Proof of Theorem poimirlem16
Step | Hyp | Ref
| Expression |
1 | | poimirlem22.2 |
. . 3
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | fveq2 6499 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
3 | 2 | breq2d 4941 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
4 | 3 | ifbid 4372 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
5 | 4 | csbeq1d 3793 |
. . . . . . . 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 | | 2fveq3 6504 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
7 | | 2fveq3 6504 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
8 | 7 | imaeq1d 5769 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
9 | 8 | xpeq1d 5436 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
10 | 7 | imaeq1d 5769 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
11 | 10 | xpeq1d 5436 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
12 | 9, 11 | uneq12d 4029 |
. . . . . . . . . 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}))) |
13 | 6, 12 | oveq12d 6994 |
. . . . . . . . 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})))) |
14 | 13 | csbeq2dv 4256 |
. . . . . . . 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})))) |
15 | 5, 14 | eqtrd 2814 |
. . . . . . 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})))) |
16 | 15 | mpteq2dv 5023 |
. . . . . 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}))))) |
17 | 16 | eqeq2d 2788 |
. . . . 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})))))) |
18 | | 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}))))} |
19 | 17, 18 | elrab2 3599 |
. . . 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})))))) |
20 | 19 | simprbi 489 |
. . 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}))))) |
21 | 1, 20 | 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}))))) |
22 | | elrabi 3590 |
. . . . . . . . . . . 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))) |
23 | 22, 18 | eleq2s 2884 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
24 | 1, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
25 | | xp1st 7533 |
. . . . . . . . . 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)})) |
26 | 24, 25 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
27 | | xp1st 7533 |
. . . . . . . . 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))) |
28 | 26, 27 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
29 | | elmapfn 8229 |
. . . . . . . 8
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
30 | 28, 29 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
31 | 30 | adantr 473 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
32 | | 1ex 10435 |
. . . . . . . . . 10
⊢ 1 ∈
V |
33 | | fnconstg 6396 |
. . . . . . . . . 10
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) |
34 | 32, 33 | ax-mp 5 |
. . . . . . . . 9
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) |
35 | | c0ex 10433 |
. . . . . . . . . 10
⊢ 0 ∈
V |
36 | | fnconstg 6396 |
. . . . . . . . . 10
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
37 | 35, 36 | ax-mp 5 |
. . . . . . . . 9
⊢
(((2nd ‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) |
38 | 34, 37 | pm3.2i 463 |
. . . . . . . 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))) |
39 | | xp2nd 7534 |
. . . . . . . . . . . . 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)}) |
40 | 26, 39 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
41 | | fvex 6512 |
. . . . . . . . . . . . 13
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
42 | | f1oeq1 6433 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
43 | 41, 42 | elab 3582 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
44 | 40, 43 | sylib 210 |
. . . . . . . . . . 11
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
45 | | dff1o3 6450 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
46 | 45 | simprbi 489 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
47 | 44, 46 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
48 | | imain 6272 |
. . . . . . . . . 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)))) |
49 | 47, 48 | 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)))) |
50 | | elfznn0 12816 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0) |
51 | | nn0p1nn 11748 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ0
→ (𝑦 + 1) ∈
ℕ) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ) |
53 | 52 | nnred 11456 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ) |
54 | 53 | ltp1d 11371 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1)) |
55 | | fzdisj 12750 |
. . . . . . . . . . . 12
⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
56 | 54, 55 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
57 | 56 | imaeq2d 5770 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
58 | | ima0 5785 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
59 | 57, 58 | syl6eq 2830 |
. . . . . . . . 9
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
60 | 49, 59 | sylan9req 2835 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
61 | | fnun 6296 |
. . . . . . . 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)))) |
62 | 38, 60, 61 | sylancr 578 |
. . . . . . 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)))) |
63 | | imaundi 5848 |
. . . . . . . . 9
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
64 | | nnuz 12095 |
. . . . . . . . . . . . . . 15
⊢ ℕ =
(ℤ≥‘1) |
65 | 52, 64 | syl6eleq 2876 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈
(ℤ≥‘1)) |
66 | | peano2uz 12115 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
67 | 65, 66 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
68 | 67 | adantl 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘1)) |
69 | | poimir.0 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℕ) |
70 | 69 | nncnd 11457 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℂ) |
71 | | npcan1 10866 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
72 | 70, 71 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
73 | 72 | adantr 473 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
74 | | elfzuz3 12721 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ≥‘𝑦)) |
75 | | eluzp1p1 12084 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
76 | 74, 75 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
77 | 76 | adantl 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1))) |
78 | 73, 77 | eqeltrrd 2867 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) |
79 | | fzsplit2 12748 |
. . . . . . . . . . . 12
⊢ ((((𝑦 + 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
80 | 68, 78, 79 | syl2anc 576 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
81 | 80 | imaeq2d 5770 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...(𝑦 + 1)) ∪
(((𝑦 + 1) + 1)...𝑁)))) |
82 | | f1ofo 6451 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
83 | | foima 6424 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
84 | 44, 82, 83 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
85 | 84 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
86 | 81, 85 | eqtr3d 2816 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁)) |
87 | 63, 86 | syl5eqr 2828 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁)) |
88 | 87 | fneq2d 6280 |
. . . . . . 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))) |
89 | 62, 88 | mpbid 224 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
