Proof of Theorem poimirlem7
Step | Hyp | Ref
| Expression |
1 | | poimirlem9.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | elrabi 3616 |
. . . . . . . . 9
⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
3 | | poimirlem22.s |
. . . . . . . . 9
⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
4 | 2, 3 | eleq2s 2903 |
. . . . . . . 8
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
5 | 1, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
6 | | xp1st 7584 |
. . . . . . 7
⊢ (𝑇 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
8 | | xp2nd 7585 |
. . . . . 6
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
9 | 7, 8 | syl 17 |
. . . . 5
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
10 | | fvex 6558 |
. . . . . 6
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
11 | | f1oeq1 6479 |
. . . . . 6
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
12 | 10, 11 | elab 3608 |
. . . . 5
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
13 | 9, 12 | sylib 219 |
. . . 4
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
14 | | f1of 6490 |
. . . 4
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
15 | 13, 14 | syl 17 |
. . 3
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
16 | | poimirlem9.2 |
. . . . . . . . 9
⊢ (𝜑 → (2nd
‘𝑇) ∈
(1...(𝑁 −
1))) |
17 | | elfznn 12790 |
. . . . . . . . 9
⊢
((2nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2nd
‘𝑇) ∈
ℕ) |
18 | 16, 17 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℕ) |
19 | 18 | peano2nnd 11509 |
. . . . . . 7
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ∈
ℕ) |
20 | 19 | peano2nnd 11509 |
. . . . . 6
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ∈
ℕ) |
21 | | nnuz 12134 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
22 | 20, 21 | syl6eleq 2895 |
. . . . 5
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ∈
(ℤ≥‘1)) |
23 | | fzss1 12800 |
. . . . 5
⊢
((((2nd ‘𝑇) + 1) + 1) ∈
(ℤ≥‘1) → ((((2nd ‘𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
24 | 22, 23 | syl 17 |
. . . 4
⊢ (𝜑 → ((((2nd
‘𝑇) + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
25 | | poimirlem7.3 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁)) |
26 | 24, 25 | sseldd 3896 |
. . 3
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
27 | 15, 26 | ffvelrnd 6724 |
. 2
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) |
28 | | xp1st 7584 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
29 | 7, 28 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
30 | | elmapfn 8286 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
32 | 31 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
33 | | 1ex 10490 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
34 | | fnconstg 6442 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1)))) |
35 | 33, 34 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) |
36 | | c0ex 10488 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
37 | | fnconstg 6442 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
38 | 36, 37 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) |
39 | 35, 38 | pm3.2i 471 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
40 | | dff1o3 6496 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
41 | 40 | simprbi 497 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
42 | 13, 41 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
43 | | imain 6316 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
44 | 42, 43 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
45 | | elfzelz 12762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → 𝑀 ∈ ℤ) |
46 | 25, 45 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
47 | 46 | zred 11941 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈ ℝ) |
48 | 47 | ltm1d 11426 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) < 𝑀) |
49 | | fzdisj 12788 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
51 | 50 | imaeq2d 5813 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
52 | | ima0 5828 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
53 | 51, 52 | syl6eq 2849 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅) |
54 | 44, 53 | eqtr3d 2835 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅) |
55 | | fnun 6340 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
56 | 39, 54, 55 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
57 | 46 | zcnd 11942 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℂ) |
58 | | npcan1 10919 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
59 | 57, 58 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
60 | | 1red 10495 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 1 ∈
ℝ) |
61 | 20 | nnred 11507 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ∈
ℝ) |
62 | 19 | nnred 11507 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ∈
ℝ) |
63 | 19 | nnge1d 11539 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 1 ≤ ((2nd
‘𝑇) +
1)) |
64 | 62 | ltp1d 11424 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((2nd
‘𝑇) + 1) <
(((2nd ‘𝑇)
+ 1) + 1)) |
65 | 60, 62, 61, 63, 64 | lelttrd 10651 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 1 < (((2nd
‘𝑇) + 1) +
1)) |
66 | | elfzle1 12764 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → (((2nd
‘𝑇) + 1) + 1) ≤
𝑀) |
67 | 25, 66 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) ≤
𝑀) |
68 | 60, 61, 47, 65, 67 | ltletrd 10653 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 1 < 𝑀) |
69 | 60, 47, 68 | ltled 10641 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 1 ≤ 𝑀) |
70 | | elnnz1 11862 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤
