Step | Hyp | Ref
| Expression |
1 | | poimirlem9.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | elrabi 3391 |
. . . . . . . . 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 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘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 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
4 | 2, 3 | eleq2s 2748 |
. . . . . . . 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 7242 |
. . . . . . 7
⊢ (𝑇 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → (1^{st}
‘𝑇) ∈
(((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
8 | | xp2nd 7243 |
. . . . . 6
⊢
((1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
9 | 7, 8 | syl 17 |
. . . . 5
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
10 | | fvex 6239 |
. . . . . 6
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
11 | | f1oeq1 6165 |
. . . . . 6
⊢ (𝑓 = (2^{nd}
‘(1^{st} ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
12 | 10, 11 | elab 3382 |
. . . . 5
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
13 | 9, 12 | sylib 208 |
. . . 4
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
14 | | f1of 6175 |
. . . 4
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
15 | 13, 14 | syl 17 |
. . 3
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
16 | | poimirlem9.2 |
. . . . . . . . 9
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
(1...(𝑁 −
1))) |
17 | | elfznn 12408 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^{nd}
‘𝑇) ∈
ℕ) |
18 | 16, 17 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℕ) |
19 | 18 | nnzd 11519 |
. . . . . . 7
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℤ) |
20 | | peano2zm 11458 |
. . . . . . 7
⊢
((2^{nd} ‘𝑇) ∈ ℤ → ((2^{nd}
‘𝑇) − 1) ∈
ℤ) |
21 | 19, 20 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((2^{nd}
‘𝑇) − 1) ∈
ℤ) |
22 | | poimir.0 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) |
23 | 22 | nnzd 11519 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℤ) |
24 | 21 | zred 11520 |
. . . . . . 7
⊢ (𝜑 → ((2^{nd}
‘𝑇) − 1) ∈
ℝ) |
25 | 18 | nnred 11073 |
. . . . . . 7
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℝ) |
26 | 22 | nnred 11073 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℝ) |
27 | 25 | lem1d 10995 |
. . . . . . 7
⊢ (𝜑 → ((2^{nd}
‘𝑇) − 1) ≤
(2^{nd} ‘𝑇)) |
28 | | nnm1nn0 11372 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ_{0}) |
29 | 22, 28 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ_{0}) |
30 | 29 | nn0red 11390 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
31 | | elfzle2 12383 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^{nd}
‘𝑇) ≤ (𝑁 − 1)) |
32 | 16, 31 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ (𝑁 − 1)) |
33 | 26 | lem1d 10995 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 − 1) ≤ 𝑁) |
34 | 25, 30, 26, 32, 33 | letrd 10232 |
. . . . . . 7
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ 𝑁) |
35 | 24, 25, 26, 27, 34 | letrd 10232 |
. . . . . 6
⊢ (𝜑 → ((2^{nd}
‘𝑇) − 1) ≤
𝑁) |
36 | | eluz2 11731 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ_{≥}‘((2^{nd} ‘𝑇) − 1)) ↔ (((2^{nd}
‘𝑇) − 1) ∈
ℤ ∧ 𝑁 ∈
ℤ ∧ ((2^{nd} ‘𝑇) − 1) ≤ 𝑁)) |
37 | 21, 23, 35, 36 | syl3anbrc 1265 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈
(ℤ_{≥}‘((2^{nd} ‘𝑇) − 1))) |
38 | | fzss2 12419 |
. . . . 5
⊢ (𝑁 ∈
(ℤ_{≥}‘((2^{nd} ‘𝑇) − 1)) → (1...((2^{nd}
‘𝑇) − 1))
⊆ (1...𝑁)) |
39 | 37, 38 | syl 17 |
. . . 4
⊢ (𝜑 → (1...((2^{nd}
‘𝑇) − 1))
⊆ (1...𝑁)) |
40 | | poimirlem6.3 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ (1...((2^{nd} ‘𝑇) − 1))) |
41 | 39, 40 | sseldd 3637 |
. . 3
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
42 | 15, 41 | ffvelrnd 6400 |
. 2
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ (1...𝑁)) |
43 | | xp1st 7242 |
. . . . . . . . . . . . 13
⊢
((1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
44 | 7, 43 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
45 | | elmapfn 7922 |
. . . . . . . . . . . 12
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
46 | 44, 45 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
47 | 46 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
48 | | 1ex 10073 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
49 | | fnconstg 6131 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1)))) |
50 | 48, 49 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) |
51 | | c0ex 10072 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
52 | | fnconstg 6131 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) |
53 | 51, 52 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) |
54 | 50, 53 | pm3.2i 470 |
. . . . . . . . . . . . 13
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) |
55 | | dff1o3 6181 |
. . . . . . . . . . . . . . . . 17
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇)))) |
56 | 55 | simprbi 479 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇))) |
57 | 13, 56 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇))) |
58 | | imain 6012 |
. . . . . . . . . . . . . . 15
