Step | Hyp | Ref
| Expression |
1 | | poimirlem22.2 |
. . . . . 6
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | elrabi 3346 |
. . . . . . 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 2716 |
. . . . . 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 7150 |
. . . . 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 7150 |
. . . . 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 7151 |
. . . . . . . 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 6163 |
. . . . . . . 8
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
12 | | f1oeq1 6089 |
. . . . . . . 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 3337 |
. . . . . . 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 208 |
. . . . . 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 12320 |
. . . . . . . . . . . . 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 10987 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℝ) |
19 | 18 | ltp1d 10906 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
20 | 18, 19 | ltned 10125 |
. . . . . . . . . 10
⊢ (𝜑 → (2^{nd}
‘𝑇) ≠
((2^{nd} ‘𝑇)
+ 1)) |
21 | 20 | necomd 2845 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ≠
(2^{nd} ‘𝑇)) |
22 | | fvex 6163 |
. . . . . . . . . . 11
⊢
(2^{nd} ‘𝑇) ∈ V |
23 | | ovex 6638 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘𝑇) + 1) ∈ V |
24 | | f1oprg 6143 |
. . . . . . . . . . 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 708 |
. . . . . . . . . 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 692 |
. . . . . . . . 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 4242 |
. . . . . . . . . 10
⊢
{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)} |
28 | | f1oeq3 6091 |
. . . . . . . . . 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 208 |
. . . . . . . 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 6136 |
. . . . . . . 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 4017 |
. . . . . . . . 9
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) =
∅ |
33 | | f1oun 6118 |
. . . . . . . . 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 720 |
. . . . . . . 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 693 |
. . . . . . 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 10988 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℂ) |
38 | | npcan1 10407 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
39 | 37, 38 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
40 | 36 | nnzd 11433 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) |
41 | | peano2zm 11372 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
42 | 40, 41 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
43 | | uzid 11654 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
44 | | peano2uz 11693 |
. . . . . . . . . . . . . 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 2699 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘(𝑁 − 1))) |
47 | | fzss2 12331 |
. . . . . . . . . . . 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 3588 |
. . . . . . . . . 10
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈ (1...𝑁)) |
50 | 17 | peano2nnd 10989 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℕ) |
51 | 42 | zred 11434 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
52 | 36 | nnred 10987 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℝ) |
53 | | elfzle2 12295 |
. . . . . . . . . . . . . 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 10908 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
56 | 18, 51, 52, 54, 55 | lelttrd 10147 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘𝑇) < 𝑁) |
57 | 17 | nnzd 11433 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℤ) |
58 | | zltp1le 11379 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘𝑇) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^{nd}
‘𝑇) < 𝑁 ↔ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁)) |
59 | 57, 40, 58 | syl2anc 692 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2^{nd}
‘𝑇) < 𝑁 ↔ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁)) |
60 | 56, 59 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ≤ 𝑁) |
61 | | fznn 12358 |
. . . . . . . . . . . 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 956 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
(1...𝑁)) |
64 | | prssi 4326 |
. . . . . . . . . 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 692 |
. . . . . . . . 9
⊢ (𝜑 → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...𝑁)) |
66 | | undif 4026 |
. . . . . . . . 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 208 |
. . . . . . . 8
⊢ (𝜑 → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...𝑁)) |
68 | | f1oeq23 6092 |
. . . . . . . 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 692 |
. . . . . . 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 222 |
. . . . . 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 6121 |
. . . . . 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 692 |
. . . . 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 4875 |
. . . . . . . 8
⊢
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∈ V |
74 | | ovex 6638 |
. . . . . . . . 9
⊢
(1...𝑁) ∈
V |
75 | | difexg 4773 |
. . . . . . . . 9
⊢
((1...𝑁) ∈ V
→ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∈ V) |
76 | | resiexg 7056 |
. . . . . . . . 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 6916 |