𝑀)) |
71 | 46, 69, 70 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℕ) |
72 | 71, 21 | syl6eleq 2895 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘1)) |
73 | 59, 72 | eqeltrd 2885 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ≥‘1)) |
74 | | peano2zm 11879 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈
ℤ) |
75 | 46, 74 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
76 | | uzid 12112 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈ ℤ
→ (𝑀 − 1) ∈
(ℤ≥‘(𝑀 − 1))) |
77 | | peano2uz 12154 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈
(ℤ≥‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
78 | 75, 76, 77 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
79 | 59, 78 | eqeltrrd 2886 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
80 | | uzss 12118 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈
(ℤ≥‘(𝑀 − 1)) →
(ℤ≥‘𝑀) ⊆
(ℤ≥‘(𝑀 − 1))) |
81 | 79, 80 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
(ℤ≥‘𝑀) ⊆
(ℤ≥‘(𝑀 − 1))) |
82 | | elfzuz3 12759 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) |
83 | 25, 82 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
84 | 81, 83 | sseldd 3896 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) |
85 | | fzsplit2 12786 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
86 | 73, 84, 85 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
87 | 59 | oveq1d 7038 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁)) |
88 | 87 | uneq2d 4066 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
89 | 86, 88 | eqtrd 2833 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
90 | 89 | imaeq2d 5813 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...(𝑀 − 1)) ∪
(𝑀...𝑁)))) |
91 | | imaundi 5891 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
92 | 90, 91 | syl6eq 2849 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) |
93 | | f1ofo 6497 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
94 | 13, 93 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
95 | | foima 6470 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
96 | 94, 95 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
97 | 92, 96 | eqtr3d 2835 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = (1...𝑁)) |
98 | 97 | fneq2d 6324 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))) |
99 | 56, 98 | mpbid 233 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
100 | 99 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
101 | | ovexd 7057 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → (1...𝑁) ∈ V) |
102 | | inidm 4121 |
. . . . . . . . . 10
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
103 | | eqidd 2798 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑛) = ((1st ‘(1st
‘𝑇))‘𝑛)) |
104 | | imaundi 5891 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ {𝑀}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
105 | | fzpred 12809 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
106 | 83, 105 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
107 | 106 | imaeq2d 5813 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁)))) |
108 | | f1ofn 6491 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
109 | 13, 108 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) Fn (1...𝑁)) |
110 | | fnsnfv 6617 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑇))‘𝑀)} = ((2nd ‘(1st
‘𝑇)) “ {𝑀})) |
111 | 109, 26, 110 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {((2nd
‘(1st ‘𝑇))‘𝑀)} = ((2nd ‘(1st
‘𝑇)) “ {𝑀})) |
112 | 111 | uneq1d 4065 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ({((2nd
‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ {𝑀}) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
113 | 104, 107,
112 | 3eqtr4a 2859 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) = ({((2nd
‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
114 | 113 | xpeq1d 5479 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0})) |
115 | | xpundir 5514 |
. . . . . . . . . . . . . . . 16
⊢
(({((2nd ‘(1st ‘𝑇))‘𝑀)} ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
116 | 114, 115 | syl6eq 2849 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
117 | 116 | uneq2d 4066 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
118 | | un12 4070 |
. . . . . . . . . . . . . 14
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
119 | 117, 118 | syl6eq 2849 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
120 | 119 | fveq1d 6547 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
121 | 120 | ad2antrr 722 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
122 | | fnconstg 6442 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
123 | 36, 122 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
124 | 35, 123 | pm3.2i 471 |
. . . . . . . . . . . . . . 15
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
125 | | imain 6316 |
. . . . . . . . . . . . . . . . 17
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
126 | 42, 125 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
127 | 75 | zred 11941 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
128 | | peano2re 10666 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
129 | 47, 128 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 + 1) ∈ ℝ) |
130 | 47 | ltp1d 11424 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
131 | 127, 47, 129, 48, 130 | lttrd 10654 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀 − 1) < (𝑀 + 1)) |
132 | | fzdisj 12788 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
133 | 131, 132 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
