Step | Hyp | Ref
| Expression |
1 | | eldif 3802 |
. . . . . . 7
⊢ (𝑦 ∈ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ↔ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
2 | | imassrn 5731 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ran (2^{nd}
‘(1^{st} ‘𝑇)) |
3 | | poimirlem12.2 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
4 | | elrabi 3567 |
. . . . . . . . . . . . . . . . . . 19
⊢ (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))) |
5 | | poimirlem22.s |
. . . . . . . . . . . . . . . . . . 19
⊢ 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}))))} |
6 | 4, 5 | eleq2s 2877 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
7 | 3, 6 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
8 | | xp1st 7477 |
. . . . . . . . . . . . . . . . 17
⊢ (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)})) |
9 | 7, 8 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1^{st}
‘𝑇) ∈
(((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
10 | | xp2nd 7478 |
. . . . . . . . . . . . . . . 16
⊢
((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)}) |
11 | 9, 10 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
12 | | fvex 6459 |
. . . . . . . . . . . . . . . 16
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
13 | | f1oeq1 6380 |
. . . . . . . . . . . . . . . 16
⊢ (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))) |
14 | 12, 13 | elab 3558 |
. . . . . . . . . . . . . . 15
⊢
((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)) |
15 | 11, 14 | sylib 210 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
16 | | f1of 6391 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
18 | 17 | frnd 6298 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (2^{nd}
‘(1^{st} ‘𝑇)) ⊆ (1...𝑁)) |
19 | 2, 18 | syl5ss 3832 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁)) |
20 | | poimirlem11.4 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
21 | | elrabi 3567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑈 ∈ {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}))))} → 𝑈 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
22 | 21, 5 | eleq2s 2877 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
23 | 20, 22 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑈 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
24 | | xp1st 7477 |
. . . . . . . . . . . . . . . 16
⊢ (𝑈 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑈) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1^{st}
‘𝑈) ∈
(((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
26 | | xp2nd 7478 |
. . . . . . . . . . . . . . 15
⊢
((1^{st} ‘𝑈) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
28 | | fvex 6459 |
. . . . . . . . . . . . . . 15
⊢
(2^{nd} ‘(1^{st} ‘𝑈)) ∈ V |
29 | | f1oeq1 6380 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (2^{nd}
‘(1^{st} ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))) |
30 | 28, 29 | elab 3558 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
31 | 27, 30 | sylib 210 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
32 | | f1ofo 6398 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁)) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁)) |
34 | | foima 6371 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
36 | 19, 35 | sseqtr4d 3861 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁))) |
37 | 36 | ssdifd 3969 |
. . . . . . . . 9
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ⊆ (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
38 | | dff1o3 6397 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑈)))) |
39 | 38 | simprbi 492 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑈))) |
40 | 31, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑈))) |
41 | | imadif 6218 |
. . . . . . . . . . 11
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑈)) →
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
42 | 40, 41 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
43 | | difun2 4272 |
. . . . . . . . . . . 12
⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)) |
44 | | poimirlem11.6 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
45 | | fzsplit 12684 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
46 | 44, 45 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
47 | | uncom 3980 |
. . . . . . . . . . . . . 14
⊢
((1...𝑀) ∪
((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)) |
48 | 46, 47 | syl6eq 2830 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))) |
49 | 48 | difeq1d 3950 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀))) |
50 | | incom 4028 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) |
51 | | elfznn 12687 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) |
52 | 44, 51 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℕ) |
53 | 52 | nnred 11391 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
54 | 53 | ltp1d 11308 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
55 | | fzdisj 12685 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
56 | 54, 55 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
57 | 50, 56 | syl5eq 2826 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅) |
58 | | disj3 4246 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
59 | 57, 58 | sylib 210 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
60 | 43, 49, 59 | 3eqtr4a 2840 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁)) |
61 | 60 | imaeq2d 5720 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
62 | 42, 61 | eqtr3d 2816 |
. . . . . . . . 9
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
63 | 37, 62 | sseqtrd 3860 |
. . . . . . . 8
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
64 | 63 | sselda 3821 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
65 | 1, 64 | sylan2br 588 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
66 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑈)) |
67 | 66 | breq2d 4898 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑈))) |
68 | 67 | ifbid 4329 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1))) |
69 | 68 | csbeq1d 3758 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋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}
‘𝑈), 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})))) |
70 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑈))) |
71 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑈 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑈))) |
72 | 71 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑗))) |
73 | 72 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1})) |