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑇)) →
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)))) |
59 | 57, 58 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)))) |
60 | | elfznn 12408 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ (1...((2^{nd}
‘𝑇) − 1))
→ 𝑀 ∈
ℕ) |
61 | 40, 60 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℕ) |
62 | 61 | nnred 11073 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈ ℝ) |
63 | 62 | ltm1d 10994 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) < 𝑀) |
64 | | fzdisj 12406 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
65 | 63, 64 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
66 | 65 | imaeq2d 5501 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ∅)) |
67 | | ima0 5516 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ∅) =
∅ |
68 | 66, 67 | syl6eq 2701 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅) |
69 | 59, 68 | eqtr3d 2687 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) = ∅) |
70 | | fnun 6035 |
. . . . . . . . . . . . 13
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)))) |
71 | 54, 69, 70 | sylancr 696 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)))) |
72 | 61 | nncnd 11074 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℂ) |
73 | | npcan1 10493 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
74 | 72, 73 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
75 | | nnuz 11761 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ℕ =
(ℤ_{≥}‘1) |
76 | 61, 75 | syl6eleq 2740 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈
(ℤ_{≥}‘1)) |
77 | 74, 76 | eqeltrd 2730 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ_{≥}‘1)) |
78 | | nnm1nn0 11372 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈
ℕ_{0}) |
79 | 61, 78 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ_{0}) |
80 | 79 | nn0zd 11518 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
81 | | uzid 11740 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑀 − 1) ∈ ℤ
→ (𝑀 − 1) ∈
(ℤ_{≥}‘(𝑀 − 1))) |
82 | 80, 81 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑀 − 1) ∈
(ℤ_{≥}‘(𝑀 − 1))) |
83 | | peano2uz 11779 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈
(ℤ_{≥}‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈
(ℤ_{≥}‘(𝑀 − 1))) |
84 | 82, 83 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ_{≥}‘(𝑀 − 1))) |
85 | 74, 84 | eqeltrrd 2731 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ (ℤ_{≥}‘(𝑀 − 1))) |
86 | | uzss 11746 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈
(ℤ_{≥}‘(𝑀 − 1)) →
(ℤ_{≥}‘𝑀) ⊆
(ℤ_{≥}‘(𝑀 − 1))) |
87 | 85, 86 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
(ℤ_{≥}‘𝑀) ⊆
(ℤ_{≥}‘(𝑀 − 1))) |
88 | 61 | nnzd 11519 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℤ) |
89 | | elfzle2 12383 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ (1...((2^{nd}
‘𝑇) − 1))
→ 𝑀 ≤
((2^{nd} ‘𝑇)
− 1)) |
90 | 40, 89 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ≤ ((2^{nd} ‘𝑇) − 1)) |
91 | 62, 24, 26, 90, 35 | letrd 10232 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
92 | | eluz2 11731 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
93 | 88, 23, 91, 92 | syl3anbrc 1265 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘𝑀)) |
94 | 87, 93 | sseldd 3637 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘(𝑀 − 1))) |
95 | | fzsplit2 12404 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ_{≥}‘1) ∧ 𝑁 ∈ (ℤ_{≥}‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
96 | 77, 94, 95 | syl2anc 694 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
97 | 74 | oveq1d 6705 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁)) |
98 | 97 | uneq2d 3800 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
99 | 96, 98 | eqtrd 2685 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
100 | 99 | imaeq2d 5501 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
((1...(𝑀 − 1)) ∪
(𝑀...𝑁)))) |
101 | | imaundi 5580 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) |
102 | 100, 101 | syl6eq 2701 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)))) |
103 | | f1ofo 6182 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
104 | 13, 103 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
105 | | foima 6158 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
106 | 104, 105 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
107 | 102, 106 | eqtr3d 2687 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) = (1...𝑁)) |
108 | 107 | fneq2d 6020 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))) |
109 | 71, 108 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
110 | 109 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
111 | | ovexd 6720 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) → (1...𝑁) ∈ V) |
112 | | inidm 3855 |
. . . . . . . . . 10
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
113 | | eqidd 2652 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑛) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑛)) |
114 | | imaundi 5580 |