90 | | ovexd 7010 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V) |
91 | | inidm 4082 |
. . . . . 6
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
92 | | eqidd 2779 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st
‘𝑇))‘𝑛)) |
93 | | eqidd 2779 |
. . . . . 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)) |
94 | 31, 89, 90, 90, 91, 92, 93 | offval 7234 |
. . . . 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)))) |
95 | | oveq1 6983 |
. . . . . . . . . 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))) |
96 | 95 | eqeq2d 2788 |
. . . . . . . . 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)))) |
97 | | oveq1 6983 |
. . . . . . . . . 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))) |
98 | 97 | eqeq2d 2788 |
. . . . . . . . 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)))) |
99 | | 1p0e1 11571 |
. . . . . . . . . . . . . 14
⊢ (1 + 0) =
1 |
100 | 99 | eqcomi 2787 |
. . . . . . . . . . . . 13
⊢ 1 = (1 +
0) |
101 | | f1ofn 6445 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
102 | 44, 101 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
103 | 102 | adantr 473 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
104 | | fzss2 12763 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
105 | 78, 104 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
106 | | eluzfz1 12730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → 1 ∈ (1...(𝑦 + 1))) |
107 | 65, 106 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1))) |
108 | 107 | adantl 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1))) |
109 | | fnfvima 6820 |
. . . . . . . . . . . . . . . 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)))) |
110 | 103, 105,
108, 109 | syl3anc 1351 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1)))) |
111 | | fvun1 6582 |
. . . . . . . . . . . . . . . 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))) |
112 | 34, 37, 111 | mp3an12 1430 |
. . . . . . . . . . . . . . 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))) |
113 | 60, 110, 112 | syl2anc 576 |
. . . . . . . . . . . . . 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))) |
114 | 32 | fvconst2 6793 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘(1st ‘𝑇))‘1) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1)) = 1) |
115 | 110, 114 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd
‘(1st ‘𝑇))‘1)) = 1) |
116 | 113, 115 | eqtrd 2814 |
. . . . . . . . . . . . 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) |
117 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1))) |
118 | 69 | nnzd 11899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝑁 ∈ ℤ) |
119 | | peano2zm 11838 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
120 | 118, 119 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
121 | | 1z 11825 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 1 ∈
ℤ |
122 | 120, 121 | jctil 512 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (1 ∈ ℤ ∧
(𝑁 − 1) ∈
ℤ)) |
123 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ) |
124 | 123, 121 | jctir 513 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
125 | | fzaddel 12757 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((1
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
→ (𝑛 ∈
(1...(𝑁 − 1)) ↔
(𝑛 + 1) ∈ ((1 +
1)...((𝑁 − 1) +
1)))) |
126 | 122, 124,
125 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))) |
127 | 117, 126 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))) |
128 | 72 | oveq2d 6992 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 +
1)...𝑁)) |
129 | 128 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 +
1)...𝑁)) |
130 | 127, 129 | eleqtrd 2868 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁)) |
131 | 130 | ralrimiva 3132 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁)) |
132 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁)) |
133 | | peano2z 11836 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (1 ∈
ℤ → (1 + 1) ∈ ℤ) |
134 | 121, 133 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (1 + 1)
∈ ℤ |
135 | 118, 134 | jctil 512 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → ((1 + 1) ∈ ℤ
∧ 𝑁 ∈
ℤ)) |
136 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ) |
137 | 136, 121 | jctir 513 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈
ℤ)) |
138 | | fzsubel 12759 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((1 +
1) ∈ ℤ ∧ 𝑁
∈ ℤ) ∧ (𝑦
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1)))) |
139 | 135, 137,
138 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1)))) |
140 | 132, 139 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) −
1)...(𝑁 −
1))) |
141 | | ax-1cn 10393 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 1 ∈
ℂ |
142 | 141, 141 | pncan3oi 10703 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((1 + 1)
− 1) = 1 |
143 | 142 | oveq1i 6986 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((1 + 1)
− 1)...(𝑁 − 1))
= (1...(𝑁 −
1)) |
144 | 140, 143 | syl6eleq 2876 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1))) |
145 | 136 | zcnd 11901 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ) |
146 | | elfznn 12752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ) |
147 | 146 | nncnd 11457 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ) |
148 | | subadd2 10690 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑦 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝑛 ∈
ℂ) → ((𝑦 −
1) = 𝑛 ↔ (𝑛 + 1) = 𝑦)) |
149 | 141, 148 | mp3an2 1428 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦)) |
150 | 149 | bicomd 215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛)) |
151 | | eqcom 2785 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑛 + 1) = 𝑦 ↔ 𝑦 = (𝑛 + 1)) |
152 | | eqcom 2785 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑦 − 1) = 𝑛 ↔ 𝑛 = (𝑦 − 1)) |
153 | 150, 151,
152 | 3bitr3g 305 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
154 | 145, 147,
153 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
155 | 154 | ralrimiva 3132 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
156 | 155 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) |
157 | | reu6i 3631 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)) |
158 | 144, 156,
157 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)) |
159 | 158 | ralrimiva 3132 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)) |
160 | | eqid 2778 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) |
161 | 160 | f1ompt 6698 |
. . . . . . . . . . . . . . . . . . . . . . 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))) |
162 | 131, 159,
161 | sylanbrc 575 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 +
1)...𝑁)) |
163 | | f1osng 6484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑁 ∈ ℕ ∧ 1 ∈
V) → {〈𝑁,
1〉}:{𝑁}–1-1-onto→{1}) |
164 | 69, 32, 163 | sylancl 577 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {〈𝑁, 1〉}:{𝑁}–1-1-onto→{1}) |
165 | 69 | nnred 11456 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ ℝ) |
166 | 165 | ltm1d 11373 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
167 | 120 | zred 11900 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
168 | 167, 165 | ltnled 10587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1))) |
169 | 166, 168 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1)) |
