𝑁)) |
382 | 380, 381 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...𝑁)) |
383 | 382 | resmptd 5371 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) |
384 | 173 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...(𝑁 − 1))) |
385 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → (1...𝑦) ⊆ (1...(𝑁 − 1))) |
386 | 75, 385 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) ⊆ (1...(𝑁 − 1))) |
387 | 386 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...(𝑁 − 1))) |
388 | 387 | sseld 3567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 ∈ (1...𝑦) → 𝑁 ∈ (1...(𝑁 − 1)))) |
389 | 384, 388 | mtod 188 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...𝑦)) |
390 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦))) |
391 | 390 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...𝑦) ↔ ¬ 𝑁 ∈ (1...𝑦))) |
392 | 389, 391 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...𝑦))) |
393 | 392 | necon2ad 2797 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) → 𝑛 ≠ 𝑁)) |
394 | 393 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ≠ 𝑁) |
395 | 394, 213 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
396 | 395 | mpteq2dva 4672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
397 | 383, 396 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
398 | 397 | rneqd 5274 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
399 | 376, 398 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
400 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑗 ∈ V |
401 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) |
402 | 401 | elrnmpt 5293 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))) |
403 | 400, 402 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)) |
404 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
405 | 404 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℤ) |
406 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ∈ (1...𝑦)) |
407 | 123 | jctl 562 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ℤ → (1 ∈
ℤ ∧ 𝑦 ∈
ℤ)) |
408 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...𝑦) → 𝑛 ∈ ℤ) |
409 | 408, 123 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 ∈ (1...𝑦) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
410 | | fzaddel 12246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((1
∈ ℤ ∧ 𝑦
∈ ℤ) ∧ (𝑛
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
411 | 407, 409,
410 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
412 | 406, 411 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))) |
413 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
414 | 412, 413 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
415 | 414 | rexlimdva 3013 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ℤ →
(∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
416 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℤ) |
417 | 416 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℂ) |
418 | | npcan1 10334 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗) |
419 | 417, 418 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
420 | 419 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (((𝑗 − 1) + 1) ∈ ((1 +
1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
421 | 420 | ibir 256 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) ∈ ((1 +
1)...(𝑦 +
1))) |
422 | 421 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) + 1) ∈ ((1 +
1)...(𝑦 +
1))) |
423 | | peano2zm 11297 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈
ℤ) |
424 | 416, 423 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑗 − 1) ∈
ℤ) |
425 | 424, 123 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) ∈ ℤ ∧
1 ∈ ℤ)) |
426 | | fzaddel 12246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((1
∈ ℤ ∧ 𝑦
∈ ℤ) ∧ ((𝑗
− 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
427 | 407, 425,
426 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
428 | 422, 427 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 − 1) ∈ (1...𝑦)) |
429 | 417 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 ∈
ℂ) |
430 | 418 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ℂ → 𝑗 = ((𝑗 − 1) + 1)) |
431 | 429, 430 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 = ((𝑗 − 1) + 1)) |
432 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = (𝑗 − 1) → (𝑛 + 1) = ((𝑗 − 1) + 1)) |
433 | 432 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = (𝑗 − 1) → (𝑗 = (𝑛 + 1) ↔ 𝑗 = ((𝑗 − 1) + 1))) |
434 | 433 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑗 − 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)) |
435 | 428, 431,
434 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)) |
436 | 435 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ℤ → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))) |
437 | 415, 436 | impbid 201 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ℤ →
(∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
438 | 405, 437 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
439 | 403, 438 | syl5bb 271 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
440 | 439 | eqrdv 2608 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1))) |
441 | 399, 440 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ((1 + 1)...(𝑦 + 1))) |
442 | 441 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦))) = ((2nd ‘(1st
‘𝑇)) “ ((1 +
1)...(𝑦 +
1)))) |
443 | 375, 442 | syl5eq 2656 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2nd ‘(1st
‘𝑇)) “ ((1 +
1)...(𝑦 +
1)))) |
444 | 443 | xpeq1d 5062 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1})) |
445 | | imaundi 5464 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1)))) |
446 | | imaco 5557 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) |
447 | | imaco 5557 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))) = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))) |
448 | 446, 447 | uneq12i 3727 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))) |
449 | 445, 448 | eqtri 2632 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))) |
450 | 196 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
451 | | fzsplit2 12237 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
452 | 78, 450, 451 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
453 | 202 | uneq2d 3729 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
454 | 453 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
455 | 452, 454 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
456 | | uncom 3719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))) |
457 | 455, 456 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) |
458 | 457 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))))) |
459 | 256 | sneqd 4137 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = {1}) |
460 | | fnsnfv 6168 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) |
461 | 259, 253,
460 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) |
462 | 459, 461 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) |
463 | 462 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {1}) = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))) |
464 | 332, 463 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))) |
465 | 464 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))) |
466 | | df-ima 5051 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) |
467 | | fzss1 12251 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1))) |
468 | 66, 467 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1))) |
469 | | fzss2 12252 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