. . . . . . 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 7072 |
. . . . . 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 6089 |
. . . . . 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 3337 |
. . . . 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 224 |
. . . 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 5113 |
. . . 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 692 |
. . 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 | | 1eluzge0 11684 |
. . . . . 6
⊢ 1 ∈
(ℤ_{≥}‘0) |
86 | | fzss1 12330 |
. . . . . 6
⊢ (1 ∈
(ℤ_{≥}‘0) → (1...𝑁) ⊆ (0...𝑁)) |
87 | 85, 86 | ax-mp 5 |
. . . . 5
⊢
(1...𝑁) ⊆
(0...𝑁) |
88 | 48, 87 | syl6ss 3599 |
. . . 4
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (0...𝑁)) |
89 | 88, 15 | sseldd 3588 |
. . 3
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈ (0...𝑁)) |
90 | | opelxpi 5113 |
. . 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))) |
91 | 84, 89, 90 | syl2anc 692 |
. 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))) |
92 | | fveq2 6153 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
93 | 92 | breq2d 4630 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
94 | 93 | ifbid 4085 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
95 | 94 | csbeq1d 3525 |
. . . . . . . . 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})))) |
96 | | fveq2 6153 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (1^{st} ‘𝑡) = (1^{st} ‘𝑇)) |
97 | 96 | fveq2d 6157 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
98 | 96 | fveq2d 6157 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
99 | 98 | imaeq1d 5429 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
100 | 99 | xpeq1d 5103 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
101 | 98 | imaeq1d 5429 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
102 | 101 | xpeq1d 5103 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
103 | 100, 102 | uneq12d 3751 |
. . . . . . . . . . 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}))) |
104 | 97, 103 | oveq12d 6628 |
. . . . . . . . . 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})))) |
105 | 104 | csbeq2dv 3969 |
. . . . . . . . 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})))) |
106 | 95, 105 | eqtrd 2655 |
. . . . . . . 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})))) |
107 | 106 | mpteq2dv 4710 |
. . . . . . 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}))))) |
108 | 107 | eqeq2d 2631 |
. . . . . 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})))))) |
109 | 108, 3 | elrab2 3352 |
. . . . 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})))))) |
110 | 109 | simprbi 480 |
. . . 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}))))) |
111 | 1, 110 | 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}))))) |
112 | | imaco 5604 |
. . . . . . . . . 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))) |
113 | | f1ofn 6100 |
. . . . . . . . . . . . . . . 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)}) |
114 | 26, 113 | 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)}) |
115 | 114 | ad2antrr 761 |
. . . . . . . . . . . . . 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)}) |
116 | | incom 3788 |
. . . . . . . . . . . . . . 15
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ (1...𝑦)) = ((1...𝑦) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
117 | | elfznn0 12382 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ_{0}) |
118 | 117 | nn0red 11304 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
119 | | ltnle 10069 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ ℝ ∧
(2^{nd} ‘𝑇)
∈ ℝ) → (𝑦
< (2^{nd} ‘𝑇) ↔ ¬ (2^{nd} ‘𝑇) ≤ 𝑦)) |
120 | 118, 18, 119 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) ↔ ¬ (2^{nd}
‘𝑇) ≤ 𝑦)) |
121 | 120 | biimpa 501 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ (2^{nd}
‘𝑇) ≤ 𝑦) |
122 | | elfzle2 12295 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2^{nd} ‘𝑇) ∈ (1...𝑦) → (2^{nd} ‘𝑇) ≤ 𝑦) |
123 | 121, 122 | nsyl 135 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ (2^{nd}
‘𝑇) ∈ (1...𝑦)) |
124 | | disjsn 4221 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑦) ∩
{(2^{nd} ‘𝑇)}) = ∅ ↔ ¬ (2^{nd}
‘𝑇) ∈ (1...𝑦)) |
125 | 123, 124 | sylibr 224 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) =
∅) |
126 | 118 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 ∈ ℝ) |
127 | 18 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ∈
ℝ) |
128 | 50 | nnred 10987 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
129 | 128 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
130 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 < (2^{nd} ‘𝑇)) |
131 | 19 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
132 | 126, 127,
129, 130, 131 | lttrd 10150 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 < ((2^{nd} ‘𝑇) + 1)) |
133 | | ltnle 10069 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℝ ∧
((2^{nd} ‘𝑇)
+ 1) ∈ ℝ) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
134 | 118, 128,
133 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
135 | 134 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
136 | 132, 135 | mpbid 222 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ ((2^{nd}
‘𝑇) + 1) ≤ 𝑦) |
137 | | elfzle2 12295 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2^{nd} ‘𝑇) + 1) ∈ (1...𝑦) → ((2^{nd} ‘𝑇) + 1) ≤ 𝑦) |