. . . . . . . . . . . . . . . . . 18
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀}) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
115 | | fzpred 12427 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
116 | 93, 115 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) |
117 | 116 | imaeq2d 5501 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ({𝑀} ∪ ((𝑀 + 1)...𝑁)))) |
118 | | f1ofn 6176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
119 | 13, 118 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
120 | | fnsnfv 6297 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁)) → {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} = ((2^{nd} ‘(1^{st}
‘𝑇)) “ {𝑀})) |
121 | 119, 41, 120 | syl2anc 694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} = ((2^{nd} ‘(1^{st}
‘𝑇)) “ {𝑀})) |
122 | 121 | uneq1d 3799 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀}) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
123 | 114, 117,
122 | 3eqtr4a 2711 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) = ({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
124 | 123 | xpeq1d 5172 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0})) |
125 | | xpundir 5206 |
. . . . . . . . . . . . . . . 16
⊢
(({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) × {0}) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
126 | 124, 125 | syl6eq 2701 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
127 | 126 | uneq2d 3800 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
128 | | un12 3804 |
. . . . . . . . . . . . . 14
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
129 | 127, 128 | syl6eq 2701 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
130 | 129 | fveq1d 6231 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
131 | 130 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
132 | | fnconstg 6131 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
133 | 51, 132 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
134 | 50, 133 | pm3.2i 470 |
. . . . . . . . . . . . . . 15
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
135 | | imain 6012 |
. . . . . . . . . . . . . . . . 17
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑇)) →
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
136 | 57, 135 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
137 | 79 | nn0red 11390 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
138 | | peano2re 10247 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
139 | 62, 138 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 + 1) ∈ ℝ) |
140 | 62 | ltp1d 10992 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
141 | 137, 62, 139, 63, 140 | lttrd 10236 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀 − 1) < (𝑀 + 1)) |
142 | | fzdisj 12406 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 − 1) < (𝑀 + 1) → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
143 | 141, 142 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
144 | 143 | imaeq2d 5501 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ∅)) |
145 | 144, 67 | syl6eq 2701 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
146 | 136, 145 | eqtr3d 2687 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
147 | | fnun 6035 |
. . . . . . . . . . . . . . 15
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
148 | 134, 146,
147 | sylancr 696 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
149 | | imaundi 5580 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
150 | | imadif 6011 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑇)) →
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀}))) |
151 | 57, 150 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀}))) |
152 | | fzsplit 12405 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
153 | 41, 152 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
154 | 153 | difeq1d 3760 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀})) |
155 | | difundir 3913 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑀) ∪
((𝑀 + 1)...𝑁)) ∖ {𝑀}) = (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) |
156 | | fzsplit2 12404 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ_{≥}‘1) ∧ 𝑀 ∈ (ℤ_{≥}‘(𝑀 − 1))) → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀))) |
157 | 77, 85, 156 | syl2anc 694 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀))) |
158 | 74 | oveq1d 6705 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = (𝑀...𝑀)) |
159 | | fzsn 12421 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
160 | 88, 159 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
161 | 158, 160 | eqtrd 2685 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑀) = {𝑀}) |
162 | 161 | uneq2d 3800 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑀)) = ((1...(𝑀 − 1)) ∪ {𝑀})) |
163 | 157, 162 | eqtrd 2685 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ {𝑀})) |
164 | 163 | difeq1d 3760 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀})) |
165 | | difun2 4081 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑀 −
