Step | Hyp | Ref
| Expression |
1 | | poimirlem22.2 |
. . . . . 6
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | elrabi 3608 |
. . . . . . 7
⊢ (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 |
. . . . . . 7
⊢ 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 2899 |
. . . . . 6
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
5 | 1, 4 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
6 | | xp1st 7568 |
. . . . 5
⊢ (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 | | xp1st 7568 |
. . . . 5
⊢
((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))) |
8 | 5, 6, 7 | 3syl 18 |
. . . 4
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
9 | | xp2nd 7569 |
. . . . . . . 8
⊢
((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)}) |
10 | 5, 6, 9 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
11 | | fvex 6543 |
. . . . . . . 8
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
12 | | f1oeq1 6464 |
. . . . . . . 8
⊢ (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))) |
13 | 11, 12 | elab 3600 |
. . . . . . 7
⊢
((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)) |
14 | 10, 13 | sylib 219 |
. . . . . 6
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
15 | | poimirlem15.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
(1...(𝑁 −
1))) |
16 | | elfznn 12775 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^{nd}
‘𝑇) ∈
ℕ) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℕ) |
18 | 17 | nnred 11490 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℝ) |
19 | 18 | ltp1d 11407 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
20 | 18, 19 | ltned 10612 |
. . . . . . . . . 10
⊢ (𝜑 → (2^{nd}
‘𝑇) ≠
((2^{nd} ‘𝑇)
+ 1)) |
21 | 20 | necomd 3037 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ≠
(2^{nd} ‘𝑇)) |
22 | | fvex 6543 |
. . . . . . . . . . 11
⊢
(2^{nd} ‘𝑇) ∈ V |
23 | | ovex 7039 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘𝑇) + 1) ∈ V |
24 | | f1oprg 6519 |
. . . . . . . . . . 11
⊢
((((2^{nd} ‘𝑇) ∈ V ∧ ((2^{nd}
‘𝑇) + 1) ∈ V)
∧ (((2^{nd} ‘𝑇) + 1) ∈ V ∧ (2^{nd}
‘𝑇) ∈ V)) →
(((2^{nd} ‘𝑇)
≠ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd} ‘𝑇) + 1) ≠ (2^{nd}
‘𝑇)) →
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)})) |
25 | 22, 23, 23, 22, 24 | mp4an 689 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ≠ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≠
(2^{nd} ‘𝑇))
→ {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)}) |
26 | 20, 21, 25 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)}) |
27 | | prcom 4569 |
. . . . . . . . . 10
⊢
{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)} |
28 | | f1oeq3 6466 |
. . . . . . . . . 10
⊢
({((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} ↔ {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
29 | 27, 28 | ax-mp 5 |
. . . . . . . . 9
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} ↔ {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
30 | 26, 29 | sylib 219 |
. . . . . . . 8
⊢ (𝜑 → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
31 | | f1oi 6512 |
. . . . . . . 8
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})):((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})–1-1-onto→((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
32 | | disjdif 4329 |
. . . . . . . . 9
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) =
∅ |
33 | | f1oun 6494 |
. . . . . . . . 9
⊢
((({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})):((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})–1-1-onto→((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ∧
(({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ∅ ∧
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ∅)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
34 | 32, 32, 33 | mpanr12 701 |
. . . . . . . 8
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})):((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})–1-1-onto→((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
35 | 30, 31, 34 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
36 | | poimir.0 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℕ) |
37 | 36 | nncnd 11491 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℂ) |
38 | | npcan1 10902 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
39 | 37, 38 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
40 | 36 | nnzd 11924 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) |
41 | | peano2zm 11863 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
42 | 40, 41 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
43 | | uzid 12097 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
44 | | peano2uz 12139 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
45 | 42, 43, 44 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
46 | 39, 45 | eqeltrrd 2882 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘(𝑁 − 1))) |
47 | | fzss2 12786 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ_{≥}‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
49 | 48, 15 | sseldd 3885 |
. . . . . . . . . 10
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈ (1...𝑁)) |
50 | 17 | peano2nnd 11492 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℕ) |
51 | 42 | zred 11925 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
52 | 36 | nnred 11490 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℝ) |
53 | | elfzle2 12750 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^{nd}
‘𝑇) ≤ (𝑁 − 1)) |
54 | 15, 53 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ (𝑁 − 1)) |
55 | 52 | ltm1d 11409 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
56 | 18, 51, 52, 54, 55 | lelttrd 10634 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘𝑇) < 𝑁) |
57 | 17 | nnzd 11924 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℤ) |
58 | | zltp1le 11870 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘𝑇) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^{nd}
‘𝑇) < 𝑁 ↔ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁)) |
59 | 57, 40, 58 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2^{nd}
‘𝑇) < 𝑁 ↔ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁)) |
60 | 56, 59 | mpbid 233 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ≤ 𝑁) |
61 | | fznn 12814 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℤ →
(((2^{nd} ‘𝑇)
+ 1) ∈ (1...𝑁) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ 𝑁))) |
62 | 40, 61 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (((2^{nd}
‘𝑇) + 1) ∈