170 | | elfzle2 12727 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
171 | 169, 170 | nsyl 138 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1))) |
172 | | disjsn 4521 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(1...(𝑁 −
1))) |
173 | 171, 172 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
174 | | 1re 10439 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 1 ∈
ℝ |
175 | 174 | ltp1i 11345 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 1 < (1
+ 1) |
176 | 174, 174 | readdcli 10455 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (1 + 1)
∈ ℝ |
177 | 174, 176 | ltnlei 10561 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (1 <
(1 + 1) ↔ ¬ (1 + 1) ≤ 1) |
178 | 175, 177 | mpbi 222 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ¬ (1
+ 1) ≤ 1 |
179 | | elfzle1 12726 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1 ∈
((1 + 1)...𝑁) → (1 +
1) ≤ 1) |
180 | 178, 179 | mto 189 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ¬ 1
∈ ((1 + 1)...𝑁) |
181 | | disjsn 4521 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((1 +
1)...𝑁) ∩ {1}) =
∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁)) |
182 | 180, 181 | mpbir 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((1 +
1)...𝑁) ∩ {1}) =
∅ |
183 | | f1oun 6463 |
. . . . . . . . . . . . . . . . . . . . . . 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})) |
184 | 182, 183 | mpanr2 691 |
. . . . . . . . . . . . . . . . . . . . . 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})) |
185 | 162, 164,
173, 184 | syl21anc 825 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1
+ 1)...𝑁) ∪
{1})) |
186 | | ssv 3881 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ℕ
⊆ V |
187 | 186, 69 | sseldi 3856 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ V) |
188 | 32 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 1 ∈
V) |
189 | 69, 64 | syl6eleq 2876 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
190 | 72, 189 | eqeltrd 2866 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘1)) |
191 | | uzid 12073 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
192 | | peano2uz 12115 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
193 | 120, 191,
192 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
194 | 72, 193 | eqeltrrd 2867 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
195 | | fzsplit2 12748 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
196 | 190, 194,
195 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
197 | 72 | oveq1d 6991 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁)) |
198 | | fzsn 12765 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) |
199 | 118, 198 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝑁...𝑁) = {𝑁}) |
200 | 197, 199 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁}) |
201 | 200 | uneq2d 4028 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
202 | 196, 201 | eqtr2d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁)) |
203 | | iftrue 4356 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1) |
204 | 203 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1) |
205 | 187, 188,
202, 204 | fmptapd 6756 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {〈𝑁, 1〉}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) |
206 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1)))) |
207 | 206 | notbid 310 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))) |
208 | 171, 207 | syl5ibrcom 239 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1)))) |
209 | 208 | necon2ad 2982 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁)) |
210 | 209 | imp 398 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁) |
211 | | ifnefalse 4362 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 ≠ 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
212 | 210, 211 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
213 | 212 | mpteq2dva 5022 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))) |
214 | 213 | uneq1d 4027 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {〈𝑁, 1〉}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉})) |
215 | 205, 214 | eqtr3d 2816 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {〈𝑁, 1〉})) |
216 | 196, 201 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
217 | | uzid 12073 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1 ∈
ℤ → 1 ∈ (ℤ≥‘1)) |
218 | | peano2uz 12115 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1 ∈
(ℤ≥‘1) → (1 + 1) ∈
(ℤ≥‘1)) |
219 | 121, 217,
218 | mp2b 10 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (1 + 1)
∈ (ℤ≥‘1) |
220 | | fzsplit2 12748 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((1 + 1)
∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘1))
→ (1...𝑁) = ((1...1)
∪ ((1 + 1)...𝑁))) |
221 | 219, 189,
220 | sylancr 578 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁))) |
222 | | fzsn 12765 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (1 ∈
ℤ → (1...1) = {1}) |
223 | 121, 222 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (1...1) =
{1} |
224 | 223 | uneq1i 4024 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((1...1)
∪ ((1 + 1)...𝑁)) = ({1}
∪ ((1 + 1)...𝑁)) |
225 | 224 | equncomi 4020 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1...1)
∪ ((1 + 1)...𝑁)) = (((1
+ 1)...𝑁) ∪
{1}) |
226 | 221, 225 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1})) |
227 | 215, 216,
226 | f1oeq123d 6439 |
. . . . . . . . . . . . . . . . . . . . 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}))) |
228 | 185, 227 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) |
229 | | f1oco 6466 |
. . . . . . . . . . . . . . . . . . . 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)) |
230 | 44, 228, 229 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁)) |
231 | | dff1o3 6450 |
. . . . . . . . . . . . . . . . . . . 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)))))) |
232 | 231 | simprbi 489 |
. . . . . . . . . . . . . . . . . . 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))))) |
233 | 230, 232 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → Fun ◡((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))) |
234 | | imain 6272 |
. . . . . . . . . . . . . . . . . 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)))) |
235 | 233, 234 | 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)))) |
236 | 50 | nn0red 11768 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
237 | 236 | ltp1d 11371 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1)) |
238 | | fzdisj 12750 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅) |
239 | 237, 238 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅) |
240 | 239 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . 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)))) “ ∅)) |
241 | | ima0 5785 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅) =
∅ |
242 | 240, 241 | syl6eq 2830 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅) |
243 | 235, 242 | sylan9req 2835 |
. . . . . . . . . . . . . . . 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))) = ∅) |
244 | | imassrn 5781 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) |
245 | | f1of 6444 |
. . . . . . . . . . . . . . . . . . . . 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)) |
246 | | frn 6350 |