74 | 71 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑗 + 1)...𝑁))) |
75 | 74 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) |
76 | 73, 75 | uneq12d 3991 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
77 | 70, 76 | oveq12d 6940 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → ((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} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
78 | 77 | csbeq2dv 4217 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^{nd} ‘𝑈), 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}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
79 | 69, 78 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑈 → ⦋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}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
80 | 79 | mpteq2dv 4980 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑈 → (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}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
81 | 80 | eqeq2d 2788 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑈 → (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}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
82 | 81, 5 | elrab2 3576 |
. . . . . . . . . . . 12
⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ ((((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}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
83 | 82 | simprbi 492 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
84 | 20, 83 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
85 | | poimirlem11.5 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2^{nd}
‘𝑈) =
0) |
86 | | breq12 4891 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 = (𝑀 − 1) ∧ (2^{nd}
‘𝑈) = 0) →
(𝑦 < (2^{nd}
‘𝑈) ↔ (𝑀 − 1) <
0)) |
87 | 85, 86 | sylan2 586 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2^{nd} ‘𝑈) ↔ (𝑀 − 1) < 0)) |
88 | 87 | ancoms 452 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2^{nd} ‘𝑈) ↔ (𝑀 − 1) < 0)) |
89 | | oveq1 6929 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1)) |
90 | 52 | nncnd 11392 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℂ) |
91 | | npcan1 10800 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
92 | 90, 91 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
93 | 89, 92 | sylan9eqr 2836 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀) |
94 | 88, 93 | ifbieq2d 4332 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀)) |
95 | 52 | nnzd 11833 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
96 | | poimir.0 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) |
97 | 96 | nnzd 11833 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℤ) |
98 | | elfzm1b 12736 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1)))) |
99 | 95, 97, 98 | syl2anc 579 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1)))) |
100 | 44, 99 | mpbid 224 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
101 | | elfzle1 12661 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) → 0 ≤
(𝑀 −
1)) |
102 | 100, 101 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 ≤ (𝑀 − 1)) |
103 | | 0red 10380 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 ∈
ℝ) |
104 | | nnm1nn0 11685 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈
ℕ_{0}) |
105 | 52, 104 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ_{0}) |
106 | 105 | nn0red 11703 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
107 | 103, 106 | lenltd 10522 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0 ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < 0)) |
108 | 102, 107 | mpbid 224 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ¬ (𝑀 − 1) < 0) |
109 | 108 | iffalsed 4318 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀) |
110 | 109 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀) |
111 | 94, 110 | eqtrd 2814 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀) |
112 | 111 | csbeq1d 3758 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
113 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
114 | 113 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀))) |
115 | 114 | xpeq1d 5384 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1})) |
116 | | oveq1 6929 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
117 | 116 | oveq1d 6937 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
118 | 117 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
119 | 118 | xpeq1d 5384 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) |
120 | 115, 119 | uneq12d 3991 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
121 | 120 | oveq2d 6938 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
122 | 121 | adantl 475 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
123 | 44, 122 | csbied 3778 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
124 | 123 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
125 | 112, 124 | eqtrd 2814 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
126 | | ovexd 6956 |
. . . . . . . . . 10
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
127 | 84, 125, 100, 126 | fvmptd 6548 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
128 | 127 | fveq1d 6448 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
129 | 128 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
130 | | imassrn 5731 |
. . . . . . . . . 10
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2^{nd}
‘(1^{st} ‘𝑈)) |
131 | | f1of 6391 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
132 | 31, 131 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
133 | 132 | frnd 6298 |
. . . . . . . . . 10
⊢ (𝜑 → ran (2^{nd}
‘(1^{st} ‘𝑈)) ⊆ (1...𝑁)) |
134 | 130, 133 | syl5ss 3832 |
. . . . . . . . 9
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁)) |
135 | 134 | sselda 3821 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁)) |
136 | | xp1st 7477 |
. . . . . . . . . . . 12
⊢
((1^{st} ‘𝑈) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^{st}
‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
137 | 25, 136 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
138 | | elmapfn 8163 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
139 | 137, 138 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
140 | 139 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
141 | | 1ex 10372 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
142 | | fnconstg 6343 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) |
143 | 141, 142 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) |
144 | | c0ex 10370 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
145 | | fnconstg 6343 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