(1...𝑁) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ 𝑁))) |
63 | 50, 60, 62 | mpbir2and 709 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
(1...𝑁)) |
64 | | prssi 4655 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ∈ (1...𝑁) ∧ ((2^{nd} ‘𝑇) + 1) ∈ (1...𝑁)) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...𝑁)) |
65 | 49, 63, 64 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...𝑁)) |
66 | | undif 4338 |
. . . . . . . . 9
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
(1...𝑁)) |
67 | 65, 66 | sylib 219 |
. . . . . . . 8
⊢ (𝜑 → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...𝑁)) |
68 | | f1oeq23 6467 |
. . . . . . . 8
⊢
((({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = (1...𝑁) ∧ ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...𝑁)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ↔
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):(1...𝑁)–1-1-onto→(1...𝑁))) |
69 | 67, 67, 68 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ↔
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):(1...𝑁)–1-1-onto→(1...𝑁))) |
70 | 35, 69 | mpbid 233 |
. . . . . 6
⊢ (𝜑 → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)) |
71 | | f1oco 6497 |
. . . . . 6
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)) |
72 | 14, 70, 71 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)) |
73 | | prex 5217 |
. . . . . . . 8
⊢
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∈ V |
74 | | ovex 7039 |
. . . . . . . . 9
⊢
(1...𝑁) ∈
V |
75 | | difexg 5115 |
. . . . . . . . 9
⊢
((1...𝑁) ∈ V
→ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∈ V) |
76 | | resiexg 7466 |
. . . . . . . . 9
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∈ V → ( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ∈
V) |
77 | 74, 75, 76 | mp2b 10 |
. . . . . . . 8
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) ∈ V |
78 | 73, 77 | unex 7317 |
. . . . . . 7
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) ∈
V |
79 | 11, 78 | coex 7482 |
. . . . . 6
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ V |
80 | | f1oeq1 6464 |
. . . . . 6
⊢ (𝑓 = ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))) |
81 | 79, 80 | elab 3600 |
. . . . 5
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)) |
82 | 72, 81 | sylibr 235 |
. . . 4
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
83 | | opelxpi 5472 |
. . . 4
⊢
(((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) ∧ ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ (((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
84 | 8, 82, 83 | syl2anc 584 |
. . 3
⊢ (𝜑 → ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ (((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
85 | | fz1ssfz0 12842 |
. . . . 5
⊢
(1...𝑁) ⊆
(0...𝑁) |
86 | 48, 85 | syl6ss 3896 |
. . . 4
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (0...𝑁)) |
87 | 86, 15 | sseldd 3885 |
. . 3
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈ (0...𝑁)) |
88 | | opelxpi 5472 |
. . 3
⊢
((⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ (((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (2^{nd} ‘𝑇) ∈ (0...𝑁)) → ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
89 | 84, 87, 88 | syl2anc 584 |
. 2
⊢ (𝜑 →
⟨⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
90 | | fveq2 6530 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
91 | 90 | breq2d 4968 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
92 | 91 | ifbid 4397 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
93 | 92 | csbeq1d 3810 |
. . . . . . . . 9
⊢ (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})))) |
94 | | 2fveq3 6535 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
95 | | 2fveq3 6535 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
96 | 95 | imaeq1d 5797 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
97 | 96 | xpeq1d 5464 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
98 | 95 | imaeq1d 5797 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
99 | 98 | xpeq1d 5464 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
100 | 97, 99 | uneq12d 4056 |
. . . . . . . . . . 11
⊢ (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}))) |
101 | 94, 100 | oveq12d 7025 |
. . . . . . . . . 10
⊢ (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})))) |
102 | 101 | csbeq2dv 3813 |
. . . . . . . . 9
⊢ (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})))) |
103 | 93, 102 | eqtrd 2829 |
. . . . . . . 8
⊢ (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})))) |
104 | 103 | mpteq2dv 5050 |
. . . . . . 7
⊢ (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}))))) |
105 | 104 | eqeq2d 2803 |
. . . . . 6
⊢ (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})))))) |
106 | 105, 3 | elrab2 3616 |
. . . . 5
⊢ (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})))))) |
107 | 106 | simprbi 497 |
. . . 4
⊢ (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}))))) |
108 | 1, 107 | syl 17 |
. . 3
⊢ (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}))))) |
109 | | imaco 5971 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
= ((2^{nd} ‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...𝑦))) |
110 | | f1ofn 6476 |
. . . . . . . . . . . . . . . 16
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
111 | 26, 110 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
112 | 111 | ad2antrr 722 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
113 | | incom 4094 |
. . . . . . . . . . . . . . 15
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ (1...𝑦)) = ((1...𝑦) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
114 | | elfznn0 12839 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ_{0}) |
115 | 114 | nn0red 11793 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
116 | | ltnle 10556 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ ℝ ∧
(2^{nd} ‘𝑇)
∈ ℝ) → (𝑦
< (2^{nd} ‘𝑇) ↔ ¬ (2^{nd} ‘𝑇) ≤ 𝑦)) |
117 | 115, 18, 116 | syl2anr 596 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) ↔ ¬ (2^{nd}
‘𝑇) ≤ 𝑦)) |
118 | 117 | biimpa 477 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ (2^{nd}
‘𝑇) ≤ 𝑦) |
119 | | elfzle2 12750 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2^{nd} ‘𝑇) ∈ (1...𝑦) → (2^{nd} ‘𝑇) ≤ 𝑦) |
120 | 118, 119 | nsyl 142 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ (2^{nd}
‘𝑇) ∈ (1...𝑦)) |
121 | | disjsn 4548 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑦) ∩
{(2^{nd} ‘𝑇)}) = ∅ ↔ ¬ (2^{nd}
‘𝑇) ∈ (1...𝑦)) |