. . . . . . . . . . . . . . . . . . . . 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)) |
247 | 228, 245,
246 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁)) |
248 | 244, 247 | syl5ss 3869 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁)) |
249 | 248 | adantr 473 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁)) |
250 | | elfz1end 12753 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) |
251 | 69, 250 | sylib 210 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ (1...𝑁)) |
252 | | eqid 2778 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) |
253 | 203, 252,
32 | fvmpt 6595 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1) |
254 | 251, 253 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1) |
255 | 254 | adantr 473 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1) |
256 | | f1ofn 6445 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
257 | 228, 256 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁)) |
258 | 257 | adantr 473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁)) |
259 | | fzss1 12762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
260 | 65, 259 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
261 | 260 | adantl 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
262 | | eluzfz2 12731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁)) |
263 | 78, 262 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁)) |
264 | | fnfvima 6820 |
. . . . . . . . . . . . . . . . . . . 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))) |
265 | 258, 261,
263, 264 | syl3anc 1351 |
. . . . . . . . . . . . . . . . . . 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))) |
266 | 255, 265 | eqeltrrd 2867 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) |
267 | | fnfvima 6820 |
. . . . . . . . . . . . . . . . . 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)))) |
268 | 103, 249,
266, 267 | syl3anc 1351 |
. . . . . . . . . . . . . . . . 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)))) |
269 | | imaco 5943 |
. . . . . . . . . . . . . . . . 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))) |
270 | 268, 269 | syl6eleqr 2877 |
. . . . . . . . . . . . . . . 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))) |
271 | | fnconstg 6396 |
. . . . . . . . . . . . . . . . . 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))) |
272 | 32, 271 | 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)) |
273 | | fnconstg 6396 |
. . . . . . . . . . . . . . . . . 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))) |
274 | 35, 273 | 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)) |
275 | | fvun2 6583 |
. . . . . . . . . . . . . . . . 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))) |
276 | 272, 274,
275 | mp3an12 1430 |
. . . . . . . . . . . . . . . 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))) |
277 | 243, 270,
276 | syl2anc 576 |
. . . . . . . . . . . . . . 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))) |
278 | 35 | fvconst2 6793 |
. . . . . . . . . . . . . . . 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) |
279 | 270, 278 | 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) |
280 | 277, 279 | eqtrd 2814 |
. . . . . . . . . . . . . 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) |
281 | 280 | oveq2d 6992 |
. . . . . . . . . . . . 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)) |
282 | 100, 116,
281 | 3eqtr4a 2840 |
. . . . . . . . . . . 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)))) |
283 | | fveq2 6499 |
. . . . . . . . . . . . 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))) |
284 | | fveq2 6499 |
. . . . . . . . . . . . . 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))) |
285 | 284 | oveq2d 6992 |
. . . . . . . . . . . . 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)))) |
286 | 283, 285 | eqeq12d 2793 |
. . . . . . . . . . . 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))))) |
287 | 282, 286 | syl5ibrcom 239 |
. . . . . . . . . . 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)))) |
288 | 287 | imp 398 |
. . . . . . . . . 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))) |
289 | 288 | adantlr 702 |
. . . . . . . . 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))) |
290 | | eldifsn 4593 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘1))) |
291 | | df-ne 2968 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ≠ ((2nd
‘(1st ‘𝑇))‘1) ↔ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1)) |
292 | 291 | anbi2i 613 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘1)) ↔
(𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1))) |
293 | 290, 292 | bitri 267 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st
‘𝑇))‘1))) |
294 | | fnconstg 6396 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) |
295 | 32, 294 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) |
296 | 295, 37 | pm3.2i 463 |
. . . . . . . . . . . . . . . 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))) |
297 | | imain 6272 |
. . . . . . . . . . . . . . . . . 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)))) |
298 | 47, 297 | 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)))) |
299 | | fzdisj 12750 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
300 | 54, 299 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
301 | 300 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
302 | 301, 58 | syl6eq 2830 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
303 | 298, 302 | sylan9req 2835 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) |
304 | | fnun 6296 |
. . . . . . . . . . . . . . . 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)))) |
305 | 296, 303,
304 | sylancr 578 |
. . . . . . . . . . . . . . 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)))) |
306 | | imaundi 5848 |
. . . . . . . . . . . . . . . . 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))) |
307 | | fzpred 12771 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈
(ℤ≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁))) |
308 | 189, 307 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁))) |
309 | | uncom 4018 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({1}
∪ ((1 + 1)...𝑁)) = (((1
+ 1)...𝑁) ∪
{1}) |
310 | 308, 309 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1})) |
311 | 310 | difeq1d 3988 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((((1 + 1)...𝑁) ∪ {1}) ∖
{1})) |
312 | | difun2 4312 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((1 +
1)...𝑁) ∪ {1}) ∖
{1}) = (((1 + 1)...𝑁)
∖ {1}) |
313 | | disj3 4286 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((1 +
1)...𝑁) ∩ {1}) =
∅ ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1})) |
314 | 182, 313 | mpbi 222 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1 +
1)...𝑁) = (((1 + 1)...𝑁) ∖ {1}) |
315 | 312, 314 | eqtr4i 2805 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((1 +
1)...𝑁) ∪ {1}) ∖
{1}) = ((1 + 1)...𝑁) |
316 | 311, 315 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁)) |
317 | 316 | adantr 473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁)) |
318 | | eluzp1p1 12084 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1))) |
319 | 65, 318 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1))) |
320 | 319 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1))) |