134 | 133 | imaeq2d 5813 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
135 | 134, 52 | syl6eq 2849 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
136 | 126, 135 | eqtr3d 2835 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
137 | | fnun 6340 |
. . . . . . . . . . . . . . 15
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
138 | 124, 136,
137 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
139 | | imaundi 5891 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
140 | | imadif 6315 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑀}))) |
141 | 42, 140 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑀}))) |
142 | | fzsplit 12787 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
143 | 26, 142 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
144 | 143 | difeq1d 4025 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀})) |
145 | | difundir 4183 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑀) ∪
((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) |
146 | | fzsplit2 12786 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀))) |
147 | 73, 79, 146 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀))) |
148 | 59 | oveq1d 7038 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀)) |
149 | | fzsn 12803 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
150 | 46, 149 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
151 | 148, 150 | eqtrd 2833 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀}) |
152 | 151 | uneq2d 4066 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀})) |
153 | 147, 152 | eqtrd 2833 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀})) |
154 | 153 | difeq1d 4025 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀})) |
155 | | difun2 4349 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑀 −
1)) ∪ {𝑀}) ∖
{𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀}) |
156 | 127, 47 | ltnled 10640 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1))) |
157 | 48, 156 | mpbid 233 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1)) |
158 | | elfzle2 12765 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1)) |
159 | 157, 158 | nsyl 142 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1))) |
160 | | difsn 4644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1))) |
161 | 159, 160 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1))) |
162 | 155, 161 | syl5eq 2845 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1))) |
163 | 154, 162 | eqtrd 2833 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1))) |
164 | 47, 129 | ltnled 10640 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
165 | 130, 164 | mpbid 233 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀) |
166 | | elfzle1 12764 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀) |
167 | 165, 166 | nsyl 142 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) |
168 | | difsn 4644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁)) |
169 | 167, 168 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁)) |
170 | 163, 169 | uneq12d 4067 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
171 | 145, 170 | syl5eq 2845 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
172 | 144, 171 | eqtrd 2833 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
173 | 172 | imaeq2d 5813 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))) |
174 | 111 | eqcomd 2803 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ {𝑀}) = {((2nd
‘(1st ‘𝑇))‘𝑀)}) |
175 | 96, 174 | difeq12d 4027 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
176 | 141, 173,
175 | 3eqtr3d 2841 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
177 | 139, 176 | syl5eqr 2847 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
178 | 177 | fneq2d 6324 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) |
179 | 138, 178 | mpbid 233 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
180 | | eldifsn 4632 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) |
181 | 180 | biimpri 229 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
182 | 181 | ancoms 459 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ≠ ((2nd
‘(1st ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
183 | | disjdif 4341 |
. . . . . . . . . . . . . 14
⊢
({((2nd ‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ |
184 | | fnconstg 6442 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
V → ({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)}) |
185 | 36, 184 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} |
186 | | fvun2 6629 |
. . . . . . . . . . . . . . 15
⊢
((({((2nd ‘(1st ‘𝑇))‘𝑀)} × {0}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
187 | 185, 186 | mp3an1 1440 |
. . . . . . . . . . . . . 14
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
188 | 183, 187 | mpanr1 699 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
189 | 179, 182,
188 | syl2an 595 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
190 | 189 | anassrs 468 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {0}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
191 | 121, 190 | eqtrd 2833 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
192 | 32, 100, 101, 101, 102, 103, 191 | ofval 7283 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))) |
193 | | fnconstg 6442 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
194 | 33, 193 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) |
195 | 194, 123 | pm3.2i 471 |
. . . . . . . . . . . . 13
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
