Proof of Theorem poimirlem11
Step | Hyp | Ref
| Expression |
1 | | eldif 3550 |
. . . . . . 7
⊢ (𝑦 ∈ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ↔ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
2 | | imassrn 5396 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ran (2nd
‘(1st ‘𝑇)) |
3 | | poimirlem12.2 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
4 | | elrabi 3328 |
. . . . . . . . . . . . . . . . . . 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 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
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 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
6 | 4, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . . . . 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 7089 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑇 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
9 | 7, 8 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
10 | | xp2nd 7090 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
11 | 9, 10 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
12 | | fvex 6113 |
. . . . . . . . . . . . . . . 16
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
13 | | f1oeq1 6040 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
14 | 12, 13 | elab 3319 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
15 | 11, 14 | sylib 207 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
16 | | f1of 6050 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
18 | | frn 5966 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁) → ran (2nd
‘(1st ‘𝑇)) ⊆ (1...𝑁)) |
19 | 17, 18 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (2nd
‘(1st ‘𝑇)) ⊆ (1...𝑁)) |
20 | 2, 19 | syl5ss 3579 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁)) |
21 | | poimirlem11.4 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
22 | | elrabi 3328 |
. . . . . . . . . . . . . . . . . 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 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
23 | 22, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
24 | 21, 23 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑈 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
25 | | xp1st 7089 |
. . . . . . . . . . . . . . . 16
⊢ (𝑈 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑈) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
26 | 24, 25 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1st
‘𝑈) ∈
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
27 | | xp2nd 7090 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑈) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
29 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢
(2nd ‘(1st ‘𝑈)) ∈ V |
30 | | f1oeq1 6040 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (2nd
‘(1st ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))) |
31 | 29, 30 | elab 3319 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
32 | 28, 31 | sylib 207 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
33 | | f1ofo 6057 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁)) |
34 | 32, 33 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁)) |
35 | | foima 6033 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
36 | 34, 35 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
37 | 20, 36 | sseqtr4d 3605 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd
‘(1st ‘𝑈)) “ (1...𝑁))) |
38 | 37 | ssdifd 3708 |
. . . . . . . . 9
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ⊆ (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
39 | | dff1o3 6056 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑈)))) |
40 | 39 | simprbi 479 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑈))) |
41 | 32, 40 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑈))) |
42 | | imadif 5887 |
. . . . . . . . . . 11
⊢ (Fun
◡(2nd ‘(1st
‘𝑈)) →
((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
43 | 41, 42 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
44 | | difun2 4000 |
. . . . . . . . . . . 12
⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)) |
45 | | poimirlem11.6 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
46 | | fzsplit 12238 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
47 | 45, 46 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
48 | | uncom 3719 |
. . . . . . . . . . . . . 14
⊢
((1...𝑀) ∪
((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)) |
49 | 47, 48 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))) |
50 | 49 | difeq1d 3689 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀))) |
51 | | incom 3767 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) |
52 | | elfznn 12241 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) |
53 | 45, 52 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℕ) |
54 | 53 | nnred 10912 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
55 | 54 | ltp1d 10833 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
56 | | fzdisj 12239 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
57 | 55, 56 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
58 | 51, 57 | syl5eq 2656 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅) |
59 | | disj3 3973 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
60 | 58, 59 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
61 | 44, 50, 60 | 3eqtr4a 2670 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁)) |
62 | 61 | imaeq2d 5385 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
63 | 43, 62 | eqtr3d 2646 |
. . . . . . . . 9
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) = ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
64 | 38, 63 | sseqtrd 3604 |
. . . . . . . 8
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ⊆ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
65 | 64 | sselda 3568 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
66 | 1, 65 | sylan2br 492 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
67 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (2nd ‘𝑡) = (2nd ‘𝑈)) |
68 | 67 | breq2d 4595 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑈))) |
69 | 68 | ifbid 4058 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1))) |
70 | 69 | csbeq1d 3506 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
71 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (1st ‘𝑡) = (1st ‘𝑈)) |
72 | 71 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑈))) |
73 | 71 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑈 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑈))) |
74 | 73 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑈)) “
(1...𝑗))) |
75 | 74 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1})) |
76 | 73 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑈)) “ ((𝑗 + 1)...𝑁))) |
77 | 76 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) |
78 | 75, 77 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
79 | 72, 78 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
80 | 79 | csbeq2dv 3944 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
81 | 70, 80 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
82 | 81 | mpteq2dv 4673 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
83 | 82 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
84 | 83, 5 | elrab2 3333 |
. . . . . . . . . . . 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 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
85 | 84 | simprbi 479 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
86 | 21, 85 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
87 | | poimirlem11.5 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2nd
‘𝑈) =
0) |
88 | | breq12 4588 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 = (𝑀 − 1) ∧ (2nd
‘𝑈) = 0) →
(𝑦 < (2nd
‘𝑈) ↔ (𝑀 − 1) <
0)) |
89 | 87, 88 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑈) ↔ (𝑀 − 1) < 0)) |
90 | 89 | ancoms 468 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2nd ‘𝑈) ↔ (𝑀 − 1) < 0)) |
91 | | oveq1 6556 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1)) |
92 | 53 | nncnd 10913 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℂ) |
93 | | npcan1 10334 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
94 | 92, 93 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