1)) ∪ {𝑀}) ∖
{𝑀}) = ((1...(𝑀 − 1)) ∖ {𝑀}) |
166 | 137, 62 | ltnled 10222 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ ¬ 𝑀 ≤ (𝑀 − 1))) |
167 | 63, 166 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ¬ 𝑀 ≤ (𝑀 − 1)) |
168 | | elfzle2 12383 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 ∈ (1...(𝑀 − 1)) → 𝑀 ≤ (𝑀 − 1)) |
169 | 167, 168 | nsyl 135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑀 ∈ (1...(𝑀 − 1))) |
170 | | difsn 4360 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
𝑀 ∈ (1...(𝑀 − 1)) → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1))) |
171 | 169, 170 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((1...(𝑀 − 1)) ∖ {𝑀}) = (1...(𝑀 − 1))) |
172 | 165, 171 | syl5eq 2697 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (((1...(𝑀 − 1)) ∪ {𝑀}) ∖ {𝑀}) = (1...(𝑀 − 1))) |
173 | 164, 172 | eqtrd 2685 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑀) ∖ {𝑀}) = (1...(𝑀 − 1))) |
174 | 62, 139 | ltnled 10222 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
175 | 140, 174 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀) |
176 | | elfzle1 12382 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ((𝑀 + 1)...𝑁) → (𝑀 + 1) ≤ 𝑀) |
177 | 175, 176 | nsyl 135 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ¬ 𝑀 ∈ ((𝑀 + 1)...𝑁)) |
178 | | difsn 4360 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (¬
𝑀 ∈ ((𝑀 + 1)...𝑁) → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁)) |
179 | 177, 178 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∖ {𝑀}) = ((𝑀 + 1)...𝑁)) |
180 | 173, 179 | uneq12d 3801 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (((1...𝑀) ∖ {𝑀}) ∪ (((𝑀 + 1)...𝑁) ∖ {𝑀})) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
181 | 155, 180 | syl5eq 2697 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
182 | 154, 181 | eqtrd 2685 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...𝑁) ∖ {𝑀}) = ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) |
183 | 182 | imaeq2d 5501 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑁) ∖ {𝑀})) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁)))) |
184 | 121 | eqcomd 2657 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀}) = {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) |
185 | 106, 184 | difeq12d 3762 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀})) = ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
186 | 151, 183,
185 | 3eqtr3d 2693 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
187 | 149, 186 | syl5eqr 2699 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
188 | 187 | fneq2d 6020 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}))) |
189 | 148, 188 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
190 | | eldifsn 4350 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) |
191 | 190 | biimpri 218 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) → 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
192 | 191 | ancoms 468 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ≠ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
193 | | disjdif 4073 |
. . . . . . . . . . . . . 14
⊢
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) = ∅ |
194 | | fnconstg 6131 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
V → ({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {0}) Fn {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) |
195 | 51, 194 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {0}) Fn {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} |
196 | | fvun2 6309 |
. . . . . . . . . . . . . . 15
⊢
((({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {0}) Fn {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∧ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ∧ (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}))) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
197 | 195, 196 | mp3an1 1451 |
. . . . . . . . . . . . . 14
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ∧ (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}))) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
198 | 193, 197 | mpanr1 719 |
. . . . . . . . . . . . 13
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
199 | 189, 192,
198 | syl2an 493 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
200 | 199 | anassrs 681 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {0}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
201 | 131, 200 | eqtrd 2685 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
202 | 47, 110, 111, 111, 112, 113, 201 | ofval 6948 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑛) + (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))) |
203 | | fnconstg 6131 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) |
204 | 48, 203 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) |
205 | 204, 133 | pm3.2i 470 |
. . . . . . . . . . . . 13
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
206 | | imain 6012 |
. . . . . . . . . . . . . . 15
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑇)) →