146 | 144, 145 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) |
147 | 143, 146 | pm3.2i 464 |
. . . . . . . . . . . 12
⊢
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
148 | | imain 6219 |
. . . . . . . . . . . . . 14
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑈)) →
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
149 | 40, 148 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
150 | 56 | imaeq2d 5720 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ∅)) |
151 | | ima0 5735 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ∅) =
∅ |
152 | 150, 151 | syl6eq 2830 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
153 | 149, 152 | eqtr3d 2816 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) |
154 | | fnun 6243 |
. . . . . . . . . . . 12
⊢
((((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
155 | 147, 153,
154 | sylancr 581 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
156 | | imaundi 5799 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
157 | 46 | imaeq2d 5720 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
158 | 157, 35 | eqtr3d 2816 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
159 | 156, 158 | syl5eqr 2828 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
160 | 159 | fneq2d 6227 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
161 | 155, 160 | mpbid 224 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
162 | 161 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
163 | | ovexd 6956 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V) |
164 | | inidm 4043 |
. . . . . . . . 9
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
165 | | eqidd 2779 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
166 | | fvun2 6530 |
. . . . . . . . . . . . 13
⊢
(((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
167 | 143, 146,
166 | mp3an12 1524 |
. . . . . . . . . . . 12
⊢
(((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
168 | 153, 167 | sylan 575 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
169 | 144 | fvconst2 6741 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
170 | 169 | adantl 475 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
171 | 168, 170 | eqtrd 2814 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
172 | 171 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
173 | 140, 162,
163, 163, 164, 165, 172 | ofval 7183 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))‘𝑦) + 0)) |
174 | 135, 173 | mpdan 677 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))‘𝑦) + 0)) |
175 | | elmapi 8162 |
. . . . . . . . . . . . 13
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
176 | 137, 175 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
177 | 176 | ffvelrnda 6623 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ (0..^𝐾)) |
178 | | elfzonn0 12832 |
. . . . . . . . . . 11
⊢
(((1^{st} ‘(1^{st} ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈
ℕ_{0}) |
179 | 177, 178 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈
ℕ_{0}) |
180 | 179 | nn0cnd 11704 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ ℂ) |
181 | 135, 180 | syldan 585 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ ℂ) |
182 | 181 | addid1d 10576 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) + 0) = ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦)) |
183 | 129, 174,
182 | 3eqtrd 2818 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
184 | 65, 183 | syldan 585 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
185 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
186 | 185 | breq2d 4898 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
187 | 186 | ifbid 4329 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
188 | 187 | csbeq1d 3758 |
. . . . . . . . . . . . . . . 16
⊢ (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})))) |
189 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
190 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
191 | 190 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
192 | 191 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
193 | 190 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
194 | 193 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
195 | 192, 194 | uneq12d 3991 |
. . . . . . . . . . . . . . . . . 18
⊢ (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}))) |
196 | 189, 195 | oveq12d 6940 |
. . . . . . . . . . . . . . . . 17
⊢ (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})))) |
197 | 196 | csbeq2dv 4217 |
. . . . . . . . . . . . . . . 16
⊢ (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})))) |
198 | 188, 197 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
199 | 198 | mpteq2dv 4980 |
. . . . . . . . . . . . . 14
⊢ (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}))))) |
200 | 199 | eqeq2d 2788 |
. . . . . . . . . . . . 13
⊢ (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})))))) |
201 | 200, 5 | elrab2 3576 |
. . . . . . . . . . . 12
⊢ (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})))))) |
202 | 201 | simprbi 492 |
. . . . . . . . . . 11
⊢ (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}))))) |
203 | 3, 202 | syl 17 |
. . . . . . . . . 10
⊢ (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}))))) |
204 | | poimirlem11.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2^{nd}
‘𝑇) =
0) |
205 | | breq12 4891 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 = (𝑀 − 1) ∧ (2^{nd}
‘𝑇) = 0) →
(𝑦 < (2^{nd}
‘𝑇) ↔ (𝑀 − 1) <
0)) |
206 | 204, 205 | sylan2 586 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2^{nd} ‘𝑇) ↔ (𝑀 − 1) < 0)) |
207 | 206 | ancoms 452 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2^{nd} ‘𝑇) ↔ (𝑀 − 1) < 0)) |
208 | 207, 93 | ifbieq2d 4332 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀)) |
209 | 208, 110 | eqtrd 2814 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
210 | 209 | csbeq1d 3758 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
211 | 113 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀))) |
212 | 211 | xpeq1d 5384 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})) |
213 | 117 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
214 | 213 | xpeq1d 5384 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
215 | 212, 214 | uneq12d 3991 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
216 | 215 | oveq2d 6938 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
217 | 216 | adantl 475 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
218 | 44, 217 | csbied 3778 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
219 | 218 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