321 | | fzsplit2 12748 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑦 + 1) + 1) ∈
(ℤ≥‘(1 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑦 + 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
322 | 320, 78, 321 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
323 | 317, 322 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) |
324 | 323 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))) |
325 | | imadif 6271 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {1}))) |
326 | 47, 325 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {1}))) |
327 | | eluzfz1 12730 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈
(ℤ≥‘1) → 1 ∈ (1...𝑁)) |
328 | 189, 327 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 1 ∈ (1...𝑁)) |
329 | | fnsnfv 6571 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ {1})) |
330 | 102, 328,
329 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ {1})) |
331 | 330 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {1}) = {((2nd
‘(1st ‘𝑇))‘1)}) |
332 | 84, 331 | difeq12d 3990 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {1})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
333 | 326, 332 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
334 | 333 | adantr 473 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
335 | 324, 334 | eqtr3d 2816 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
336 | 306, 335 | syl5eqr 2828 |
. . . . . . . . . . . . . . . 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)})) |
337 | 336 | fneq2d 6280 |
. . . . . . . . . . . . . . 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)}))) |
338 | 305, 337 | mpbid 224 |
. . . . . . . . . . . . . 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)})) |
339 | | incom 4066 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ∖
{((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ({((2nd
‘(1st ‘𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) |
340 | | disjdif 4304 |
. . . . . . . . . . . . . . . 16
⊢
({((2nd ‘(1st ‘𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘1)})) = ∅ |
341 | 339, 340 | eqtri 2802 |
. . . . . . . . . . . . . . 15
⊢
(((1...𝑁) ∖
{((2nd ‘(1st ‘𝑇))‘1)}) ∩ {((2nd
‘(1st ‘𝑇))‘1)}) = ∅ |
342 | | fnconstg 6396 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
V → ({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn
{((2nd ‘(1st ‘𝑇))‘1)}) |
343 | 32, 342 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
({((2nd ‘(1st ‘𝑇))‘1)} × {1}) Fn
{((2nd ‘(1st ‘𝑇))‘1)} |
344 | | fvun1 6582 |
. . . . . . . . . . . . . . . . 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)) |
345 | 343, 344 | mp3an2 1428 |
. . . . . . . . . . . . . . . 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)) |
346 | | fnconstg 6396 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ∈
V → ({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn
{((2nd ‘(1st ‘𝑇))‘1)}) |
347 | 35, 346 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢
({((2nd ‘(1st ‘𝑇))‘1)} × {0}) Fn
{((2nd ‘(1st ‘𝑇))‘1)} |
348 | | fvun1 6582 |
. . . . . . . . . . . . . . . . 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)) |
349 | 347, 348 | mp3an2 1428 |
. . . . . . . . . . . . . . . 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)) |
350 | 345, 349 | eqtr4d 2817 |
. . . . . . . . . . . . . . 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)) |
351 | 341, 350 | mpanr1 690 |
. . . . . . . . . . . . . 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)) |
352 | 338, 351 | sylan 572 |
. . . . . . . . . . . . 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)) |
353 | 293, 352 | sylan2br 585 |
. . . . . . . . . . . 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)) |
354 | 353 | anassrs 460 |
. . . . . . . . . . 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)) |
355 | | fzpred 12771 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1)))) |
356 | 65, 355 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1)))) |
357 | 356 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
358 | 357 | adantl 474 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
359 | 330 | uneq1d 4027 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) = (((2nd
‘(1st ‘𝑇)) “ {1}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))))) |
360 | | uncom 4018 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) = ({((2nd
‘(1st ‘𝑇))‘1)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) |
361 | | imaundi 5848 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))) = (((2nd
‘(1st ‘𝑇)) “ {1}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1)))) |
362 | 359, 360,
361 | 3eqtr4g 2839 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)}) = ((2nd
‘(1st ‘𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1))))) |
363 | 362 | adantr 473 |
. . . . . . . . . . . . . . . . . 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))))) |
364 | 358, 363 | eqtr4d 2817 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) = (((2nd
‘(1st ‘𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd
‘(1st ‘𝑇))‘1)})) |
365 | 364 | xpeq1d 5436 |
. . . . . . . . . . . . . . . 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})) |
366 | | xpundir 5471 |
. . . . . . . . . . . . . . . 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})) |
367 | 365, 366 | syl6eq 2830 |
. . . . . . . . . . . . . . 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}))) |
368 | 367 | uneq1d 4027 |
. . . . . . . . . . . . . 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}))) |
369 | | un23 4033 |
. . . . . . . . . . . . . 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})) |
370 | 368, 369 | syl6eq 2830 |
. . . . . . . . . . . . 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}))) |
371 | 370 | fveq1d 6501 |
. . . . . . . . . . . 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)) |
372 | 371 | ad2antrr 713 |
. . . . . . . . . . 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)) |
373 | | imaco 5943 |
. . . . . . . . . . . . . . . . 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))) |
374 | | df-ima 5420 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) |
375 | | peano2uz 12115 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
376 | 74, 375 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
377 | 376 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘𝑦)) |
378 | 73, 377 | eqeltrrd 2867 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘𝑦)) |
379 | | fzss2 12763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈
(ℤ≥‘𝑦) → (1...𝑦) ⊆ (1...𝑁)) |
380 | 378, 379 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...𝑁)) |
381 | 380 | resmptd 5753 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
382 | 171 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...(𝑁 − 1))) |
383 | | fzss2 12763 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑦) → (1...𝑦) ⊆ (1...(𝑁 − 1))) |
384 | 74, 383 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) ⊆ (1...(𝑁 − 1))) |
385 | 384 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...(𝑁 − 1))) |
386 | 385 | sseld 3857 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 ∈ (1...𝑦) → 𝑁 ∈ (1...(𝑁 − 1)))) |
387 | 382, 386 | mtod 190 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...𝑦)) |
388 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦))) |
389 | 388 | notbid 310 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...𝑦) ↔ ¬ 𝑁 ∈ (1...𝑦))) |