470 | 196, 469 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
471 | 468, 470 | sylan9ssr 3582 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...𝑁)) |
472 | 471 | resmptd 5371 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) |
473 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
474 | 171, 473 | nsyl 134 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))) |
475 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝑁 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))) |
476 | 475 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))) |
477 | 474, 476 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)))) |
478 | 477 | con2d 128 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → ¬ 𝑛 = 𝑁)) |
479 | 478 | imp 444 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → ¬ 𝑛 = 𝑁) |
480 | 479 | iffalsed 4047 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
481 | 480 | mpteq2dva 4672 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
482 | 481 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
483 | 472, 482 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
484 | 483 | rneqd 5274 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
485 | 466, 484 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
486 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℤ) |
487 | 486 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℂ) |
488 | 487, 418 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
489 | 488 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
490 | 489 | ibir 256 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) |
491 | 490 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) |
492 | 53 | nnzd 11357 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ) |
493 | 122, 492 | anim12ci 589 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈
ℤ)) |
494 | 486, 423 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (𝑗 − 1) ∈ ℤ) |
495 | 494, 123 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) ∈ ℤ ∧
1 ∈ ℤ)) |
496 | | fzaddel 12246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑦 + 1) ∈ ℤ ∧
(𝑁 − 1) ∈
ℤ) ∧ ((𝑗 −
1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
497 | 493, 495,
496 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
498 | 491, 497 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → (𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1))) |
499 | 487, 430 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 = ((𝑗 − 1) + 1)) |
500 | 499 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → 𝑗 = ((𝑗 − 1) + 1)) |
501 | 433 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)) |
502 | 498, 500,
501 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)) |
503 | 502 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))) |
504 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) |
505 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑛 ∈ ℤ) |
506 | 505, 123 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
507 | | fzaddel 12246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑦 + 1) ∈ ℤ ∧
(𝑁 − 1) ∈
ℤ) ∧ (𝑛 ∈
ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
508 | 493, 506,
507 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
509 | 504, 508 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) |
510 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
511 | 509, 510 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
512 | 511 | rexlimdva 3013 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
513 | 503, 512 | impbid 201 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))) |
514 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) |
515 | 514 | elrnmpt 5293 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))) |
516 | 400, 515 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)) |
517 | 513, 516 | syl6bbr 277 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))) |
518 | 517 | eqrdv 2608 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
519 | 73 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁)) |
520 | 519 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁)) |
521 | 485, 518,
520 | 3eqtr2rd 2651 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))) |
522 | 521 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))) |
523 | 465, 522 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))) |
524 | 449, 458,
523 | 3eqtr4a 2670 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))) |
525 | 524 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0})) |
526 | | xpundir 5095 |
. . . . . . . . . . . . . . . 16
⊢
(({((2nd ‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}) = (({((2nd
‘(1st ‘𝑇))‘1)} × {0}) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
527 | 525, 526 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2nd
‘(1st ‘𝑇))‘1)} × {0}) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
528 | 444, 527 | uneq12d 3730 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
529 | | unass 3732 |
. . . . . . . . . . . . . . 15
⊢
(((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0})) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
530 | | un23 3734 |
. . . . . . . . . . . . . . 15
⊢
(((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0})) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0})) |
531 | 529, 530 | eqtr3i 2634 |
. . . . . . . . . . . . . 14
⊢
((((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘1)} × {0}) ∪
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0})) |
532 | 528, 531 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))) |
533 | 532 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
534 | 533 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘1)} × {0}))‘𝑛)) |
535 | 356, 374,
534 | 3eqtr4d 2654 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
536 | | snssi 4280 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
ℂ → {1} ⊆ ℂ) |
537 | 143, 536 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ {1}
⊆ ℂ |
538 | | 0cn 9911 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℂ |
539 | | snssi 4280 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
ℂ → {0} ⊆ ℂ) |
540 | 538, 539 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ {0}
⊆ ℂ |
541 | 537, 540 | unssi 3750 |
. . . . . . . . . . . . 13
⊢ ({1}
∪ {0}) ⊆ ℂ |
542 | 33 | fconst 6004 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} |
543 | 36 | fconst 6004 |
. . . . . . . . . . . . . . . . 17
⊢
((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0} |
544 | 542, 543 | pm3.2i 470 |
. . . . . . . . . . . . . . . 16
⊢
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} ∧ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}) |
545 | | fun 5979 |
. . . . . . . . . . . . . . . 16
⊢
(((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} ∧ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}) ∧ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0})) |
546 | 544, 245,
545 | sylancr 694 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0})) |
547 | | imaundi 5464 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) |
548 | 66 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈
(ℤ≥‘1)) |
549 | | fzsplit2 12237 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) |
550 | 548, 380,
549 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) |
551 | 550 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))) |
552 | | f1ofo 6057 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁)) |
553 | | foima 6033 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁)) |
554 | 232, 552,