122 | 120, 121 | sylibr 235 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) =
∅) |
123 | 115 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 ∈ ℝ) |
124 | 18 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ∈
ℝ) |
125 | 50 | nnred 11490 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
126 | 125 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
127 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 < (2^{nd} ‘𝑇)) |
128 | 19 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
129 | 123, 124,
126, 127, 128 | lttrd 10637 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 < ((2^{nd} ‘𝑇) + 1)) |
130 | | ltnle 10556 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℝ ∧
((2^{nd} ‘𝑇)
+ 1) ∈ ℝ) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
131 | 115, 125,
130 | syl2anr 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
132 | 131 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
133 | 129, 132 | mpbid 233 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ ((2^{nd}
‘𝑇) + 1) ≤ 𝑦) |
134 | | elfzle2 12750 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2^{nd} ‘𝑇) + 1) ∈ (1...𝑦) → ((2^{nd} ‘𝑇) + 1) ≤ 𝑦) |
135 | 133, 134 | nsyl 142 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ ((2^{nd}
‘𝑇) + 1) ∈
(1...𝑦)) |
136 | | disjsn 4548 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)}) = ∅ ↔ ¬ ((2^{nd} ‘𝑇) + 1) ∈ (1...𝑦)) |
137 | 135, 136 | sylibr 235 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {((2^{nd}
‘𝑇) + 1)}) =
∅) |
138 | 122, 137 | uneq12d 4056 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) ∪
((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)})) = (∅ ∪ ∅)) |
139 | | df-pr 4469 |
. . . . . . . . . . . . . . . . . 18
⊢
{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} = ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) +
1)}) |
140 | 139 | ineq2i 4101 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ∩
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ((1...𝑦) ∩
({(2^{nd} ‘𝑇)} ∪ {((2^{nd} ‘𝑇) + 1)})) |
141 | | indi 4165 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ∩
({(2^{nd} ‘𝑇)} ∪ {((2^{nd} ‘𝑇) + 1)})) = (((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) ∪
((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)})) |
142 | 140, 141 | eqtr2i 2818 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑦) ∩
{(2^{nd} ‘𝑇)}) ∪ ((1...𝑦) ∩ {((2^{nd} ‘𝑇) + 1)})) = ((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
143 | | un0 4258 |
. . . . . . . . . . . . . . . 16
⊢ (∅
∪ ∅) = ∅ |
144 | 138, 142,
143 | 3eqtr3g 2852 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) =
∅) |
145 | 113, 144 | syl5eq 2841 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(1...𝑦)) =
∅) |
146 | | fnimadisj 6340 |
. . . . . . . . . . . . . 14
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} Fn {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(1...𝑦)) = ∅) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) = ∅) |
147 | 112, 145,
146 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) = ∅) |
148 | 39 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
149 | | elfzuz3 12744 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ_{≥}‘𝑦)) |
150 | | peano2uz 12139 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
151 | 149, 150 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
152 | 151 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
153 | 148, 152 | eqeltrrd 2882 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_{≥}‘𝑦)) |
154 | | fzss2 12786 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑦) → (1...𝑦) ⊆ (1...𝑁)) |
155 | | reldisj 4310 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ⊆
(1...𝑁) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ∅
↔ (1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
156 | 153, 154,
155 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ∅
↔ (1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
157 | 156 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ∅
↔ (1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
158 | 144, 157 | mpbid 233 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
159 | | resiima 5812 |
. . . . . . . . . . . . . 14
⊢
((1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) → (( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦)) |
160 | 158, 159 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (1...𝑦)) =
(1...𝑦)) |
161 | 147, 160 | uneq12d 4056 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) “ (1...𝑦))) = (∅ ∪ (1...𝑦))) |
162 | | imaundir 5877 |
. . . . . . . . . . . 12
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...𝑦)) =
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) “ (1...𝑦))) |
163 | | uncom 4045 |
. . . . . . . . . . . . 13
⊢ (∅
∪ (1...𝑦)) =
((1...𝑦) ∪
∅) |
164 | | un0 4258 |
. . . . . . . . . . . . 13
⊢
((1...𝑦) ∪
∅) = (1...𝑦) |
165 | 163, 164 | eqtr2i 2818 |
. . . . . . . . . . . 12
⊢
(1...𝑦) = (∅
∪ (1...𝑦)) |
166 | 161, 162,
165 | 3eqtr4g 2854 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...𝑦)) = (1...𝑦)) |
167 | 166 | imaeq2d 5798 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...𝑦)))
= ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦))) |
168 | 109, 167 | syl5eq 2841 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
= ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦))) |
169 | 168 | xpeq1d 5464 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) = (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) ×
{1})) |
170 | | imaco 5971 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ ((𝑦 +
1)...𝑁))) |
171 | | imaundir 5877 |
. . . . . . . . . . . . 13
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
((𝑦 + 1)...𝑁)) = (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((𝑦 + 1)...𝑁))) |
172 | | imassrn 5809 |
. . . . . . . . . . . . . . . . 17
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} |
173 | 172 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}) |
174 | | fnima 6338 |
. . . . . . . . . . . . . . . . . . 19
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} Fn {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
175 | 26, 110, 174 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ran {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
176 | 175 | ad2antrr 722 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
177 | | elfzelz 12747 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
178 | | zltp1le 11870 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℤ ∧
(2^{nd} ‘𝑇)