196 | | imain 6316 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
197 | 42, 196 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
198 | | fzdisj 12788 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
199 | 130, 198 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
200 | 199 | imaeq2d 5813 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
201 | 200, 52 | syl6eq 2849 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
202 | 197, 201 | eqtr3d 2835 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
203 | | fnun 6340 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
204 | 195, 202,
203 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
205 | 143 | imaeq2d 5813 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
206 | | imaundi 5891 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
207 | 205, 206 | syl6eq 2849 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
208 | 207, 96 | eqtr3d 2835 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
209 | 208 | fneq2d 6324 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
210 | 204, 209 | mpbid 233 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
211 | 210 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
212 | | imaundi 5891 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ {𝑀})) |
213 | 153 | imaeq2d 5813 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) = ((2nd ‘(1st
‘𝑇)) “
((1...(𝑀 − 1)) ∪
{𝑀}))) |
214 | 111 | uneq2d 4066 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2nd
‘(1st ‘𝑇)) “ {𝑀}))) |
215 | 212, 213,
214 | 3eqtr4a 2859 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)})) |
216 | 215 | xpeq1d 5479 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)}) × {1})) |
217 | | xpundir 5514 |
. . . . . . . . . . . . . . . 16
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2nd
‘(1st ‘𝑇))‘𝑀)}) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) |
218 | 216, 217 | syl6eq 2849 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}))) |
219 | 218 | uneq1d 4065 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
220 | | un23 4071 |
. . . . . . . . . . . . . . 15
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) ∪ ({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1})) |
221 | 220 | equncomi 4058 |
. . . . . . . . . . . . . 14
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1})) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
222 | 219, 221 | syl6eq 2849 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
223 | 222 | fveq1d 6547 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
224 | 223 | ad2antrr 722 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
225 | | fnconstg 6442 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
V → ({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)}) |
226 | 33, 225 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} |
227 | | fvun2 6629 |
. . . . . . . . . . . . . . 15
⊢
((({((2nd ‘(1st ‘𝑇))‘𝑀)} × {1}) Fn {((2nd
‘(1st ‘𝑇))‘𝑀)} ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
228 | 226, 227 | mp3an1 1440 |
. . . . . . . . . . . . . 14
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ (({((2nd
‘(1st ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
229 | 183, 228 | mpanr1 699 |
. . . . . . . . . . . . 13
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd
‘(1st ‘𝑇))‘𝑀)})) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
230 | 179, 182,
229 | syl2an 595 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
231 | 230 | anassrs 468 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2nd
‘(1st ‘𝑇))‘𝑀)} × {1}) ∪ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
232 | 224, 231 | eqtrd 2833 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
233 | 32, 211, 101, 101, 102, 103, 232 | ofval 7283 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))‘𝑛) + (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))) |
234 | 192, 233 | eqtr4d 2836 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
235 | 234 | an32s 648 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
236 | 235 | anasss 467 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
237 | | fveq2 6545 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
238 | 237 | breq2d 4980 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
239 | 238 | ifbid 4409 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
240 | 239 | csbeq1d 3821 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
241 | | 2fveq3 6550 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
242 | | 2fveq3 6550 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
243 | 242 | imaeq1d 5812 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
244 | 243 | xpeq1d 5479 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
245 | 242 | imaeq1d 5812 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
246 | 245 | xpeq1d 5479 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
247 | 244, 246 | uneq12d 4067 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
248 | 241, 247 | oveq12d 7041 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
249 | 248 | csbeq2dv 3824 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
250 | 240, 249 | eqtrd 2833 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
251 | 250 | mpteq2dv 5063 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
252 | 251 | eqeq2d 2807 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
253 | 252, 3 | elrab2 3624 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
254 | 253 | simprbi 497 |
. . . . . . . . . 10
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
255 | 1, 254 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