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
207 | 57, 206 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
208 | | fzdisj 12406 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
209 | 140, 208 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
210 | 209 | imaeq2d 5501 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ∅)) |
211 | 210, 67 | syl6eq 2701 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
212 | 207, 211 | eqtr3d 2687 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
213 | | fnun 6035 |
. . . . . . . . . . . . 13
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
214 | 205, 212,
213 | sylancr 696 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
215 | 153 | imaeq2d 5501 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
216 | | imaundi 5580 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
217 | 215, 216 | syl6eq 2701 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
218 | 217, 106 | eqtr3d 2687 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
219 | 218 | fneq2d 6020 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
220 | 214, 219 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
221 | 220 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
222 | | imaundi 5580 |
. . . . . . . . . . . . . . . . . 18
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...(𝑀 − 1)) ∪ {𝑀})) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀})) |
223 | 163 | imaeq2d 5501 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
((1...(𝑀 − 1)) ∪
{𝑀}))) |
224 | 121 | uneq2d 3800 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ {𝑀}))) |
225 | 222, 223,
224 | 3eqtr4a 2711 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) |
226 | 225 | xpeq1d 5172 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) × {1})) |
227 | | xpundir 5206 |
. . . . . . . . . . . . . . . 16
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∪ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) × {1}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1})) |
228 | 226, 227 | syl6eq 2701 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1}))) |
229 | 228 | uneq1d 3799 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1})) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
230 | | un23 3805 |
. . . . . . . . . . . . . . 15
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1})) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) ∪ ({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1})) |
231 | 230 | equncomi 3792 |
. . . . . . . . . . . . . 14
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1})) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
232 | 229, 231 | syl6eq 2701 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) = (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
233 | 232 | fveq1d 6231 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
234 | 233 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
235 | | fnconstg 6131 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
V → ({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1}) Fn {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) |
236 | 48, 235 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1}) Fn {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} |
237 | | fvun2 6309 |
. . . . . . . . . . . . . . 15
⊢
((({((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀)} × {1}) Fn {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∧ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ∧ (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}))) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
238 | 236, 237 | mp3an1 1451 |
. . . . . . . . . . . . . 14
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ∧ (({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} ∩ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}))) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
239 | 193, 238 | mpanr1 719 |
. . . . . . . . . . . . 13
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)})) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
240 | 189, 192,
239 | syl2an 493 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀) ∧ 𝑛 ∈ (1...𝑁))) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
241 | 240 | anassrs 681 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → ((({((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)} × {1}) ∪ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
242 | 234, 241 | eqtrd 2685 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛) = (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛)) |
243 | 47, 221, 111, 111, 112, 113, 242 | ofval 6948 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑛) + (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑛))) |
244 | 202, 243 | eqtr4d 2688 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) ∧ 𝑛 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
245 | 244 | an32s 863 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
246 | 245 | anasss 680 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