∈ ℤ) → (𝑦
< (2^{nd} ‘𝑇) ↔ (𝑦 + 1) ≤ (2^{nd} ‘𝑇))) |
179 | 177, 57, 178 | syl2anr 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) ↔ (𝑦 + 1) ≤ (2^{nd} ‘𝑇))) |
180 | 179 | biimpa 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 + 1) ≤ (2^{nd} ‘𝑇)) |
181 | 18, 52, 56 | ltled 10624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ 𝑁) |
182 | 181 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ≤ 𝑁) |
183 | 57 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^{nd}
‘𝑇) ∈
ℤ) |
184 | | nn0p1nn 11773 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℕ_{0}
→ (𝑦 + 1) ∈
ℕ) |
185 | 114, 184 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ) |
186 | 185 | nnzd 11924 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ) |
187 | 186 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ ℤ) |
188 | 40 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℤ) |
189 | | elfz 12737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((2^{nd} ‘𝑇) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
190 | 183, 187,
188, 189 | syl3anc 1362 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
191 | 190 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
192 | 180, 182,
191 | mpbir2and 709 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁)) |
193 | | 1red 10477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 1 ∈
ℝ) |
194 | | ltle 10565 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 ∈ ℝ ∧
(2^{nd} ‘𝑇)
∈ ℝ) → (𝑦
< (2^{nd} ‘𝑇) → 𝑦 ≤ (2^{nd} ‘𝑇))) |
195 | 115, 18, 194 | syl2anr 596 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) → 𝑦 ≤ (2^{nd} ‘𝑇))) |
196 | 195 | imp 407 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 ≤ (2^{nd} ‘𝑇)) |
197 | 123, 124,
193, 196 | leadd1dd 11091 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1)) |
198 | 60 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ≤ 𝑁) |
199 | 57 | peano2zd 11928 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℤ) |
200 | 199 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) + 1) ∈
ℤ) |
201 | | elfz 12737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((2^{nd} ‘𝑇) + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(((2^{nd} ‘𝑇)
+ 1) ∈ ((𝑦 +
1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd}
‘𝑇) + 1) ∧
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑁))) |
202 | 200, 187,
188, 201 | syl3anc 1362 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁))) |
203 | 202 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁))) |
204 | 197, 198,
203 | mpbir2and 709 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁)) |
205 | | prssi 4655 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2^{nd} ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ∧ ((2^{nd} ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁)) → {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁)) |
206 | 192, 204,
205 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
((𝑦 + 1)...𝑁)) |
207 | | imass2 5833 |
. . . . . . . . . . . . . . . . . 18
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ⊆ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁))) |
208 | 206, 207 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ⊆
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁))) |
209 | 176, 208 | eqsstrrd 3922 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ⊆ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁))) |
210 | 173, 209 | eqssd 3901 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = ran {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}) |
211 | | f1ofo 6482 |
. . . . . . . . . . . . . . . . . 18
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–onto→{((2^{nd} ‘𝑇) + 1), (2^{nd}
‘𝑇)}) |
212 | | forn 6453 |
. . . . . . . . . . . . . . . . . 18
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} = {((2^{nd} ‘𝑇) + 1), (2^{nd}
‘𝑇)}) |
213 | 26, 211, 212 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} = {((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)}) |
214 | 213, 27 | syl6eq 2845 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
215 | 214 | ad2antrr 722 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} = {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
216 | 210, 215 | eqtrd 2829 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
217 | | undif 4338 |
. . . . . . . . . . . . . . . . 17
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) ↔ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ((𝑦 + 1)...𝑁)) |
218 | 206, 217 | sylib 219 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) = ((𝑦 + 1)...𝑁)) |
219 | 218 | imaeq2d 5798 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((𝑦 +
1)...𝑁))) |
220 | | fnresi 6337 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) Fn ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) |
221 | | incom 4094 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
222 | 221, 32 | eqtri 2817 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ |
223 | | fnimadisj 6340 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) Fn ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∧ (((1...𝑁)
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) → (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) |
224 | 220, 222,
223 | mp2an 688 |
. . . . . . . . . . . . . . . . . 18
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ |
225 | 224 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) |
226 | | nnuz 12119 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ_{≥}‘1) |
227 | 185, 226 | syl6eleq 2891 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈
(ℤ_{≥}‘1)) |
228 | | fzss1 12785 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 + 1) ∈
(ℤ_{≥}‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
229 | 227, 228 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
230 | 229 | ssdifd 4033 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
231 | | resiima 5812 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) → ((
I ↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 +
1)...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
232 | 230, 231 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
233 | 232 | ad2antlr 723 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 +
1)...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
234 | 225, 233 | uneq12d 4056 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) =
(∅ ∪ (((𝑦 +
1)...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
235 | | imaundi 5876 |
. . . . . . . . . . . . . . . 16