138 | 136, 137 | nsyl 135 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ ((2^{nd}
‘𝑇) + 1) ∈
(1...𝑦)) |
139 | | disjsn 4221 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)}) = ∅ ↔ ¬ ((2^{nd} ‘𝑇) + 1) ∈ (1...𝑦)) |
140 | 138, 139 | sylibr 224 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {((2^{nd}
‘𝑇) + 1)}) =
∅) |
141 | 125, 140 | uneq12d 3751 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) ∪
((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)})) = (∅ ∪ ∅)) |
142 | | df-pr 4156 |
. . . . . . . . . . . . . . . . . 18
⊢
{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} = ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) +
1)}) |
143 | 142 | ineq2i 3794 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ∩
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ((1...𝑦) ∩
({(2^{nd} ‘𝑇)} ∪ {((2^{nd} ‘𝑇) + 1)})) |
144 | | indi 3854 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ∩
({(2^{nd} ‘𝑇)} ∪ {((2^{nd} ‘𝑇) + 1)})) = (((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) ∪
((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)})) |
145 | 143, 144 | eqtr2i 2644 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑦) ∩
{(2^{nd} ‘𝑇)}) ∪ ((1...𝑦) ∩ {((2^{nd} ‘𝑇) + 1)})) = ((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
146 | | un0 3944 |
. . . . . . . . . . . . . . . 16
⊢ (∅
∪ ∅) = ∅ |
147 | 141, 145,
146 | 3eqtr3g 2678 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) =
∅) |
148 | 116, 147 | syl5eq 2667 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(1...𝑦)) =
∅) |
149 | | fnimadisj 5974 |
. . . . . . . . . . . . . 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)) = ∅) |
150 | 115, 148,
149 | syl2anc 692 |
. . . . . . . . . . . . 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)) = ∅) |
151 | 39 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
152 | | elfzuz3 12289 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ_{≥}‘𝑦)) |
153 | | peano2uz 11693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
154 | 152, 153 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
155 | 154 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
156 | 151, 155 | eqeltrrd 2699 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_{≥}‘𝑦)) |
157 | | fzss2 12331 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑦) → (1...𝑦) ⊆ (1...𝑁)) |
158 | | reldisj 3997 |
. . . . . . . . . . . . . . . . 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)}))) |
159 | 156, 157,
158 | 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)}))) |
160 | 159 | 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)}))) |
161 | 147, 160 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
162 | | resiima 5444 |
. . . . . . . . . . . . . 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)) |
163 | 161, 162 | 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)) |
164 | 150, 163 | uneq12d 3751 |
. . . . . . . . . . . 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))) |
165 | | imaundir 5510 |
. . . . . . . . . . . 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))) |
166 | | uncom 3740 |
. . . . . . . . . . . . 13
⊢ (∅
∪ (1...𝑦)) =
((1...𝑦) ∪
∅) |
167 | | un0 3944 |
. . . . . . . . . . . . 13
⊢
((1...𝑦) ∪
∅) = (1...𝑦) |
168 | 166, 167 | eqtr2i 2644 |
. . . . . . . . . . . 12
⊢
(1...𝑦) = (∅
∪ (1...𝑦)) |
169 | 164, 165,
168 | 3eqtr4g 2680 |
. . . . . . . . . . 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)) |
170 | 169 | imaeq2d 5430 |
. . . . . . . . . 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))) |
171 | 112, 170 | syl5eq 2667 |
. . . . . . . . 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))) |
172 | 171 | xpeq1d 5103 |
. . . . . . . 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})) |
173 | | imaco 5604 |
. . . . . . . . . 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))) |
174 | | imaundir 5510 |
. . . . . . . . . . . . 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))) |
175 | | imassrn 5441 |
. . . . . . . . . . . . . . . . 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)⟩} |
176 | 175 | 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)⟩}) |
177 | | fnima 5972 |
. . . . . . . . . . . . . . . . . . 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)⟩}) |
178 | 26, 113, 177 | 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)⟩}) |
179 | 178 | ad2antrr 761 |
. . . . . . . . . . . . . . . . 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)⟩}) |
180 | | elfzelz 12292 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
181 | | zltp1le 11379 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℤ ∧
(2^{nd} ‘𝑇)
∈ ℤ) → (𝑦
< (2^{nd} ‘𝑇) ↔ (𝑦 + 1) ≤ (2^{nd} ‘𝑇))) |
182 | 180, 57, 181 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) ↔ (𝑦 + 1) ≤ (2^{nd} ‘𝑇))) |
183 | 182 | biimpa 501 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 + 1) ≤ (2^{nd} ‘𝑇)) |
184 | 18, 52, 56 | ltled 10137 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ 𝑁) |
185 | 184 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ≤ 𝑁) |
186 | 57 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^{nd}
‘𝑇) ∈
ℤ) |
187 | | nn0p1nn 11284 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℕ_{0}
→ (𝑦 + 1) ∈
ℕ) |
188 | 117, 187 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ) |
189 | 188 | nnzd 11433 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ) |
190 | 189 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ ℤ) |