220 | 210, 219 | eqtrd 2814 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
221 | | ovexd 6956 |
. . . . . . . . . 10
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
222 | 203, 220,
100, 221 | fvmptd 6548 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
223 | 222 | fveq1d 6448 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
224 | 223 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
225 | 19 | sselda 3821 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁)) |
226 | | xp1st 7477 |
. . . . . . . . . . . 12
⊢
((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))) |
227 | 9, 226 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
228 | | elmapfn 8163 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
229 | 227, 228 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
230 | 229 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
231 | | fnconstg 6343 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) |
232 | 141, 231 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) |
233 | | fnconstg 6343 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
234 | 144, 233 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
235 | 232, 234 | pm3.2i 464 |
. . . . . . . . . . . 12
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
236 | | dff1o3 6397 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇)))) |
237 | 236 | simprbi 492 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇))) |
238 | 15, 237 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇))) |
239 | | imain 6219 |
. . . . . . . . . . . . . 14
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑇)) →
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
240 | 238, 239 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
241 | 56 | imaeq2d 5720 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ∅)) |
242 | | ima0 5735 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ∅) =
∅ |
243 | 241, 242 | syl6eq 2830 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
244 | 240, 243 | eqtr3d 2816 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
245 | | fnun 6243 |
. . . . . . . . . . . 12
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
246 | 235, 244,
245 | sylancr 581 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
247 | | imaundi 5799 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
248 | 46 | imaeq2d 5720 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
249 | | f1ofo 6398 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
250 | 15, 249 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
251 | | foima 6371 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
252 | 250, 251 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
253 | 248, 252 | eqtr3d 2816 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
254 | 247, 253 | syl5eqr 2828 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
255 | 254 | fneq2d 6227 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
256 | 246, 255 | mpbid 224 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
257 | 256 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
258 | | ovexd 6956 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V) |
259 | | eqidd 2779 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
260 | | fvun1 6529 |
. . . . . . . . . . . . 13
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
261 | 232, 234,
260 | mp3an12 1524 |
. . . . . . . . . . . 12
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
262 | 244, 261 | sylan 575 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
263 | 141 | fvconst2 6741 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
264 | 263 | adantl 475 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
265 | 262, 264 | eqtrd 2814 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
266 | 265 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
267 | 230, 257,
258, 258, 164, 259, 266 | ofval 7183 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
268 | 225, 267 | mpdan 677 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
269 | 224, 268 | eqtrd 2814 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
270 | 269 | adantrr 707 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
271 | | poimirlem22.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_{𝑚} (1...𝑁))) |
272 | 96, 5, 271, 20, 85 | poimirlem10 34045 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘_{𝑓}
− ((1...𝑁) ×
{1})) = (1^{st} ‘(1^{st} ‘𝑈))) |
273 | 96, 5, 271, 3, 204 | poimirlem10 34045 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘_{𝑓}
− ((1...𝑁) ×
{1})) = (1^{st} ‘(1^{st} ‘𝑇))) |
274 | 272, 273 | eqtr3d 2816 |
. . . . . . 7
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) = (1^{st} ‘(1^{st}
‘𝑇))) |
275 | 274 | fveq1d 6448 |
. . . . . 6
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
276 | 275 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
277 | 184, 270,
276 | 3eqtr3d 2822 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
278 | | elmapi 8162 |
. . . . . . . . . . . 12
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
279 | 227, 278 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
280 | 279 | ffvelrnda 6623 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈ (0..^𝐾)) |
281 | | elfzonn0 12832 |
. . . . . . . . . 10
⊢
(((1^{st} ‘(1^{st} ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈
ℕ_{0}) |
282 | 280, 281 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈
ℕ_{0}) |
283 | 282 | nn0red 11703 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈ ℝ) |
284 | 283 | ltp1d 11308 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) < (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1)) |
285 | 283, 284 | gtned 10511 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) ≠ ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
286 | 225, 285 | syldan 585 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) ≠ ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
287 | 286 | neneqd 2974 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ¬ (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
288 | 287 | adantrr 707 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ¬ (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
289 | 277, 288 | pm2.65da 807 |
. . 3
⊢ (𝜑 → ¬ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
290 | | iman 392 |
. . 3
⊢ ((𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ↔ ¬ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
291 | 289, 290 | sylibr 226 |
. 2
⊢ (𝜑 → (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
292 | 291 | ssrdv 3827 |
1
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) |