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
236 | | uncom 4045 |
. . . . . . . . . . . . . . . . 17
⊢ (∅
∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
((((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ∪
∅) |
237 | | un0 4258 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ ∅) =
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
238 | 236, 237 | eqtr2i 2818 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = (∅ ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
239 | 234, 235,
238 | 3eqtr4g 2854 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
240 | 219, 239 | eqtr3d 2831 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((𝑦 +
1)...𝑁)) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
241 | 216, 240 | uneq12d 4056 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((𝑦 + 1)...𝑁))) = ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
242 | 171, 241 | syl5eq 2841 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
((𝑦 + 1)...𝑁)) = ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
243 | 242, 218 | eqtrd 2829 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
((𝑦 + 1)...𝑁)) = ((𝑦 + 1)...𝑁)) |
244 | 243 | imaeq2d 5798 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ ((𝑦 +
1)...𝑁))) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
245 | 170, 244 | syl5eq 2841 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
246 | 245 | xpeq1d 5464 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) × {0}) =
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) |
247 | 169, 246 | uneq12d 4056 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) × {0})) =
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
248 | 247 | oveq2d 7023 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
249 | | iftrue 4381 |
. . . . . . . . 9
⊢ (𝑦 < (2^{nd}
‘𝑇) → if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦) |
250 | 249 | csbeq1d 3810 |
. . . . . . . 8
⊢ (𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋𝑦 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
251 | | vex 3435 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
252 | | oveq2 7015 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦)) |
253 | 252 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
= (((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))) |
254 | 253 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) = ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(1...𝑦)) ×
{1})) |
255 | | oveq1 7014 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1)) |
256 | 255 | oveq1d 7022 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁)) |
257 | 256 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) =
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁))) |
258 | 257 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) × {0}) =
((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) ×
{0})) |
259 | 254, 258 | uneq12d 4056 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0})) =
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0}))) |
260 | 259 | oveq2d 7023 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0})))) |
261 | 251, 260 | csbie 3838 |
. . . . . . . 8
⊢
⦋𝑦 /
𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0}))) |
262 | 250, 261 | syl6eq 2845 |
. . . . . . 7
⊢ (𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0})))) |
263 | 262 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0})))) |
264 | 249 | csbeq1d 3810 |
. . . . . . . 8
⊢ (𝑦 < (2^{nd}
‘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}))) = ⦋𝑦 / 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 | 252 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑦))) |
266 | 265 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1})) |
267 | 256 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
268 | 267 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) |
269 | 266, 268 | uneq12d 4056 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
270 | 269 | oveq2d 7023 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((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...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
271 | 251, 270 | csbie 3838 |
. . . . . . . 8
⊢
⦋𝑦 /
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...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
272 | 264, 271 | syl6eq 2845 |
. . . . . . 7
⊢ (𝑦 < (2^{nd}
‘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}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
273 | 272 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘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}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
274 | 248, 263,
273 | 3eqtr4d 2839 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((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})))) |
275 | | lenlt 10555 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^{nd}
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^{nd} ‘𝑇))) |
276 | 18, 115, 275 | syl2an 595 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^{nd} ‘𝑇))) |
277 | 276 | biimpar 478 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ≤ 𝑦) |
278 | | imaco 5971 |
. . . . . . . . . . 11
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) = ((2^{nd} ‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...(𝑦 +
1)))) |
279 | | imaundir 5877 |
. . . . . . . . . . . . . 14
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...(𝑦 + 1))) =
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(1...(𝑦 +
1)))) |
280 | | imassrn 5809 |
. . . . . . . . . . . . . . . . . 18
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} |
281 | 280 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
282 | 175 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ran {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
283 | 17 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
ℕ) |
284 | 18 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
ℝ) |
285 | 115 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 ∈ ℝ) |
286 | 185 | nnred 11490 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ) |
287 | 286 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (𝑦 + 1) ∈ ℝ) |
288 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ≤ 𝑦) |
289 | 115 | lep1d 11408 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑦 + 1)) |
290 | 289 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 ≤ (𝑦 + 1)) |
291 | 284, 285,
287, 288, 290 | letrd 10633 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ≤ (𝑦 + 1)) |