256 | | breq1 4971 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 2) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 2) < (2nd
‘𝑇))) |
257 | 256 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 2) < (2nd
‘𝑇))) |
258 | | oveq1 7030 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 2) → (𝑦 + 1) = ((𝑀 − 2) + 1)) |
259 | | sub1m1 11743 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) − 1) = (𝑀 − 2)) |
260 | 57, 259 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑀 − 1) − 1) = (𝑀 − 2)) |
261 | 260 | oveq1d 7038 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑀 − 1) − 1) + 1) = ((𝑀 − 2) +
1)) |
262 | 75 | zcnd 11942 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 − 1) ∈ ℂ) |
263 | | npcan1 10919 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 − 1) ∈ ℂ
→ (((𝑀 − 1)
− 1) + 1) = (𝑀
− 1)) |
264 | 262, 263 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑀 − 1) − 1) + 1) = (𝑀 − 1)) |
265 | 261, 264 | eqtr3d 2835 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑀 − 2) + 1) = (𝑀 − 1)) |
266 | 258, 265 | sylan9eqr 2855 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → (𝑦 + 1) = (𝑀 − 1)) |
267 | 257, 266 | ifbieq2d 4412 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 2) < (2nd
‘𝑇), 𝑦, (𝑀 − 1))) |
268 | 18 | nncnd 11508 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℂ) |
269 | | add1p1 11742 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑇) ∈ ℂ → (((2nd
‘𝑇) + 1) + 1) =
((2nd ‘𝑇)
+ 2)) |
270 | 268, 269 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘𝑇) + 1) + 1) =
((2nd ‘𝑇)
+ 2)) |
271 | 270, 67 | eqbrtrrd 4992 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘𝑇) + 2) ≤ 𝑀) |
272 | 18 | nnred 11507 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (2nd
‘𝑇) ∈
ℝ) |
273 | | 2re 11565 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℝ |
274 | | leaddsub 10970 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 2 ∈ ℝ
∧ 𝑀 ∈ ℝ)
→ (((2nd ‘𝑇) + 2) ≤ 𝑀 ↔ (2nd ‘𝑇) ≤ (𝑀 − 2))) |
275 | 273, 274 | mp3an2 1441 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd
‘𝑇) + 2) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 2))) |
276 | 272, 47, 275 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘𝑇) + 2) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 2))) |
277 | 60, 47 | posdifd 11081 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (1 < 𝑀 ↔ 0 < (𝑀 − 1))) |
278 | 68, 277 | mpbid 233 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 0 < (𝑀 − 1)) |
279 | | elnnz 11845 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑀 − 1) ∈ ℕ
↔ ((𝑀 − 1)
∈ ℤ ∧ 0 < (𝑀 − 1))) |
280 | 75, 278, 279 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 − 1) ∈ ℕ) |
281 | | nnm1nn0 11792 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 − 1) ∈ ℕ
→ ((𝑀 − 1)
− 1) ∈ ℕ0) |
282 | 280, 281 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑀 − 1) − 1) ∈
ℕ0) |
283 | 260, 282 | eqeltrrd 2886 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 − 2) ∈
ℕ0) |
284 | 283 | nn0red 11810 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 2) ∈ ℝ) |
285 | 272, 284 | lenltd 10639 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑇) ≤ (𝑀 − 2) ↔ ¬ (𝑀 − 2) < (2nd
‘𝑇))) |
286 | 276, 285 | bitrd 280 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘𝑇) + 2) ≤ 𝑀 ↔ ¬ (𝑀 − 2) < (2nd
‘𝑇))) |
287 | 271, 286 | mpbid 233 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ (𝑀 − 2) < (2nd
‘𝑇)) |
288 | 287 | iffalsed 4398 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if((𝑀 − 2) < (2nd
‘𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1)) |
289 | 288 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if((𝑀 − 2) < (2nd
‘𝑇), 𝑦, (𝑀 − 1)) = (𝑀 − 1)) |
290 | 267, 289 | eqtrd 2833 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1)) |
291 | 290 | csbeq1d 3821 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 − 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
292 | | oveq2 7031 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1))) |
293 | 292 | imaeq2d 5813 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑀 − 1) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...(𝑀 −
1)))) |
294 | 293 | xpeq1d 5479 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑀 − 1) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1})) |
295 | 294 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1})) |
296 | | oveq1 7030 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1)) |
297 | 296, 59 | sylan9eqr 2855 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀) |
298 | 297 | oveq1d 7038 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁)) |
299 | 298 | imaeq2d 5813 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ (𝑀...𝑁))) |
300 | 299 | xpeq1d 5479 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})) |
301 | 295, 300 | uneq12d 4067 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))) |
302 | 301 | oveq2d 7039 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
303 | 75, 302 | csbied 3850 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋(𝑀 − 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
304 | 303 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋(𝑀 − 1) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
305 | 291, 304 | eqtrd 2833 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 2)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
306 | | poimir.0 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℕ) |
307 | | nnm1nn0 11792 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
308 | 306, 307 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ0) |