247 | | fveq2 6229 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
248 | 247 | breq2d 4697 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
249 | 248 | ifbid 4141 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
250 | 249 | csbeq1d 3573 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
251 | | fveq2 6229 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1^{st} ‘𝑡) = (1^{st} ‘𝑇)) |
252 | 251 | fveq2d 6233 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
253 | 251 | fveq2d 6233 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
254 | 253 | imaeq1d 5500 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
255 | 254 | xpeq1d 5172 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
256 | 253 | imaeq1d 5500 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
257 | 256 | xpeq1d 5172 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
258 | 255, 257 | uneq12d 3801 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
259 | 252, 258 | oveq12d 6708 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
260 | 259 | csbeq2dv 4025 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
261 | 250, 260 | eqtrd 2685 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
262 | 261 | mpteq2dv 4778 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
263 | 262 | eqeq2d 2661 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
264 | 263, 3 | elrab2 3399 |
. . . . . . . . . . 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 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
265 | 264 | simprbi 479 |
. . . . . . . . . 10
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
266 | 1, 265 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
267 | | breq1 4688 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑀 − 1) → (𝑦 < (2^{nd} ‘𝑇) ↔ (𝑀 − 1) < (2^{nd}
‘𝑇))) |
268 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1)) |
269 | 267, 268 | ifbieq1d 4142 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑀 − 1) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < (2^{nd}
‘𝑇), (𝑀 − 1), (𝑦 + 1))) |
270 | 25 | ltm1d 10994 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((2^{nd}
‘𝑇) − 1) <
(2^{nd} ‘𝑇)) |
271 | 62, 24, 25, 90, 270 | lelttrd 10233 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2^{nd} ‘𝑇)) |
272 | 137, 62, 25, 63, 271 | lttrd 10236 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀 − 1) < (2^{nd}
‘𝑇)) |
273 | 272 | iftrued 4127 |
. . . . . . . . . . . 12
⊢ (𝜑 → if((𝑀 − 1) < (2^{nd}
‘𝑇), (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1)) |
274 | 269, 273 | sylan9eqr 2707 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑀 − 1)) |
275 | 274 | csbeq1d 3573 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 − 1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
276 | | oveq2 6698 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1))) |
277 | 276 | imaeq2d 5501 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑀 − 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...(𝑀 −
1)))) |
278 | 277 | xpeq1d 5172 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑀 − 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1})) |
279 | 278 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1})) |
280 | | oveq1 6697 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1)) |
281 | 280, 74 | sylan9eqr 2707 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀) |
282 | 281 | oveq1d 6705 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁)) |
283 | 282 | imaeq2d 5501 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ (𝑀...𝑁))) |
284 | 283 | xpeq1d 5172 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})) |
285 | 279, 284 | uneq12d 3801 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))) |
286 | 285 | oveq2d 6706 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
287 | 79, 286 | csbied 3593 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋(𝑀 − 1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
288 | 287 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋(𝑀 − 1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
289 | 275, 288 | eqtrd 2685 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
290 | | 1red 10093 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℝ) |
291 | 62, 26, 290, 91 | lesub1dd 10681 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 − 1) ≤ (𝑁 − 1)) |
292 | | elfz2nn0 12469 |
. . . . . . . . . 10
⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) ↔ ((𝑀 − 1) ∈
ℕ_{0} ∧ (𝑁 − 1) ∈ ℕ_{0} ∧
(𝑀 − 1) ≤ (𝑁 − 1))) |
293 | 79, 29, 291, 292 | syl3anbrc 1265 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
294 | | ovexd 6720 |
. . . . . . . . 9
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))) ∈ V) |
295 | 266, 289,
293, 294 | fvmptd 6327 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))) |
296 | 295 | fveq1d 6231 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...(𝑀 − 1)))
× {1}) ∪ (((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛)) |
297 | 296 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...(𝑀 − 1)))