292 | | fznn 12814 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈ ℤ →
((2^{nd} ‘𝑇)
∈ (1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
293 | 186, 292 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^{nd}
‘𝑇) ∈
(1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
294 | 293 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) ∈
(1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
295 | 283, 291,
294 | mpbir2and 709 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
(1...(𝑦 +
1))) |
296 | 50 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
ℕ) |
297 | | 1red 10477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 1 ∈
ℝ) |
298 | 284, 285,
297, 288 | leadd1dd 11091 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ≤ (𝑦 + 1)) |
299 | | fznn 12814 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈ ℤ →
(((2^{nd} ‘𝑇)
+ 1) ∈ (1...(𝑦 + 1))
↔ (((2^{nd} ‘𝑇) + 1) ∈ ℕ ∧ ((2^{nd}
‘𝑇) + 1) ≤ (𝑦 + 1)))) |
300 | 186, 299 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 + 1)) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ (𝑦 + 1)))) |
301 | 300 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 + 1)) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ (𝑦 + 1)))) |
302 | 296, 298,
301 | mpbir2and 709 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 +
1))) |
303 | | prssi 4655 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘𝑇) ∈ (1...(𝑦 + 1)) ∧ ((2^{nd} ‘𝑇) + 1) ∈ (1...(𝑦 + 1))) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...(𝑦 +
1))) |
304 | 295, 302,
303 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...(𝑦 +
1))) |
305 | | imass2 5833 |
. . . . . . . . . . . . . . . . . . 19
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ⊆ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1)))) |
306 | 304, 305 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ⊆ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1)))) |
307 | 282, 306 | eqsstrrd 3922 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ⊆ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1)))) |
308 | 281, 307 | eqssd 3901 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) = ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
309 | 214 | ad2antrr 722 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} = {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
310 | 308, 309 | eqtrd 2829 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
311 | | undif 4338 |
. . . . . . . . . . . . . . . . . 18
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)) ↔ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
(1...(𝑦 +
1))) |
312 | 304, 311 | sylib 219 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...(𝑦 + 1)) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...(𝑦 +
1))) |
313 | 312 | imaeq2d 5798 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (1...(𝑦 +
1)))) |
314 | 224 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) |
315 | | eluzp1p1 12108 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
316 | 149, 315 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
317 | 316 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
318 | 148, 317 | eqeltrrd 2882 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_{≥}‘(𝑦 + 1))) |
319 | | fzss2 12786 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ_{≥}‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
320 | 318, 319 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
321 | 320 | ssdifd 4033 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) |
322 | 321 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) |
323 | | resiima 5812 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((1...(𝑦 +
1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
324 | 322, 323 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((1...(𝑦 +
1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
325 | 314, 324 | uneq12d 4056 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((1...(𝑦 + 1)) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
326 | | imaundi 5876 |
. . . . . . . . . . . . . . . . 17
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((1...(𝑦 + 1)) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
327 | | uncom 4045 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ ((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = (((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪
∅) |
328 | | un0 4258 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ ∅) = ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
329 | 327, 328 | eqtr2i 2818 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
330 | 325, 326,
329 | 3eqtr4g 2854 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
331 | 313, 330 | eqtr3d 2831 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (1...(𝑦 +
1))) = ((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
332 | 310, 331 | uneq12d 4056 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(1...(𝑦 + 1)))) =
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
333 | 279, 332 | syl5eq 2841 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...(𝑦 + 1))) =
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
334 | 333, 312 | eqtrd 2829 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...(𝑦 + 1))) =
(1...(𝑦 +
1))) |
335 | 334 | imaeq2d 5798 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...(𝑦 +
1)))) = ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1)))) |
336 | 278, 335 | syl5eq 2841 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) = ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1)))) |
337 | 336 | xpeq1d 5464 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) = (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) ×
{1})) |
338 | | imaco 5971 |
. . . . . . . . . . 11
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (((𝑦 + 1)
+ 1)...𝑁))) |
339 | 111 | ad2antrr 722 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
340 | | incom 4094 |
. . . . . . . . . . . . . . . 16
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
341 | 125 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
342 | 185 | peano2nnd 11492 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ) |
343 | 342 | nnred 11490 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ) |
344 | 343 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((𝑦 + 1) + 1) ∈ ℝ) |
345 | 19 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
346 | 115 | ltp1d 11407 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1)) |