191 | 40 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℤ) |
192 | | elfz 12282 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
193 | 186, 190,
191, 192 | syl3anc 1323 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
194 | 193 | 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))) |
195 | 183, 185,
194 | mpbir2and 956 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁)) |
196 | | 1red 10007 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 1 ∈
ℝ) |
197 | | ltle 10078 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 ∈ ℝ ∧
(2^{nd} ‘𝑇)
∈ ℝ) → (𝑦
< (2^{nd} ‘𝑇) → 𝑦 ≤ (2^{nd} ‘𝑇))) |
198 | 118, 18, 197 | syl2anr 495 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) → 𝑦 ≤ (2^{nd} ‘𝑇))) |
199 | 198 | imp 445 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 ≤ (2^{nd} ‘𝑇)) |
200 | 126, 127,
196, 199 | leadd1dd 10593 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1)) |
201 | 60 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ≤ 𝑁) |
202 | 57 | peano2zd 11437 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℤ) |
203 | 202 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) + 1) ∈
ℤ) |
204 | | elfz 12282 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
205 | 203, 190,
191, 204 | syl3anc 1323 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁))) |
206 | 205 | 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))) |
207 | 200, 201,
206 | mpbir2and 956 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁)) |
208 | | prssi 4326 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2^{nd} ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ∧ ((2^{nd} ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁)) → {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁)) |
209 | 195, 207,
208 | syl2anc 692 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
((𝑦 + 1)...𝑁)) |
210 | | imass2 5465 |
. . . . . . . . . . . . . . . . . 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))) |
211 | 209, 210 | 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))) |
212 | 179, 211 | eqsstr3d 3624 |
. . . . . . . . . . . . . . . 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))) |
213 | 176, 212 | eqssd 3604 |
. . . . . . . . . . . . . . 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)⟩}) |
214 | | f1ofo 6106 |
. . . . . . . . . . . . . . . . . 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)}) |
215 | | forn 6080 |
. . . . . . . . . . . . . . . . . 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)}) |
216 | 26, 214, 215 | 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)}) |
217 | 216, 27 | syl6eq 2671 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
218 | 217 | ad2antrr 761 |
. . . . . . . . . . . . . . 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)}) |
219 | 213, 218 | eqtrd 2655 |
. . . . . . . . . . . . . 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)}) |
220 | | undif 4026 |
. . . . . . . . . . . . . . . . 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)) |
221 | 209, 220 | sylib 208 |
. . . . . . . . . . . . . . . 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)) |
222 | 221 | imaeq2d 5430 |
. . . . . . . . . . . . . . 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))) |
223 | | fnresi 5971 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) Fn ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) |
224 | | incom 3788 |
. . . . . . . . . . . . . . . . . . . 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)})) |
225 | 224, 32 | eqtri 2643 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ |
226 | | fnimadisj 5974 |
. . . . . . . . . . . . . . . . . . 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)}) = ∅) |
227 | 223, 225,
226 | mp2an 707 |
. . . . . . . . . . . . . . . . . 18
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ |
228 | 227 | 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)}) = ∅) |
229 | | nnuz 11675 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ_{≥}‘1) |
230 | 188, 229 | syl6eleq 2708 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈
(ℤ_{≥}‘1)) |
231 | | fzss1 12330 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 + 1) ∈
(ℤ_{≥}‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
232 | 230, 231 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
233 | 232 | ssdifd 3729 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
234 | | resiima 5444 |
. . . . . . . . . . . . . . . . . . 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)})) |
235 | 233, 234 | 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)})) |
236 | 235 | ad2antlr 762 |
. . . . . . . . . . . . . . . . 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)})) |
237 | 228, 236 | uneq12d 3751 |
. . . . . . . . . . . . . . . 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)}))) |
238 | | imaundi 5509 |
. . . . . . . . . . . . . . . 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)}))) |
239 | | uncom 3740 |
. . . . . . . . . . . . . . . . 17
⊢ (∅
∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
((((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ∪
∅) |
240 | | un0 3944 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ ∅) =
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
241 | 239, 240 | eqtr2i 2644 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = (∅ ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