× {1}) ∪ (((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘𝑛)) |
298 | | breq1 4688 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → (𝑦 < (2^{nd} ‘𝑇) ↔ 𝑀 < (2^{nd} ‘𝑇))) |
299 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) |
300 | 298, 299 | ifbieq1d 4142 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑀 → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2^{nd} ‘𝑇), 𝑀, (𝑦 + 1))) |
301 | 271 | iftrued 4127 |
. . . . . . . . . . . 12
⊢ (𝜑 → if(𝑀 < (2^{nd} ‘𝑇), 𝑀, (𝑦 + 1)) = 𝑀) |
302 | 300, 301 | sylan9eqr 2707 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
303 | 302 | csbeq1d 3573 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
304 | | oveq2 6698 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
305 | 304 | imaeq2d 5501 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀))) |
306 | 305 | xpeq1d 5172 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})) |
307 | | oveq1 6697 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
308 | 307 | oveq1d 6705 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
309 | 308 | imaeq2d 5501 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
310 | 309 | xpeq1d 5172 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
311 | 306, 310 | uneq12d 3801 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
312 | 311 | oveq2d 6706 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑀 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
313 | 312 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
314 | 40, 313 | csbied 3593 |
. . . . . . . . . . 11
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
315 | 314 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
316 | 303, 315 | eqtrd 2685 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
317 | 29 | nn0zd 11518 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
318 | 25, 26, 290, 34 | lesub1dd 10681 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘𝑇) − 1) ≤
(𝑁 −
1)) |
319 | | eluz2 11731 |
. . . . . . . . . . . . 13
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘((2^{nd} ‘𝑇) − 1)) ↔ (((2^{nd}
‘𝑇) − 1) ∈
ℤ ∧ (𝑁 − 1)
∈ ℤ ∧ ((2^{nd} ‘𝑇) − 1) ≤ (𝑁 − 1))) |
320 | 21, 317, 318, 319 | syl3anbrc 1265 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 − 1) ∈
(ℤ_{≥}‘((2^{nd} ‘𝑇) − 1))) |
321 | | fzss2 12419 |
. . . . . . . . . . . 12
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘((2^{nd} ‘𝑇) − 1)) → (1...((2^{nd}
‘𝑇) − 1))
⊆ (1...(𝑁 −
1))) |
322 | 320, 321 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1...((2^{nd}
‘𝑇) − 1))
⊆ (1...(𝑁 −
1))) |
323 | | fz1ssfz0 12474 |
. . . . . . . . . . 11
⊢
(1...(𝑁 − 1))
⊆ (0...(𝑁 −
1)) |
324 | 322, 323 | syl6ss 3648 |
. . . . . . . . . 10
⊢ (𝜑 → (1...((2^{nd}
‘𝑇) − 1))
⊆ (0...(𝑁 −
1))) |
325 | 324, 40 | sseldd 3637 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1))) |
326 | | ovexd 6720 |
. . . . . . . . 9
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
327 | 266, 316,
325, 326 | fvmptd 6327 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑀) = ((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
328 | 327 | fveq1d 6231 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑀)‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
329 | 328 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) → ((𝐹‘𝑀)‘𝑛) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑛)) |
330 | 246, 297,
329 | 3eqtr4d 2695 |
. . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘𝑀)‘𝑛)) |
331 | 330 | expr 642 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ≠ ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘𝑀)‘𝑛))) |
332 | 331 | necon1d 2845 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) → 𝑛 = ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) |
333 | | elmapi 7921 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
334 | 44, 333 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
335 | 334, 42 | ffvelrnd 6400 |
. . . . . . . . 9
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ∈ (0..^𝐾)) |
336 | | elfzonn0 12552 |
. . . . . . . . 9
⊢
(((1^{st} ‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ∈
ℕ_{0}) |
337 | 335, 336 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ∈
ℕ_{0}) |
338 | 337 | nn0red 11390 |
. . . . . . 7
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ∈ ℝ) |
339 | 338 | ltp1d 10992 |
. . . . . . 7
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) < (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 1)) |
340 | 338, 339 | ltned 10211 |
. . . . . 6
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ≠ (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 1)) |
341 | 295 | fveq1d 6231 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
342 | | ovexd 6720 |
. . . . . . . . 9
⊢ (𝜑 → (1...𝑁) ∈ V) |
343 | | eqidd 2652 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((1^{st} ‘(1^{st}
‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
344 | | fzss1 12418 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈
(ℤ_{≥}‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) |