347 | 346 | ad2antlr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 < (𝑦 + 1)) |
348 | 284, 285,
287, 288, 347 | lelttrd 10634 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) < (𝑦 + 1)) |
349 | 284, 287,
297, 348 | ltadd1dd 11088 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1)) |
350 | 284, 341,
344, 345, 349 | lttrd 10637 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) < ((𝑦 + 1) + 1)) |
351 | | ltnle 10556 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2^{nd} ‘𝑇) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) →
((2^{nd} ‘𝑇)
< ((𝑦 + 1) + 1) ↔
¬ ((𝑦 + 1) + 1) ≤
(2^{nd} ‘𝑇))) |
352 | 18, 343, 351 | syl2an 595 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇))) |
353 | 352 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇))) |
354 | 350, 353 | mpbid 233 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇)) |
355 | | elfzle1 12749 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2^{nd} ‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ (2^{nd} ‘𝑇)) |
356 | 354, 355 | nsyl 142 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ (2^{nd}
‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁)) |
357 | | disjsn 4548 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{(2^{nd} ‘𝑇)}) = ∅ ↔ ¬ (2^{nd}
‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁)) |
358 | 356, 357 | sylibr 235 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇)}) = ∅) |
359 | | ltnle 10556 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2^{nd} ‘𝑇) + 1) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ)
→ (((2^{nd} ‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd} ‘𝑇) + 1))) |
360 | 125, 343,
359 | syl2an 595 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1))) |
361 | 360 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1))) |
362 | 349, 361 | mpbid 233 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1)) |
363 | | elfzle1 12749 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ ((2^{nd} ‘𝑇) + 1)) |
364 | 362, 363 | nsyl 142 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((2^{nd}
‘𝑇) + 1) ∈
(((𝑦 + 1) + 1)...𝑁)) |
365 | | disjsn 4548 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{((2^{nd} ‘𝑇)
+ 1)}) = ∅ ↔ ¬ ((2^{nd} ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁)) |
366 | 364, 365 | sylibr 235 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)}) =
∅) |
367 | 358, 366 | uneq12d 4056 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)})) = (∅ ∪
∅)) |
368 | 139 | ineq2i 4101 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) +
1)})) |
369 | | indi 4165 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) + 1)})) =
(((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd}
‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)})) |
370 | 368, 369 | eqtr2i 2818 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{(2^{nd} ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)})) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
371 | 367, 370,
143 | 3eqtr3g 2852 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) =
∅) |
372 | 340, 371 | syl5eq 2841 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(((𝑦 + 1) + 1)...𝑁)) = ∅) |
373 | | fnimadisj 6340 |
. . . . . . . . . . . . . . 15
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} Fn {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(((𝑦 + 1) + 1)...𝑁)) = ∅) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
374 | 339, 372,
373 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
375 | 342, 226 | syl6eleq 2891 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ_{≥}‘1)) |
376 | | fzss1 12785 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑦 + 1) + 1) ∈
(ℤ_{≥}‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
377 | | reldisj 4310 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ ↔
(((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
378 | 375, 376,
377 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ ↔
(((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
379 | 378 | ad2antlr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ ↔
(((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
380 | 371, 379 | mpbid 233 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
381 | | resiima 5812 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 + 1) +
1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁)) |
382 | 380, 381 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 + 1) +
1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁)) |
383 | 374, 382 | uneq12d 4056 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1) + 1)...𝑁))) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁))) |
384 | | imaundir 5877 |
. . . . . . . . . . . . 13
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(((𝑦 + 1) + 1)...𝑁)) = (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1) + 1)...𝑁))) |
385 | | uncom 4045 |
. . . . . . . . . . . . . 14
⊢ (∅
∪ (((𝑦 + 1) +
1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) |
386 | | un0 4258 |
. . . . . . . . . . . . . 14
⊢ ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) = (((𝑦 + 1) + 1)...𝑁) |
387 | 385, 386 | eqtr2i 2818 |
. . . . . . . . . . . . 13
⊢ (((𝑦 + 1) + 1)...𝑁) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁)) |
388 | 383, 384,
387 | 3eqtr4g 2854 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁)) |
389 | 388 | imaeq2d 5798 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (((𝑦 + 1)
+ 1)...𝑁))) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
390 | 338, 389 | syl5eq 2841 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
391 | 390 | xpeq1d 5464 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) × {0}) =
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
392 | 337, 391 | uneq12d 4056 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) × {0})) =
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
393 | 277, 392 | syldan 591 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) × {0})) =
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
394 | 393 | oveq2d 7023 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
395 | | iffalse 4384 |
. . . . . . . . 9
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) → if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1)) |
396 | 395 | csbeq1d 3810 |
. . . . . . . 8
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋(𝑦 + 1) /
𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
397 | | ovex 7039 |
. . . . . . . . 9
⊢ (𝑦 + 1) ∈ V |