553 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁)) |
555 | 554 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁)) |
556 | 551, 555 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁)) |
557 | 547, 556 | syl5eqr 2658 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = (1...𝑁)) |
558 | 557 | feq2d 5944 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}) ↔
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))) |
559 | 546, 558 | mpbid 221 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})) |
560 | 559 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0})) |
561 | 541, 560 | sseldi 3566 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ) |
562 | 561 | addid2d 10116 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (0 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
563 | 562 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
(0 + ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) |
564 | 535, 563 | eqtr4d 2647 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) →
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (0 + ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
565 | 98, 100, 291, 564 | ifbothda 4073 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
566 | 565 | oveq2d 6565 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st
‘𝑇))‘𝑛) + (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
567 | | elmapi 7765 |
. . . . . . . . . . . . 13
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
568 | 29, 567 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
569 | 568 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾)) |
570 | | elfzonn0 12380 |
. . . . . . . . . . 11
⊢
(((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈
ℕ0) |
571 | 569, 570 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈
ℕ0) |
572 | 571 | nn0cnd 11230 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ ℂ) |
573 | 572 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ ℂ) |
574 | 143, 538 | keepel 4105 |
. . . . . . . . 9
⊢ if(𝑛 = ((2nd
‘(1st ‘𝑇))‘1), 1, 0) ∈
ℂ |
575 | 574 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
∈ ℂ) |
576 | 573, 575,
561 | addassd 9941 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st
‘𝑇))‘𝑛) + (if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
577 | 566, 576 | eqtr4d 2647 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
578 | 577 | mpteq2dva 4672 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
579 | 96, 578 | eqtrd 2644 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
580 | | poimirlem18.4 |
. . . . . . . . . 10
⊢ (𝜑 → (2nd
‘𝑇) =
0) |
581 | 580 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑇) =
0) |
582 | | elfzle1 12215 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦) |
583 | 582 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦) |
584 | 581, 583 | eqbrtrd 4605 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑇) ≤ 𝑦) |
585 | | 0re 9919 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
586 | 580, 585 | syl6eqel 2696 |
. . . . . . . . 9
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℝ) |
587 | | lenlt 9995 |
. . . . . . . . 9
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd ‘𝑇))) |
588 | 586, 238,
587 | syl2an 493 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd ‘𝑇))) |
589 | 584, 588 | mpbid 221 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd
‘𝑇)) |
590 | 589 | iffalsed 4047 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1)) |
591 | 590 | csbeq1d 3506 |
. . . . 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 + 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
592 | | ovex 6577 |
. . . . . 6
⊢ (𝑦 + 1) ∈ V |
593 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1))) |
594 | 593 | imaeq2d 5385 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...(𝑦 +
1)))) |
595 | 594 | xpeq1d 5062 |
. . . . . . . 8
⊢ (𝑗 = (𝑦 + 1) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})) |
596 | | oveq1 6556 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1)) |
597 | 596 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁)) |
598 | 597 | imaeq2d 5385 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
599 | 598 | xpeq1d 5062 |
. . . . . . . 8
⊢ (𝑗 = (𝑦 + 1) → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
600 | 595, 599 | uneq12d 3730 |
. . . . . . 7
⊢ (𝑗 = (𝑦 + 1) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
601 | 600 | oveq2d 6565 |
. . . . . 6
⊢ (𝑗 = (𝑦 + 1) → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
602 | 592, 601 | csbie 3525 |
. . . . 5
⊢
⦋(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))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
603 | 591, 602 | syl6eq 2660 |
. . . 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))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
604 | | ovex 6577 |
. . . . . 6
⊢
(((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
∈ V |
605 | 604 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
∈ V) |
606 | | fvex 6113 |
. . . . . 6
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ V |
607 | 606 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ V) |
608 | | eqidd 2611 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1,
0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1,
0)))) |
609 | | ffn 5958 |
. . . . . . 7
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) →
(((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
610 | 559, 609 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
611 | | nfcv 2751 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(2nd ‘(1st
‘𝑇)) |
612 | | nfmpt1 4675 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) |
613 | 611, 612 | nfco 5209 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) |
614 | | nfcv 2751 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(1...𝑦) |
615 | 613, 614 | nfima 5393 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) |
616 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑛{1} |
617 | 615, 616 | nfxp 5066 |
. . . . . . . 8
⊢
Ⅎ𝑛((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) |
618 | | nfcv 2751 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((𝑦 + 1)...𝑁) |
619 | 613, 618 | nfima 5393 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) |
620 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑛{0} |
621 | 619, 620 | nfxp 5066 |
. . . . . . . 8
⊢
Ⅎ𝑛((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) |
622 | 617, 621 | nfun 3731 |
. . . . . . 7
⊢
Ⅎ𝑛(((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) |
623 | 622 | dffn5f 6162 |
. . . . . 6
⊢
((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
624 | 610, 623 | sylib 207 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) |
625 | 92, 605, 607, 608, 624 | offval2 6812 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1,
0))) ∘𝑓 + (((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
+ ((((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))) |
626 | 579, 603,
625 | 3eqtr4rd 2655 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1,
0))) ∘𝑓 + (((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 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})))) |
627 | 626 | mpteq2dva 4672 |
. 2
⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1,
0))) ∘𝑓 + (((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 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}))))) |
628 | 22, 627 | eqtr4d 2647 |
1
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1,
0))) ∘𝑓 + (((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))) |