242 | 237, 238,
241 | 3eqtr4g 2680 |
. . . . . . . . . . . . . . 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)})) |
243 | 222, 242 | eqtr3d 2657 |
. . . . . . . . . . . . . 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)})) |
244 | 219, 243 | uneq12d 3751 |
. . . . . . . . . . . . 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)}))) |
245 | 174, 244 | syl5eq 2667 |
. . . . . . . . . . . 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)}))) |
246 | 245, 221 | eqtrd 2655 |
. . . . . . . . . . 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)) |
247 | 246 | imaeq2d 5430 |
. . . . . . . . . 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))) |
248 | 173, 247 | syl5eq 2667 |
. . . . . . . . 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))) |
249 | 248 | xpeq1d 5103 |
. . . . . . . 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})) |
250 | 172, 249 | uneq12d 3751 |
. . . . . . 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}))) |
251 | 250 | oveq2d 6626 |
. . . . . 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})))) |
252 | | iftrue 4069 |
. . . . . . . . 9
⊢ (𝑦 < (2^{nd}
‘𝑇) → if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦) |
253 | 252 | csbeq1d 3525 |
. . . . . . . 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})))) |
254 | | vex 3192 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
255 | | oveq2 6618 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦)) |
256 | 255 | imaeq2d 5430 |
. . . . . . . . . . . 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))) |
257 | 256 | xpeq1d 5103 |
. . . . . . . . . . 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})) |
258 | | oveq1 6617 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1)) |
259 | 258 | oveq1d 6625 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁)) |
260 | 259 | imaeq2d 5430 |
. . . . . . . . . . . 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))) |
261 | 260 | xpeq1d 5103 |
. . . . . . . . . . 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})) |
262 | 257, 261 | uneq12d 3751 |
. . . . . . . . . 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}))) |
263 | 262 | oveq2d 6626 |
. . . . . . . . 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})))) |
264 | 254, 263 | csbie 3544 |
. . . . . . . 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}))) |
265 | 253, 264 | syl6eq 2671 |
. . . . . . 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})))) |
266 | 265 | 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})))) |
267 | 252 | csbeq1d 3525 |
. . . . . . . 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})))) |
268 | 255 | imaeq2d 5430 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑦))) |
269 | 268 | xpeq1d 5103 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1})) |
270 | 259 | imaeq2d 5430 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
271 | 270 | xpeq1d 5103 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) |
272 | 269, 271 | uneq12d 3751 |
. . . . . . . . . 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}))) |
273 | 272 | oveq2d 6626 |
. . . . . . . . 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})))) |
274 | 254, 273 | csbie 3544 |
. . . . . . . 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}))) |
275 | 267, 274 | syl6eq 2671 |
. . . . . . 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})))) |
276 | 275 | 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})))) |
277 | 251, 266,
276 | 3eqtr4d 2665 |
. . . . 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})))) |
278 | | lenlt 10068 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^{nd}
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^{nd} ‘𝑇))) |
279 | 18, 118, 278 | syl2an 494 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^{nd} ‘𝑇))) |
280 | 279 | biimpar 502 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ≤ 𝑦) |
281 | | imaco 5604 |
. . . . . . . . . . 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)))) |
282 | | imaundir 5510 |
. . . . . . . . . . . . . 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)))) |
283 | | imassrn 5441 |
. . . . . . . . . . . . . . . . . 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)⟩} |
284 | 283 | 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)⟩}) |
285 | 178 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . 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)⟩}) |
286 | 17 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
ℕ) |
287 | 18 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
ℝ) |
288 | 118 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 ∈ ℝ) |
289 | 188 | nnred 10987 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ) |
290 | 289 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (𝑦 + 1) ∈ ℝ) |
291 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ≤ 𝑦) |
292 | 118 | lep1d 10907 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑦 + 1)) |
293 | 292 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 ≤ (𝑦 + 1)) |
294 | 287, 288,
290, 291, 293 | letrd 10146 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ≤ (𝑦 + 1)) |
295 | | fznn 12358 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈ ℤ →
((2^{nd} ‘𝑇)
∈ (1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
296 | 189, 295 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^{nd}
‘𝑇) ∈