390 | 387, 389 | syl5ibrcom 239 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...𝑦))) |
391 | 390 | necon2ad 2982 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) → 𝑛 ≠ 𝑁)) |
392 | 391 | imp 398 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ≠ 𝑁) |
393 | 392, 211 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
394 | 393 | mpteq2dva 5022 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
395 | 381, 394 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
396 | 395 | rneqd 5651 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
397 | 374, 396 | syl5eq 2826 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))) |
398 | | vex 3418 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑗 ∈ V |
399 | | eqid 2778 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) |
400 | 399 | elrnmpt 5671 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))) |
401 | 398, 400 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)) |
402 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
403 | 402 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℤ) |
404 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ∈ (1...𝑦)) |
405 | 121 | jctl 516 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ℤ → (1 ∈
ℤ ∧ 𝑦 ∈
ℤ)) |
406 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...𝑦) → 𝑛 ∈ ℤ) |
407 | 406, 121 | jctir 513 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 ∈ (1...𝑦) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
408 | | fzaddel 12757 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((1
∈ ℤ ∧ 𝑦
∈ ℤ) ∧ (𝑛
∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
409 | 405, 407,
408 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
410 | 404, 409 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))) |
411 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
412 | 410, 411 | syl5ibrcom 239 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
413 | 412 | rexlimdva 3229 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ℤ →
(∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
414 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℤ) |
415 | 414 | zcnd 11901 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℂ) |
416 | | npcan1 10866 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗) |
417 | 415, 416 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
418 | 417 | eleq1d 2850 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (((𝑗 − 1) + 1) ∈ ((1 +
1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
419 | 418 | ibir 260 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) ∈ ((1 +
1)...(𝑦 +
1))) |
420 | 419 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) + 1) ∈ ((1 +
1)...(𝑦 +
1))) |
421 | | peano2zm 11838 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈
ℤ) |
422 | 414, 421 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑗 − 1) ∈
ℤ) |
423 | 422, 121 | jctir 513 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) ∈ ℤ ∧
1 ∈ ℤ)) |
424 | | fzaddel 12757 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((1
∈ ℤ ∧ 𝑦
∈ ℤ) ∧ ((𝑗
− 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
425 | 405, 423,
424 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))) |
426 | 420, 425 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 − 1) ∈ (1...𝑦)) |
427 | 415 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 ∈
ℂ) |
428 | 416 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ℂ → 𝑗 = ((𝑗 − 1) + 1)) |
429 | 427, 428 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 = ((𝑗 − 1) + 1)) |
430 | | oveq1 6983 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = (𝑗 − 1) → (𝑛 + 1) = ((𝑗 − 1) + 1)) |
431 | 430 | rspceeqv 3553 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑗 − 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)) |
432 | 426, 429,
431 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)) |
433 | 432 | ex 405 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ℤ → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))) |
434 | 413, 433 | impbid 204 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ℤ →
(∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
435 | 403, 434 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
436 | 401, 435 | syl5bb 275 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1)))) |
437 | 436 | eqrdv 2776 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1))) |
438 | 397, 437 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ((1 + 1)...(𝑦 + 1))) |
439 | 438 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . 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)))) |
440 | 373, 439 | syl5eq 2826 |
. . . . . . . . . . . . . . . 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)))) |
441 | 440 | xpeq1d 5436 |
. . . . . . . . . . . . . . 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})) |
442 | | imaundi 5848 |
. . . . . . . . . . . . . . . . . . 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)))) |
443 | | imaco 5943 |
. . . . . . . . . . . . . . . . . . . 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})) |
444 | | imaco 5943 |
. . . . . . . . . . . . . . . . . . . 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)))) |
445 | 443, 444 | uneq12i 4026 |
. . . . . . . . . . . . . . . . . . 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))))) |
446 | 442, 445 | eqtri 2802 |
. . . . . . . . . . . . . . . . . 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))))) |
447 | 194 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
448 | | fzsplit2 12748 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
449 | 77, 447, 448 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁))) |
450 | 200 | uneq2d 4028 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
451 | 450 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
452 | 449, 451 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) |
453 | | uncom 4018 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))) |
454 | 452, 453 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) |
455 | 454 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . 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))))) |
456 | 254 | sneqd 4453 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = {1}) |
457 | | fnsnfv 6571 |
. . . . . . . . . . . . . . . . . . . . . . . 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})) |
458 | 257, 251,
457 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) |
459 | 456, 458 | eqtr3d 2816 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) |
460 | 459 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {1}) = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))) |
461 | 330, 460 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))) |
462 | 461 | adantr 473 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → {((2nd
‘(1st ‘𝑇))‘1)} = ((2nd
‘(1st ‘𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))) |
463 | | df-ima 5420 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
464 | | fzss1 12762 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑦 + 1) ∈
(ℤ≥‘1) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1))) |
465 | 65, 464 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1))) |
466 | | fzss2 12763 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
467 | 194, 466 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