95 | 91, 94 | sylan9eqr 2666 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀) |
96 | 90, 95 | ifbieq2d 4061 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀)) |
97 | 53 | nnzd 11357 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
98 | | poimir.0 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) |
99 | 98 | nnzd 11357 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℤ) |
100 | | elfzm1b 12287 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1)))) |
101 | 97, 99, 100 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1)))) |
102 | 45, 101 | mpbid 221 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
103 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) → 0 ≤
(𝑀 −
1)) |
104 | 102, 103 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 ≤ (𝑀 − 1)) |
105 | | 0red 9920 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 ∈
ℝ) |
106 | | nnm1nn0 11211 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈
ℕ0) |
107 | 53, 106 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ0) |
108 | 107 | nn0red 11229 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
109 | 105, 108 | lenltd 10062 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0 ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < 0)) |
110 | 104, 109 | mpbid 221 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ¬ (𝑀 − 1) < 0) |
111 | 110 | iffalsed 4047 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀) |
112 | 111 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀) |
113 | 96, 112 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀) |
114 | 113 | csbeq1d 3506 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
115 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
116 | 115 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑈)) “
(1...𝑀))) |
117 | 116 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1})) |
118 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
119 | 118 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
120 | 119 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
121 | 120 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) |
122 | 117, 121 | uneq12d 3730 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
123 | 122 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
124 | 123 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
125 | 45, 124 | csbied 3526 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
126 | 125 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
127 | 114, 126 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
128 | | ovex 6577 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V |
129 | 128 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
130 | 86, 127, 102, 129 | fvmptd 6197 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
131 | 130 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑈))
∘𝑓 + ((((2nd ‘(1st
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
132 | 131 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑈))
∘𝑓 + ((((2nd ‘(1st
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
133 | | imassrn 5396 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2nd
‘(1st ‘𝑈)) |
134 | | f1of 6050 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
135 | 32, 134 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
136 | | frn 5966 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁) → ran (2nd
‘(1st ‘𝑈)) ⊆ (1...𝑁)) |
137 | 135, 136 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ran (2nd
‘(1st ‘𝑈)) ⊆ (1...𝑁)) |
138 | 133, 137 | syl5ss 3579 |
. . . . . . . . 9
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁)) |
139 | 138 | sselda 3568 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁)) |
140 | | xp1st 7089 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑈) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
141 | 26, 140 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
142 | | elmapfn 7766 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
143 | 141, 142 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
144 | 143 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
145 | | 1ex 9914 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
146 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) |
147 | 145, 146 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) |
148 | | c0ex 9913 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
149 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
150 | 148, 149 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) |
151 | 147, 150 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
152 | | imain 5888 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡(2nd ‘(1st
‘𝑈)) →
((2nd ‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
153 | 41, 152 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
154 | 57 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑈)) “ ∅)) |
155 | | ima0 5400 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑈)) “ ∅) =
∅ |
156 | 154, 155 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
157 | 153, 156 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) |
158 | | fnun 5911 |
. . . . . . . . . . . 12
⊢
((((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
159 | 151, 157,
158 | sylancr 694 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
160 | | imaundi 5464 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
161 | 47 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑈)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
162 | 161, 36 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
163 | 160, 162 | syl5eqr 2658 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
164 | 163 | fneq2d 5896 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
165 | 159, 164 | mpbid 221 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
166 | 165 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
167 | | ovex 6577 |
. . . . . . . . . 10
⊢
(1...𝑁) ∈
V |
168 | 167 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V) |
169 | | inidm 3784 |
. . . . . . . . 9
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
170 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
171 | | fvun2 6180 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
172 | 147, 150,
171 | mp3an12 1406 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
173 | 157, 172 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
174 | 148 | fvconst2 6374 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
175 | 174 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
176 | 173, 175 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
177 | 176 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
178 | 144, 166,
168, 168, 169, 170, 177 | ofval 6804 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑈))‘𝑦) + 0)) |
179 | 139, 178 | mpdan 699 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑈))‘𝑦) + 0)) |
180 | | elmapi 7765 |
. . . . . . . . . . . . 13
⊢
((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
181 | 141, 180 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
182 | 181 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ (0..^𝐾)) |
183 | | elfzonn0 12380 |
. . . . . . . . . . 11
⊢
(((1st ‘(1st ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈
ℕ0) |
184 | 182, 183 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈
ℕ0) |
185 | 184 | nn0cnd 11230 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ ℂ) |
186 | 139, 185 | syldan 486 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ ℂ) |
187 | 186 | addid1d 10115 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st
‘(1st ‘𝑈))‘𝑦) + 0) = ((1st
‘(1st ‘𝑈))‘𝑦)) |
188 | 132, 179,
187 | 3eqtrd 2648 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
189 | 66, 188 | syldan 486 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
190 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
191 | 190 | breq2d 4595 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
192 | 191 | ifbid 4058 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
193 | 192 | csbeq1d 3506 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
194 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇)) |
195 | 194 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
196 | 194 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