(1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
297 | 296 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
298 | 286, 294,
297 | mpbir2and 956 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
(1...(𝑦 +
1))) |
299 | 50 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
ℕ) |
300 | | 1red 10007 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 1 ∈
ℝ) |
301 | 287, 288,
300, 291 | leadd1dd 10593 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ≤ (𝑦 + 1)) |
302 | | fznn 12358 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈ ℤ →
(((2^{nd} ‘𝑇)
+ 1) ∈ (1...(𝑦 + 1))
↔ (((2^{nd} ‘𝑇) + 1) ∈ ℕ ∧ ((2^{nd}
‘𝑇) + 1) ≤ (𝑦 + 1)))) |
303 | 189, 302 | 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)))) |
304 | 303 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
305 | 299, 301,
304 | mpbir2and 956 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 +
1))) |
306 | | prssi 4326 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘𝑇) ∈ (1...(𝑦 + 1)) ∧ ((2^{nd} ‘𝑇) + 1) ∈ (1...(𝑦 + 1))) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...(𝑦 +
1))) |
307 | 298, 305,
306 | syl2anc 692 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...(𝑦 +
1))) |
308 | | imass2 5465 |
. . . . . . . . . . . . . . . . . . 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)))) |
309 | 307, 308 | 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)))) |
310 | 285, 309 | eqsstr3d 3624 |
. . . . . . . . . . . . . . . . 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)))) |
311 | 284, 310 | eqssd 3604 |
. . . . . . . . . . . . . . . 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)⟩}) |
312 | 217 | ad2antrr 761 |
. . . . . . . . . . . . . . . 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)}) |
313 | 311, 312 | eqtrd 2655 |
. . . . . . . . . . . . . . 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)}) |
314 | | undif 4026 |
. . . . . . . . . . . . . . . . . 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))) |
315 | 307, 314 | sylib 208 |
. . . . . . . . . . . . . . . . 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))) |
316 | 315 | imaeq2d 5430 |
. . . . . . . . . . . . . . . 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)))) |
317 | 227 | 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)}) = ∅) |
318 | | eluzp1p1 11665 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
319 | 152, 318 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
320 | 319 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
321 | 151, 320 | eqeltrrd 2699 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_{≥}‘(𝑦 + 1))) |
322 | | fzss2 12331 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ_{≥}‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
323 | 321, 322 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
324 | 323 | ssdifd 3729 |
. . . . . . . . . . . . . . . . . . . 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)})) |
325 | 324 | 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)})) |
326 | | resiima 5444 |
. . . . . . . . . . . . . . . . . . 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)})) |
327 | 325, 326 | 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)})) |
328 | 317, 327 | uneq12d 3751 |
. . . . . . . . . . . . . . . . 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)}))) |
329 | | imaundi 5509 |
. . . . . . . . . . . . . . . . 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)}))) |
330 | | uncom 3740 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ ((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = (((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪
∅) |
331 | | un0 3944 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ ∅) = ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
332 | 330, 331 | eqtr2i 2644 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
333 | 328, 329,
332 | 3eqtr4g 2680 |
. . . . . . . . . . . . . . . 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)})) |
334 | 316, 333 | eqtr3d 2657 |
. . . . . . . . . . . . . . 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)})) |
335 | 313, 334 | uneq12d 3751 |
. . . . . . . . . . . . . 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)}))) |
336 | 282, 335 | syl5eq 2667 |
. . . . . . . . . . . . 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)}))) |
337 | 336, 315 | eqtrd 2655 |
. . . . . . . . . . . 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))) |
338 | 337 | imaeq2d 5430 |
. . . . . . . . . . 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)))) |
339 | 281, 338 | syl5eq 2667 |
. . . . . . . . . 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)))) |
340 | 339 | xpeq1d 5103 |
. . . . . . . . 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})) |
341 | | imaco 5604 |
. . . . . . . . . . 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))) |
342 | 114 | ad2antrr 761 |
. . . . . . . . . . . . . . 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)}) |
343 | | incom 3788 |
. . . . . . . . . . . . . . . 16
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
344 | 128 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
345 | 188 | peano2nnd 10989 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ) |
346 | 345 | nnred 10987 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ) |
347 | 346 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((𝑦 + 1) + 1) ∈ ℝ) |