309 | 306 | nnred 11507 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℝ) |
310 | 47 | lem1d 11427 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀 − 1) ≤ 𝑀) |
311 | | elfzle2 12765 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ((((2nd
‘𝑇) + 1) + 1)...𝑁) → 𝑀 ≤ 𝑁) |
312 | 25, 311 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
313 | 127, 47, 309, 310, 312 | letrd 10650 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑀 − 1) ≤ 𝑁) |
314 | 127, 309,
60, 313 | lesub1dd 11110 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑀 − 1) − 1) ≤ (𝑁 − 1)) |
315 | 260, 314 | eqbrtrrd 4992 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 2) ≤ (𝑁 − 1)) |
316 | | elfz2nn0 12852 |
. . . . . . . . . 10
⊢ ((𝑀 − 2) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 2) ∈
ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧
(𝑀 − 2) ≤ (𝑁 − 1))) |
317 | 283, 308,
315, 316 | syl3anbrc 1336 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 − 2) ∈ (0...(𝑁 − 1))) |
318 | | ovexd 7057 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V) |
319 | 255, 305,
317, 318 | fvmptd 6648 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑀 − 2)) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
320 | 319 | fveq1d 6547 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...(𝑀 − 1)))
× {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛)) |
321 | 320 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...(𝑀 − 1)))
× {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛)) |
322 | | breq1 4971 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 1) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < (2nd
‘𝑇))) |
323 | 322 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < (2nd
‘𝑇))) |
324 | | oveq1 7030 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1)) |
325 | 324, 59 | sylan9eqr 2855 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀) |
326 | 323, 325 | ifbieq2d 4412 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2nd
‘𝑇), 𝑦, 𝑀)) |
327 | 62 | lep1d 11425 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ≤
(((2nd ‘𝑇)
+ 1) + 1)) |
328 | 62, 61, 47, 327, 67 | letrd 10650 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2nd
‘𝑇) + 1) ≤ 𝑀) |
329 | | 1re 10494 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℝ |
330 | | leaddsub 10970 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 1 ∈ ℝ
∧ 𝑀 ∈ ℝ)
→ (((2nd ‘𝑇) + 1) ≤ 𝑀 ↔ (2nd ‘𝑇) ≤ (𝑀 − 1))) |
331 | 329, 330 | mp3an2 1441 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘𝑇) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((2nd
‘𝑇) + 1) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 1))) |
332 | 272, 47, 331 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2nd
‘𝑇) + 1) ≤ 𝑀 ↔ (2nd
‘𝑇) ≤ (𝑀 − 1))) |
333 | 272, 127 | lenltd 10639 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑇) ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < (2nd
‘𝑇))) |
334 | 332, 333 | bitrd 280 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2nd
‘𝑇) + 1) ≤ 𝑀 ↔ ¬ (𝑀 − 1) < (2nd
‘𝑇))) |
335 | 328, 334 | mpbid 233 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ (𝑀 − 1) < (2nd
‘𝑇)) |
336 | 335 | iffalsed 4398 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if((𝑀 − 1) < (2nd
‘𝑇), 𝑦, 𝑀) = 𝑀) |
337 | 336 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < (2nd
‘𝑇), 𝑦, 𝑀) = 𝑀) |
338 | 326, 337 | eqtrd 2833 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
339 | 338 | csbeq1d 3821 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
340 | | oveq2 7031 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
341 | 340 | imaeq2d 5813 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑀))) |
342 | 341 | xpeq1d 5479 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})) |
343 | | oveq1 7030 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
344 | 343 | oveq1d 7038 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
345 | 344 | imaeq2d 5813 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
346 | 345 | xpeq1d 5479 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
347 | 342, 346 | uneq12d 4067 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
348 | 347 | oveq2d 7039 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑀 → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
349 | 348 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
350 | 25, 349 | csbied 3850 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
351 | 350 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
352 | 339, 351 | eqtrd 2833 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
353 | 280 | nnnn0d 11809 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ0) |
354 | 47, 309, 60, 312 | lesub1dd 11110 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1)) |
355 | | elfz2nn0 12852 |
. . . . . . . . . 10
⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈
ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧
(𝑀 − 1) ≤ (𝑁 − 1))) |
356 | 353, 308,
354, 355 | syl3anbrc 1336 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
357 | | ovexd 7057 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
358 | 255, 352,
356, 357 | fvmptd 6648 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
359 | 358 | fveq1d 6547 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
360 | 359 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
361 | 236, 321,
360 | 3eqtr4d 2843 |
. . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛)) |
362 | 361 | expr 457 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2nd ‘(1st
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑛))) |