398 | | oveq2 7015 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1))) |
399 | 398 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
= (((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1)))) |
400 | 399 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) = ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(1...(𝑦 + 1))) ×
{1})) |
401 | | oveq1 7014 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1)) |
402 | 401 | oveq1d 7022 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁)) |
403 | 402 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) =
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁))) |
404 | 403 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) × {0}) =
((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) ×
{0})) |
405 | 400, 404 | uneq12d 4056 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0})) =
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0}))) |
406 | 405 | oveq2d 7023 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0})))) |
407 | 397, 406 | csbie 3838 |
. . . . . . . 8
⊢
⦋(𝑦 +
1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0}))) |
408 | 396, 407 | syl6eq 2845 |
. . . . . . 7
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0})))) |
409 | 408 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0})))) |
410 | 395 | csbeq1d 3810 |
. . . . . . . 8
⊢ (¬
𝑦 < (2^{nd}
‘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}))) = ⦋(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})))) |
411 | 398 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...(𝑦 +
1)))) |
412 | 411 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1})) |
413 | 402 | imaeq2d 5798 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
414 | 413 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
415 | 412, 414 | uneq12d 4056 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
416 | 415 | oveq2d 7023 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 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...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
417 | 397, 416 | csbie 3838 |
. . . . . . . 8
⊢
⦋(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...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
418 | 410, 417 | syl6eq 2845 |
. . . . . . 7
⊢ (¬
𝑦 < (2^{nd}
‘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}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
419 | 418 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘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}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
420 | 394, 409,
419 | 3eqtr4d 2839 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((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})))) |
421 | 274, 420 | pm2.61dan 809 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((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})))) |
422 | 421 | mpteq2dva 5049 |
. . 3
⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((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}))))) |
423 | 108, 422 | eqtr4d 2832 |
. 2
⊢ (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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0}))))) |
424 | | opex 5241 |
. . . . . . 7
⊢
⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ V |
425 | 424, 22 | op1std 7546 |
. . . . . 6
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → (1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))))⟩) |
426 | 424, 22 | op2ndd 7547 |
. . . . . 6
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
427 | | breq2 4960 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
428 | 427 | ifbid 4397 |
. . . . . . . 8
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
429 | 428 | csbeq1d 3810 |
. . . . . . 7
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘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})))) |
430 | | fvex 6543 |
. . . . . . . . . 10
⊢
(1^{st} ‘(1^{st} ‘𝑇)) ∈ V |
431 | 430, 79 | op1std 7546 |
. . . . . . . . 9
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
→ (1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
432 | 430, 79 | op2ndd 7547 |
. . . . . . . . 9
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
→ (2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))))) |
433 | | id 22 |
. . . . . . . . . 10
⊢
((1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇)) →
(1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
434 | | imaeq1 5793 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) = (((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(1...𝑗))) |
435 | 434 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
(((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1})) |
436 | | imaeq1 5793 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
((2^{nd} ‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁))) |
437 | 436 | xpeq1d 5464 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
(((2^{nd} ‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) ×
{0})) |
438 | 435, 437 | uneq12d 4056 |
. . . . . . . . . 10
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0}))) |
439 | 433, 438 | oveqan12d 7026 |
. . . . . . . . 9
⊢
(((1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇)) ∧
(2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
440 | 431, 432,
439 | syl2anc 584 |
. . . . . . . 8
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
441 | 440 | csbeq2dv 3813 |
. . . . . . 7
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 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}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
442 | 429, 441 | sylan9eqr 2851 |
. . . . . 6
⊢
(((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∧ (2^{nd} ‘𝑡) = (2^{nd} ‘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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
443 | 425, 426,
442 | syl2anc 584 |
. . . . 5
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
444 | 443 | mpteq2dv 5050 |
. . . 4
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0}))))) |
445 | 444 | eqeq2d 2803 |
. . 3
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))))) |
446 | 445, 3 | elrab2 3616 |
. 2
⊢
(⟨⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))))⟩, (2^{nd} ‘𝑇)⟩ ∈ 𝑆 ↔ (⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘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)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))))) |
447 | 89, 423, 446 | sylanbrc 583 |
1
⊢ (𝜑 →
⟨⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ 𝑆) |