345 | 76, 344 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑀...𝑁) ⊆ (1...𝑁)) |
346 | | eluzfz1 12386 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
347 | 93, 346 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
348 | | fnfvima 6536 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) Fn (1...𝑁) ∧ (𝑀...𝑁) ⊆ (1...𝑁) ∧ 𝑀 ∈ (𝑀...𝑁)) → ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) |
349 | 119, 345,
347, 348 | syl3anc 1366 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) |
350 | | fvun2 6309 |
. . . . . . . . . . . . 13
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) Fn
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) ∧ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
351 | 50, 53, 350 | mp3an12 1454 |
. . . . . . . . . . . 12
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) = ∅ ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
352 | 69, 349, 351 | syl2anc 694 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
353 | 51 | fvconst2 6510 |
. . . . . . . . . . . 12
⊢
(((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 0) |
354 | 349, 353 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 0) |
355 | 352, 354 | eqtrd 2685 |
. . . . . . . . . 10
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 0) |
356 | 355 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 0) |
357 | 46, 109, 342, 342, 112, 343, 356 | ofval 6948 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 0)) |
358 | 42, 357 | mpdan 703 |
. . . . . . 7
⊢ (𝜑 → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑀 − 1))) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (𝑀...𝑁)) × {0})))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 0)) |
359 | 337 | nn0cnd 11391 |
. . . . . . . 8
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ∈ ℂ) |
360 | 359 | addid1d 10274 |
. . . . . . 7
⊢ (𝜑 → (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 0) = ((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
361 | 341, 358,
360 | 3eqtrd 2689 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((1^{st} ‘(1^{st}
‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
362 | 327 | fveq1d 6231 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝑀)‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
363 | | fzss2 12419 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) → (1...𝑀) ⊆ (1...𝑁)) |
364 | 93, 363 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑀) ⊆ (1...𝑁)) |
365 | | elfz1end 12409 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀)) |
366 | 61, 365 | sylib 208 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
367 | | fnfvima 6536 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) Fn (1...𝑁) ∧ (1...𝑀) ⊆ (1...𝑁) ∧ 𝑀 ∈ (1...𝑀)) → ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) |
368 | 119, 364,
366, 367 | syl3anc 1366 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) |
369 | | fvun1 6308 |
. . . . . . . . . . . . 13
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
370 | 204, 133,
369 | mp3an12 1454 |
. . . . . . . . . . . 12
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
371 | 212, 368,
370 | syl2anc 694 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
372 | 48 | fvconst2 6510 |
. . . . . . . . . . . 12
⊢
(((2^{nd} ‘(1^{st} ‘𝑇))‘𝑀) ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 1) |
373 | 368, 372 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 1) |
374 | 371, 373 | eqtrd 2685 |
. . . . . . . . . 10
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 1) |
375 | 374 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = 1) |
376 | 46, 220, 342, 342, 112, 343, 375 | ofval 6948 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 1)) |
377 | 42, 376 | mpdan 703 |
. . . . . . 7
⊢ (𝜑 → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 1)) |
378 | 362, 377 | eqtrd 2685 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑀)‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) = (((1^{st}
‘(1^{st} ‘𝑇))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) + 1)) |
379 | 340, 361,
378 | 3netr4d 2900 |
. . . . 5
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ≠ ((𝐹‘𝑀)‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
380 | | fveq2 6229 |
. . . . . 6
⊢ (𝑛 = ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
381 | | fveq2 6229 |
. . . . . 6
⊢ (𝑛 = ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀))) |
382 | 380, 381 | neeq12d 2884 |
. . . . 5
⊢ (𝑛 = ((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)) ≠ ((𝐹‘𝑀)‘((2^{nd}
‘(1^{st} ‘𝑇))‘𝑀)))) |
383 | 379, 382 | syl5ibrcom 237 |
. . . 4
⊢ (𝜑 → (𝑛 = ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))) |
384 | 383 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 = ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))) |
385 | 332, 384 | impbid 202 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ 𝑛 = ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀))) |
386 | 42, 385 | riota5 6677 |
1
⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) = ((2^{nd} ‘(1^{st}
‘𝑇))‘𝑀)) |