468 | 465, 467 | sylan9ssr 3872 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...𝑁)) |
469 | 468 | resmptd 5753 |
. . . . . . . . . . . . . . . . . . . . . . . 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)))) |
470 | | elfzle2 12727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
471 | 169, 470 | nsyl 138 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))) |
472 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝑁 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))) |
473 | 472 | notbid 310 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))) |
474 | 471, 473 | syl5ibrcom 239 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)))) |
475 | 474 | con2d 132 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → ¬ 𝑛 = 𝑁)) |
476 | 475 | imp 398 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → ¬ 𝑛 = 𝑁) |
477 | 476 | iffalsed 4361 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1)) |
478 | 477 | mpteq2dva 5022 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
479 | 478 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
480 | 469, 479 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . 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))) |
481 | 480 | rneqd 5651 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
482 | 463, 481 | syl5eq 2826 |
. . . . . . . . . . . . . . . . . . . . 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))) |
483 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℤ) |
484 | 483 | zcnd 11901 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℂ) |
485 | 484, 416 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) = 𝑗) |
486 | 485 | eleq1d 2850 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
487 | 486 | ibir 260 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) |
488 | 487 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) |
489 | 52 | nnzd 11899 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ) |
490 | 120, 489 | anim12ci 604 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈
ℤ)) |
491 | 483, 421 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (𝑗 − 1) ∈ ℤ) |
492 | 491, 121 | jctir 513 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) ∈ ℤ ∧
1 ∈ ℤ)) |
493 | | fzaddel 12757 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
494 | 490, 492,
493 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
495 | 488, 494 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → (𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1))) |
496 | 484, 428 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 = ((𝑗 − 1) + 1)) |
497 | 496 | adantl 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → 𝑗 = ((𝑗 − 1) + 1)) |
498 | 430 | rspceeqv 3553 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)) |
499 | 495, 497,
498 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)) |
500 | 499 | ex 405 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))) |
501 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) |
502 | | elfzelz 12724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑛 ∈ ℤ) |
503 | 502, 121 | jctir 513 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈
ℤ)) |
504 | | fzaddel 12757 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑦 + 1) ∈ ℤ ∧
(𝑁 − 1) ∈
ℤ) ∧ (𝑛 ∈
ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
505 | 490, 503,
504 | syl2an 586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
506 | 501, 505 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) |
507 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = (𝑛 + 1) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
508 | 506, 507 | syl5ibrcom 239 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
509 | 508 | rexlimdva 3229 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))) |
510 | 500, 509 | impbid 204 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))) |
511 | | eqid 2778 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) |
512 | 511 | elrnmpt 5671 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))) |
513 | 398, 512 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)) |
514 | 510, 513 | syl6bbr 281 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))) |
515 | 514 | eqrdv 2776 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))) |
516 | 72 | oveq2d 6992 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁)) |
517 | 516 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁)) |
518 | 482, 515,
517 | 3eqtr2rd 2821 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))) |
519 | 518 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . . 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))))) |
520 | 462, 519 | uneq12d 4029 |
. . . . . . . . . . . . . . . . . 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)))))) |
521 | 446, 455,
520 | 3eqtr4a 2840 |
. . . . . . . . . . . . . . . . 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)))) |
522 | 521 | xpeq1d 5436 |
. . . . . . . . . . . . . . . 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})) |
523 | | xpundir 5471 |
. . . . . . . . . . . . . . . 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})) |
524 | 522, 523 | syl6eq 2830 |
. . . . . . . . . . . . . . 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}))) |
525 | 441, 524 | uneq12d 4029 |
. . . . . . . . . . . . . 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})))) |
526 | | unass 4031 |
. . . . . . . . . . . . . . 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}))) |
527 | | un23 4033 |
. . . . . . . . . . . . . . 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})) |
528 | 526, 527 | eqtr3i 2804 |
. . . . . . . . . . . . . 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})) |
529 | 525, 528 | syl6eq 2830 |
. . . . . . . . . . . . 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}))) |
530 | 529 | fveq1d 6501 |
. . . . . . . . . . . 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)) |
531 | 530 | ad2antrr 713 |
. . . . . . . . . . 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)) |
532 | 354, 372,
531 | 3eqtr4d 2824 |
. . . . . . . . . 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)) |
533 | | snssi 4615 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
ℂ → {1} ⊆ ℂ) |
534 | 141, 533 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ {1}
⊆ ℂ |
535 | | 0cn 10431 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℂ |
536 | | snssi 4615 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
ℂ → {0} ⊆ ℂ) |
537 | 535, 536 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ {0}
⊆ ℂ |
538 | 534, 537 | unssi 4049 |
. . . . . . . . . . . . 13
⊢ ({1}
∪ {0}) ⊆ ℂ |
539 | 32 | fconst 6394 |
. . . . . . . . . . . . . . . . 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} |
540 | 35 | fconst 6394 |
. . . . . . . . . . . . . . . . 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} |
541 | 539, 540 | pm3.2i 463 |
. . . . . . . . . . . . . . . 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}) |
542 | | fun 6369 |
. . . . . . . . . . . . . . . 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})) |
543 | 541, 243,
542 | sylancr 578 |
. . . . . . . . . . . . . . 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})) |
544 | | imaundi 5848 |
. . . . . . . . . . . . . . . . 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))) |