348 | 19 | ad2antrr 761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
349 | 118 | ltp1d 10906 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1)) |
350 | 349 | ad2antlr 762 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 < (𝑦 + 1)) |
351 | 287, 288,
290, 291, 350 | lelttrd 10147 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) < (𝑦 + 1)) |
352 | 287, 290,
300, 351 | ltadd1dd 10590 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1)) |
353 | 287, 344,
347, 348, 352 | lttrd 10150 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) < ((𝑦 + 1) + 1)) |
354 | | ltnle 10069 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2^{nd} ‘𝑇) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) →
((2^{nd} ‘𝑇)
< ((𝑦 + 1) + 1) ↔
¬ ((𝑦 + 1) + 1) ≤
(2^{nd} ‘𝑇))) |
355 | 18, 346, 354 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇))) |
356 | 355 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇))) |
357 | 353, 356 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇)) |
358 | | elfzle1 12294 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2^{nd} ‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ (2^{nd} ‘𝑇)) |
359 | 357, 358 | nsyl 135 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ (2^{nd}
‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁)) |
360 | | disjsn 4221 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{(2^{nd} ‘𝑇)}) = ∅ ↔ ¬ (2^{nd}
‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁)) |
361 | 359, 360 | sylibr 224 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇)}) = ∅) |
362 | | ltnle 10069 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2^{nd} ‘𝑇) + 1) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ)
→ (((2^{nd} ‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd} ‘𝑇) + 1))) |
363 | 128, 346,
362 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1))) |
364 | 363 | 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))) |
365 | 352, 364 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1)) |
366 | | elfzle1 12294 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ ((2^{nd} ‘𝑇) + 1)) |
367 | 365, 366 | nsyl 135 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((2^{nd}
‘𝑇) + 1) ∈
(((𝑦 + 1) + 1)...𝑁)) |
368 | | disjsn 4221 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{((2^{nd} ‘𝑇)
+ 1)}) = ∅ ↔ ¬ ((2^{nd} ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁)) |
369 | 367, 368 | sylibr 224 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)}) =
∅) |
370 | 361, 369 | uneq12d 3751 |
. . . . . . . . . . . . . . . . 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)})) = (∅ ∪
∅)) |
371 | 142 | ineq2i 3794 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) +
1)})) |
372 | | indi 3854 |
. . . . . . . . . . . . . . . . . 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)})) |
373 | 371, 372 | eqtr2i 2644 |
. . . . . . . . . . . . . . . . 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)}) |
374 | 370, 373,
146 | 3eqtr3g 2678 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) =
∅) |
375 | 343, 374 | syl5eq 2667 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(((𝑦 + 1) + 1)...𝑁)) = ∅) |
376 | | fnimadisj 5974 |
. . . . . . . . . . . . . . 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)) = ∅) |
377 | 342, 375,
376 | syl2anc 692 |
. . . . . . . . . . . . . 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)) = ∅) |
378 | 345, 229 | syl6eleq 2708 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ_{≥}‘1)) |
379 | | fzss1 12330 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑦 + 1) + 1) ∈
(ℤ_{≥}‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
380 | | reldisj 3997 |
. . . . . . . . . . . . . . . . . 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)}))) |
381 | 378, 379,
380 | 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)}))) |
382 | 381 | ad2antlr 762 |
. . . . . . . . . . . . . . . 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)}))) |
383 | 374, 382 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
384 | | resiima 5444 |
. . . . . . . . . . . . . . 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)) |
385 | 383, 384 | 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)) |
386 | 377, 385 | uneq12d 3751 |
. . . . . . . . . . . . 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))) |
387 | | imaundir 5510 |
. . . . . . . . . . . . 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))) |
388 | | uncom 3740 |
. . . . . . . . . . . . . 14
⊢ (∅
∪ (((𝑦 + 1) +
1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) |
389 | | un0 3944 |
. . . . . . . . . . . . . 14
⊢ ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) = (((𝑦 + 1) + 1)...𝑁) |
390 | 388, 389 | eqtr2i 2644 |
. . . . . . . . . . . . 13
⊢ (((𝑦 + 1) + 1)...𝑁) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁)) |
391 | 386, 387,
390 | 3eqtr4g 2680 |
. . . . . . . . . . . 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)) |
392 | 391 | imaeq2d 5430 |
. . . . . . . . . . 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))) |
393 | 341, 392 | syl5eq 2667 |
. . . . . . . . . 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))) |
394 | 393 | xpeq1d 5103 |
. . . . . . . . 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})) |
395 | 340, 394 | uneq12d 3751 |
. . . . . . . 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}))) |
396 | 280, 395 | syldan 487 |