363 | 362 | necon1d 3008 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) → 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀))) |
364 | | elmapi 8285 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
365 | 29, 364 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
366 | 365, 27 | ffvelrnd 6724 |
. . . . . . . . 9
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ (0..^𝐾)) |
367 | | elfzonn0 12936 |
. . . . . . . . 9
⊢
(((1st ‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈
ℕ0) |
368 | 366, 367 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈
ℕ0) |
369 | 368 | nn0red 11810 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ ℝ) |
370 | 369 | ltp1d 11424 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) < (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
371 | 369, 370 | ltned 10629 |
. . . . . 6
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ≠ (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
372 | 319 | fveq1d 6547 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
373 | | ovexd 7057 |
. . . . . . . . 9
⊢ (𝜑 → (1...𝑁) ∈ V) |
374 | | eqidd 2798 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((1st ‘(1st
‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
375 | | fzss1 12800 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈
(ℤ≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) |
376 | 72, 375 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁)) |
377 | | eluzfz1 12768 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
378 | 83, 377 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
379 | | fnfvima 6867 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
380 | 109, 376,
378, 379 | syl3anc 1364 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) |
381 | | fvun2 6629 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
382 | 35, 38, 381 | mp3an12 1443 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁))) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
383 | 54, 380, 382 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
384 | 36 | fvconst2 6840 |
. . . . . . . . . . . 12
⊢
(((2nd ‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) → ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
385 | 380, 384 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
386 | 383, 385 | eqtrd 2833 |
. . . . . . . . . 10
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
387 | 386 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 0) |
388 | 31, 99, 373, 373, 102, 374, 387 | ofval 7283 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 0)) |
389 | 27, 388 | mpdan 683 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 0)) |
390 | 368 | nn0cnd 11811 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ∈ ℂ) |
391 | 390 | addid1d 10693 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 0) = ((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
392 | 372, 389,
391 | 3eqtrd 2837 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((1st ‘(1st
‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
393 | 358 | fveq1d 6547 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
394 | | fzss2 12801 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (1...𝑀) ⊆ (1...𝑁)) |
395 | 83, 394 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑀) ⊆ (1...𝑁)) |
396 | | elfz1end 12791 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀)) |
397 | 71, 396 | sylib 219 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
398 | | fnfvima 6867 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
399 | 109, 395,
397, 398 | syl3anc 1364 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
400 | | fvun1 6628 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
401 | 194, 123,
400 | mp3an12 1443 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
402 | 202, 399,
401 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
403 | 33 | fvconst2 6840 |
. . . . . . . . . . . 12
⊢
(((2nd ‘(1st ‘𝑇))‘𝑀) ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
404 | 399, 403 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
405 | 402, 404 | eqtrd 2833 |
. . . . . . . . . 10
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
406 | 405 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = 1) |
407 | 31, 210, 373, 373, 102, 374, 406 | ofval 7283 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((2nd
‘(1st ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
408 | 27, 407 | mpdan 683 |
. . . . . . 7
⊢ (𝜑 → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
409 | 393, 408 | eqtrd 2833 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀)) = (((1st
‘(1st ‘𝑇))‘((2nd
‘(1st ‘𝑇))‘𝑀)) + 1)) |
410 | 371, 392,
409 | 3netr4d 3063 |
. . . . 5
⊢ (𝜑 → ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
411 | | fveq2 6545 |
. . . . . 6
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) = ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
412 | | fveq2 6545 |
. . . . . 6
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀))) |
413 | 411, 412 | neeq12d 3047 |
. . . . 5
⊢ (𝑛 = ((2nd
‘(1st ‘𝑇))‘𝑀) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ ((𝐹‘(𝑀 − 2))‘((2nd
‘(1st ‘𝑇))‘𝑀)) ≠ ((𝐹‘(𝑀 − 1))‘((2nd
‘(1st ‘𝑇))‘𝑀)))) |
414 | 410, 413 | syl5ibrcom 248 |
. . . 4
⊢ (𝜑 → (𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛))) |
415 | 414 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛))) |
416 | 363, 415 | impbid 213 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛) ↔ 𝑛 = ((2nd ‘(1st
‘𝑇))‘𝑀))) |
417 | 27, 416 | riota5 7010 |
1
⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st
‘𝑇))‘𝑀)) |