545 | 65 | adantl 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈
(ℤ≥‘1)) |
546 | | fzsplit2 12748 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) |
547 | 545, 378,
546 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) |
548 | 547 | imaeq2d 5770 |
. . . . . . . . . . . . . . . . . 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)))) |
549 | | f1ofo 6451 |
. . . . . . . . . . . . . . . . . . . 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)) |
550 | | foima 6424 |
. . . . . . . . . . . . . . . . . . . 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)) |
551 | 230, 549,
550 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁)) |
552 | 551 | adantr 473 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2nd
‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁)) |
553 | 548, 552 | eqtr3d 2816 |
. . . . . . . . . . . . . . . . 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)) |
554 | 544, 553 | syl5eqr 2828 |
. . . . . . . . . . . . . . . 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)) |
555 | 554 | feq2d 6330 |
. . . . . . . . . . . . . . 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}))) |
556 | 543, 555 | mpbid 224 |
. . . . . . . . . . . . . 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})) |
557 | 556 | ffvelrnda 6676 |
. . . . . . . . . . . . 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})) |
558 | 538, 557 | sseldi 3856 |
. . . . . . . . . . . 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) |
559 | 558 | addid2d 10641 |
. . . . . . . . . . 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)) |
560 | 559 | adantr 473 |
. . . . . . . . . 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)) |
561 | 532, 560 | eqtr4d 2817 |
. . . . . . . . 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))) |
562 | 96, 98, 289, 561 | ifbothda 4387 |
. . . . . . . 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))) |
563 | 562 | oveq2d 6992 |
. . . . . . 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)))) |
564 | | elmapi 8228 |
. . . . . . . . . . . . 13
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
565 | 28, 564 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
566 | 565 | ffvelrnda 6676 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾)) |
567 | | elfzonn0 12897 |
. . . . . . . . . . 11
⊢
(((1st ‘(1st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈
ℕ0) |
568 | 566, 567 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈
ℕ0) |
569 | 568 | nn0cnd 11769 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ ℂ) |
570 | 569 | adantlr 702 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) ∈ ℂ) |
571 | 141, 535 | ifcli 4396 |
. . . . . . . . 9
⊢ if(𝑛 = ((2nd
‘(1st ‘𝑇))‘1), 1, 0) ∈
ℂ |
572 | 571 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0)
∈ ℂ) |
573 | 570, 572,
558 | addassd 10462 |
. . . . . . 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)))) |
574 | 563, 573 | eqtr4d 2817 |
. . . . . 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))) |
575 | 574 | mpteq2dva 5022 |
. . . . 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)))) |
576 | 94, 575 | eqtrd 2814 |
. . . 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)))) |
577 | | poimirlem18.4 |
. . . . . . . . . 10
⊢ (𝜑 → (2nd
‘𝑇) =
0) |
578 | 577 | adantr 473 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑇) =
0) |
579 | | elfzle1 12726 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦) |
580 | 579 | adantl 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦) |
581 | 578, 580 | eqbrtrd 4951 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑇) ≤ 𝑦) |
582 | | 0re 10441 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
583 | 577, 582 | syl6eqel 2874 |
. . . . . . . . 9
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℝ) |
584 | | lenlt 10519 |
. . . . . . . . 9
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd ‘𝑇))) |
585 | 583, 236,
584 | syl2an 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2nd
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd ‘𝑇))) |
586 | 581, 585 | mpbid 224 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd
‘𝑇)) |
587 | 586 | iffalsed 4361 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1)) |
588 | 587 | csbeq1d 3793 |
. . . . 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})))) |
589 | | ovex 7008 |
. . . . . 6
⊢ (𝑦 + 1) ∈ V |
590 | | oveq2 6984 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1))) |
591 | 590 | imaeq2d 5770 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...(𝑦 +
1)))) |
592 | 591 | xpeq1d 5436 |
. . . . . . . 8
⊢ (𝑗 = (𝑦 + 1) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})) |
593 | | oveq1 6983 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1)) |
594 | 593 | oveq1d 6991 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁)) |
595 | 594 | imaeq2d 5770 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
596 | 595 | xpeq1d 5436 |
. . . . . . . 8
⊢ (𝑗 = (𝑦 + 1) → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
597 | 592, 596 | uneq12d 4029 |
. . . . . . 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}))) |
598 | 597 | oveq2d 6992 |
. . . . . 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})))) |
599 | 589, 598 | csbie 3814 |
. . . . 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}))) |
600 | 588, 599 | syl6eq 2830 |
. . . 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})))) |
601 | | ovexd 7010 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st
‘𝑇))‘1), 1, 0))
∈ V) |
602 | | fvexd 6514 |
. . . . 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) |
603 | | eqidd 2779 |
. . . . 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)))) |
604 | 556 | ffnd 6345 |
. . . . . 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)) |
605 | | nfcv 2932 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(2nd ‘(1st
‘𝑇)) |
606 | | nfmpt1 5025 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) |
607 | 605, 606 | nfco 5586 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) |
608 | | nfcv 2932 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(1...𝑦) |
609 | 607, 608 | nfima 5778 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) |
610 | | nfcv 2932 |
. . . . . . . . 9
⊢
Ⅎ𝑛{1} |
611 | 609, 610 | nfxp 5440 |
. . . . . . . 8
⊢
Ⅎ𝑛((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) |
612 | | nfcv 2932 |
. . . . . . . . . 10
⊢
Ⅎ𝑛((𝑦 + 1)...𝑁) |
613 | 607, 612 | nfima 5778 |
. . . . . . . . 9
⊢
Ⅎ𝑛(((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) |
614 | | nfcv 2932 |
. . . . . . . . 9
⊢
Ⅎ𝑛{0} |
615 | 613, 614 | nfxp 5440 |
. . . . . . . 8
⊢
Ⅎ𝑛((((2nd ‘(1st
‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) |
616 | 611, 615 | nfun 4030 |
. . . . . . 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})) |
617 | 616 | dffn5f 6565 |
. . . . . 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))) |
618 | 604, 617 | sylib 210 |
. . . . 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))) |
619 | 90, 601, 602, 603, 618 | offval2 7244 |
. . . 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)))) |
620 | 576, 600,
619 | 3eqtr4rd 2825 |
. . 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})))) |
621 | 620 | mpteq2dva 5022 |
. 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}))))) |
622 | 21, 621 | eqtr4d 2817 |
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}))))) |