197 | 196 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
198 | 197 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
199 | 196 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
200 | 199 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
201 | 198, 200 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
202 | 195, 201 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
203 | 202 | csbeq2dv 3944 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
204 | 193, 203 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
205 | 204 | mpteq2dv 4673 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
206 | 205 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
207 | 206, 5 | elrab2 3333 |
. . . . . . . . . . . 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 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
208 | 207 | simprbi 479 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
209 | 3, 208 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
210 | | poimirlem11.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2nd
‘𝑇) =
0) |
211 | | breq12 4588 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 = (𝑀 − 1) ∧ (2nd
‘𝑇) = 0) →
(𝑦 < (2nd
‘𝑇) ↔ (𝑀 − 1) <
0)) |
212 | 210, 211 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < 0)) |
213 | 212 | ancoms 468 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < 0)) |
214 | 213, 95 | ifbieq2d 4061 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀)) |
215 | 214, 112 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
216 | 215 | csbeq1d 3506 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
217 | 115 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑀))) |
218 | 217 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})) |
219 | 119 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
220 | 219 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
221 | 218, 220 | uneq12d 3730 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
222 | 221 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
223 | 222 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
224 | 45, 223 | csbied 3526 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
225 | 224 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
226 | 216, 225 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
227 | | ovex 6577 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V |
228 | 227 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
229 | 209, 226,
102, 228 | fvmptd 6197 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
230 | 229 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
231 | 230 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
232 | 20 | sselda 3568 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁)) |
233 | | xp1st 7089 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
234 | 9, 233 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
235 | | elmapfn 7766 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
236 | 234, 235 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
237 | 236 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
238 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
239 | 145, 238 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) |
240 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
241 | 148, 240 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
242 | 239, 241 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
243 | | dff1o3 6056 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
244 | 243 | simprbi 479 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
245 | 15, 244 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
246 | | imain 5888 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
247 | 245, 246 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
248 | 57 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
249 | | ima0 5400 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
250 | 248, 249 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
251 | 247, 250 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
252 | | fnun 5911 |
. . . . . . . . . . . 12
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
253 | 242, 251,
252 | sylancr 694 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
254 | | imaundi 5464 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
255 | 47 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
256 | | f1ofo 6057 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
257 | 15, 256 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
258 | | foima 6033 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
259 | 257, 258 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
260 | 255, 259 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
261 | 254, 260 | syl5eqr 2658 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
262 | 261 | fneq2d 5896 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
263 | 253, 262 | mpbid 221 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
264 | 263 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
265 | 167 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V) |
266 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
267 | | fvun1 6179 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
268 | 239, 241,
267 | mp3an12 1406 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
269 | 251, 268 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
270 | 145 | fvconst2 6374 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
271 | 270 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
272 | 269, 271 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
273 | 272 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
274 | 237, 264,
265, 265, 169, 266, 273 | ofval 6804 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
275 | 232, 274 | mpdan 699 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
276 | 231, 275 | eqtrd 2644 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
277 | 276 | adantrr 749 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
278 | | poimirlem22.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁))) |
279 | 98, 5, 278, 21, 87 | poimirlem10 32589 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓
− ((1...𝑁) ×
{1})) = (1st ‘(1st ‘𝑈))) |
280 | 98, 5, 278, 3, 210 | poimirlem10 32589 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓
− ((1...𝑁) ×
{1})) = (1st ‘(1st ‘𝑇))) |
281 | 279, 280 | eqtr3d 2646 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) = (1st ‘(1st
‘𝑇))) |
282 | 281 | fveq1d 6105 |
. . . . . 6
⊢ (𝜑 → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
283 | 282 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
284 | 189, 277,
283 | 3eqtr3d 2652 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
285 | | elmapi 7765 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
286 | 234, 285 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
287 | 286 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈ (0..^𝐾)) |
288 | | elfzonn0 12380 |
. . . . . . . . . 10
⊢
(((1st ‘(1st ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈
ℕ0) |
289 | 287, 288 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈
ℕ0) |
290 | 289 | nn0red 11229 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈ ℝ) |
291 | 290 | ltp1d 10833 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) < (((1st
‘(1st ‘𝑇))‘𝑦) + 1)) |
292 | 290, 291 | gtned 10051 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) ≠ ((1st
‘(1st ‘𝑇))‘𝑦)) |
293 | 232, 292 | syldan 486 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) ≠ ((1st
‘(1st ‘𝑇))‘𝑦)) |
294 | 293 | neneqd 2787 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ¬ (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
295 | 294 | adantrr 749 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ¬ (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
296 | 284, 295 | pm2.65da 598 |
. . 3
⊢ (𝜑 → ¬ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
297 | | iman 439 |
. . 3
⊢ ((𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ↔ ¬ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
298 | 296, 297 | sylibr 223 |
. 2
⊢ (𝜑 → (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
299 | 298 | ssrdv 3574 |
1
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) |