. . . . . . 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}))) |
397 | 396 | oveq2d 6626 |
. . . . . 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})))) |
398 | | iffalse 4072 |
. . . . . . . . 9
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) → if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1)) |
399 | 398 | csbeq1d 3525 |
. . . . . . . 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})))) |
400 | | ovex 6638 |
. . . . . . . . 9
⊢ (𝑦 + 1) ∈ V |
401 | | oveq2 6618 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1))) |
402 | 401 | imaeq2d 5430 |
. . . . . . . . . . . 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)))) |
403 | 402 | xpeq1d 5103 |
. . . . . . . . . . 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})) |
404 | | oveq1 6617 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1)) |
405 | 404 | oveq1d 6625 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁)) |
406 | 405 | imaeq2d 5430 |
. . . . . . . . . . . 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))) |
407 | 406 | xpeq1d 5103 |
. . . . . . . . . . 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})) |
408 | 403, 407 | uneq12d 3751 |
. . . . . . . . . 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}))) |
409 | 408 | oveq2d 6626 |
. . . . . . . . 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})))) |
410 | 400, 409 | csbie 3544 |
. . . . . . . 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}))) |
411 | 399, 410 | syl6eq 2671 |
. . . . . . 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})))) |
412 | 411 | 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})))) |
413 | 398 | csbeq1d 3525 |
. . . . . . . 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})))) |
414 | 401 | imaeq2d 5430 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...(𝑦 +
1)))) |
415 | 414 | xpeq1d 5103 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1})) |
416 | 405 | imaeq2d 5430 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
417 | 416 | xpeq1d 5103 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
418 | 415, 417 | uneq12d 3751 |
. . . . . . . . . 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}))) |
419 | 418 | oveq2d 6626 |
. . . . . . . . 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})))) |
420 | 400, 419 | csbie 3544 |
. . . . . . . 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}))) |
421 | 413, 420 | syl6eq 2671 |
. . . . . . 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})))) |
422 | 421 | 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})))) |
423 | 397, 412,
422 | 3eqtr4d 2665 |
. . . . 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})))) |
424 | 277, 423 | pm2.61dan 831 |
. . . 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})))) |
425 | 424 | mpteq2dva 4709 |
. . 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}))))) |
426 | 111, 425 | eqtr4d 2658 |
. 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}))))) |
427 | | opex 4898 |
. . . . . . 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 |
428 | 427, 22 | op1std 7130 |
. . . . . 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)}))))⟩) |
429 | 427, 22 | op2ndd 7131 |
. . . . . 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)) |
430 | | breq2 4622 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
431 | 430 | ifbid 4085 |
. . . . . . . 8
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
432 | 431 | csbeq1d 3525 |
. . . . . . 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})))) |
433 | | fvex 6163 |
. . . . . . . . . 10
⊢
(1^{st} ‘(1^{st} ‘𝑇)) ∈ V |
434 | 433, 79 | op1std 7130 |
. . . . . . . . 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))) |
435 | 433, 79 | op2ndd 7131 |
. . . . . . . . 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)}))))) |
436 | | id 22 |
. . . . . . . . . 10
⊢
((1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇)) →
(1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
437 | | imaeq1 5425 |
. . . . . . . . . . . 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))) |
438 | 437 | xpeq1d 5103 |
. . . . . . . . . . 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})) |
439 | | imaeq1 5425 |
. . . . . . . . . . . 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))) |
440 | 439 | xpeq1d 5103 |
. . . . . . . . . . 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})) |
441 | 438, 440 | uneq12d 3751 |
. . . . . . . . . 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}))) |
442 | 436, 441 | oveqan12d 6629 |
. . . . . . . . 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})))) |
443 | 434, 435,
442 | syl2anc 692 |
. . . . . . . 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})))) |
444 | 443 | csbeq2dv 3969 |
. . . . . . 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})))) |
445 | 432, 444 | sylan9eqr 2677 |
. . . . . 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})))) |
446 | 428, 429,
445 | syl2anc 692 |
. . . . 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})))) |
447 | 446 | mpteq2dv 4710 |
. . . 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}))))) |
448 | 447 | eqeq2d 2631 |
. . 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})))))) |
449 | 448, 3 | elrab2 3352 |
. 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})))))) |
450 | 91, 426, 449 | sylanbrc 697 |
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) |