Proof of Theorem poimirlem11
Step | Hyp | Ref
| Expression |
1 | | eldif 3584 |
. . . . . . 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 5477 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ran (2nd
‘(1st ‘𝑇)) |
3 | | poimirlem12.2 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
4 | | elrabi 3359 |
. . . . . . . . . . . . . . . . . . 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 2719 |
. . . . . . . . . . . . . . . . . 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 7198 |
. . . . . . . . . . . . . . . . 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 7199 |
. . . . . . . . . . . . . . . 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 6201 |
. . . . . . . . . . . . . . . 16
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
13 | | f1oeq1 6127 |
. . . . . . . . . . . . . . . 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 3350 |
. . . . . . . . . . . . . . 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 208 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
16 | | f1of 6137 |
. . . . . . . . . . . . . 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 6053 |
. . . . . . . . . . . . 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 3614 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁)) |
21 | | poimirlem11.4 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
22 | | elrabi 3359 |
. . . . . . . . . . . . . . . . . 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 2719 |
. . . . . . . . . . . . . . . . 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 7198 |
. . . . . . . . . . . . . . . 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 7199 |
. . . . . . . . . . . . . . 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 6201 |
. . . . . . . . . . . . . . 15
⊢
(2nd ‘(1st ‘𝑈)) ∈ V |
30 | | f1oeq1 6127 |
. . . . . . . . . . . . . . 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 3350 |
. . . . . . . . . . . . . 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 208 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
33 | | f1ofo 6144 |
. . . . . . . . . . . . 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 6120 |
. . . . . . . . . . . 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 3642 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd
‘(1st ‘𝑈)) “ (1...𝑁))) |
38 | 37 | ssdifd 3746 |
. . . . . . . . 9
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ⊆ (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
39 | | dff1o3 6143 |
. . . . . . . . . . . . 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 480 |
. . . . . . . . . . . 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 5973 |
. . . . . . . . . . 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 4048 |
. . . . . . . . . . . 12
⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)) |
45 | | poimirlem11.6 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
46 | | fzsplit 12367 |
. . . . . . . . . . . . . . 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 3757 |
. . . . . . . . . . . . . 14
⊢
((1...𝑀) ∪
((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)) |
49 | 47, 48 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))) |
50 | 49 | difeq1d 3727 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀))) |
51 | | incom 3805 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) |
52 | | elfznn 12370 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) |
53 | 45, 52 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℕ) |
54 | 53 | nnred 11035 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
55 | 54 | ltp1d 10954 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
56 | | fzdisj 12368 |
. . . . . . . . . . . . . . 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 2668 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅) |
59 | | disj3 4021 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
60 | 58, 59 | sylib 208 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
61 | 44, 50, 60 | 3eqtr4a 2682 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁)) |
62 | 61 | imaeq2d 5466 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
63 | 43, 62 | eqtr3d 2658 |
. . . . . . . . 9
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) = ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
64 | 38, 63 | sseqtrd 3641 |
. . . . . . . 8
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ⊆ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
65 | 64 | sselda 3603 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
66 | 1, 65 | sylan2br 493 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
67 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (2nd ‘𝑡) = (2nd ‘𝑈)) |
68 | 67 | breq2d 4665 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑈))) |
69 | 68 | ifbid 4108 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1))) |
70 | 69 | csbeq1d 3540 |
. . . . . . . . . . . . . . . 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 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (1st ‘𝑡) = (1st ‘𝑈)) |
72 | 71 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑈))) |
73 | 71 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑈 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑈))) |
74 | 73 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑈)) “
(1...𝑗))) |
75 | 74 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1})) |
76 | 73 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑈)) “ ((𝑗 + 1)...𝑁))) |
77 | 76 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) |
78 | 75, 77 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . 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 6668 |
. . . . . . . . . . . . . . . . 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 3992 |
. . . . . . . . . . . . . . . 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 2656 |
. . . . . . . . . . . . . . 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 4745 |
. . . . . . . . . . . . . 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 2632 |
. . . . . . . . . . . . 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 3366 |
. . . . . . . . . . . 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 480 |
. . . . . . . . . . 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 4658 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 = (𝑀 − 1) ∧ (2nd
‘𝑈) = 0) →
(𝑦 < (2nd
‘𝑈) ↔ (𝑀 − 1) <
0)) |
89 | 87, 88 | sylan2 491 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑈) ↔ (𝑀 − 1) < 0)) |
90 | 89 | ancoms 469 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2nd ‘𝑈) ↔ (𝑀 − 1) < 0)) |
91 | | oveq1 6657 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1)) |
92 | 53 | nncnd 11036 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℂ) |
93 | | npcan1 10455 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
94 | 92, 93 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
95 | 91, 94 | sylan9eqr 2678 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀) |
96 | 90, 95 | ifbieq2d 4111 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀)) |
97 | 53 | nnzd 11481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
98 | | poimir.0 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) |
99 | 98 | nnzd 11481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ ℤ) |
100 | | elfzm1b 12418 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1)))) |
101 | 97, 99, 100 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1)))) |
102 | 45, 101 | mpbid 222 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
103 | | elfzle1 12344 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) → 0 ≤
(𝑀 −
1)) |
104 | 102, 103 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 ≤ (𝑀 − 1)) |
105 | | 0red 10041 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 ∈
ℝ) |
106 | | nnm1nn0 11334 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈
ℕ0) |
107 | 53, 106 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 − 1) ∈
ℕ0) |
108 | 107 | nn0red 11352 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
109 | 105, 108 | lenltd 10183 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0 ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < 0)) |
110 | 104, 109 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ¬ (𝑀 − 1) < 0) |
111 | 110 | iffalsed 4097 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀) |
112 | 111 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀) |
113 | 96, 112 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀) |
114 | 113 | csbeq1d 3540 |
. . . . . . . . . . 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 6658 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
116 | 115 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑈)) “
(1...𝑀))) |
117 | 116 | xpeq1d 5138 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1})) |
118 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
119 | 118 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
120 | 119 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
121 | 120 | xpeq1d 5138 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) |
122 | 117, 121 | uneq12d 3768 |
. . . . . . . . . . . . . . 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 6666 |
. . . . . . . . . . . . . 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 482 |
. . . . . . . . . . . . 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 3560 |
. . . . . . . . . . . 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 481 |
. . . . . . . . . . 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 2656 |
. . . . . . . . . 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 | | ovexd 6680 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
129 | 86, 127, 102, 128 | fvmptd 6288 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st
‘(1st ‘𝑈)) ∘𝑓 +
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
130 | 129 | fveq1d 6193 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑈))
∘𝑓 + ((((2nd ‘(1st
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
131 | 130 | adantr 481 |
. . . . . . 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)) |
132 | | imassrn 5477 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2nd
‘(1st ‘𝑈)) |
133 | | f1of 6137 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
134 | 32, 133 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
135 | | frn 6053 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁) → ran (2nd
‘(1st ‘𝑈)) ⊆ (1...𝑁)) |
136 | 134, 135 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ran (2nd
‘(1st ‘𝑈)) ⊆ (1...𝑁)) |
137 | 132, 136 | syl5ss 3614 |
. . . . . . . . 9
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁)) |
138 | 137 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁)) |
139 | | xp1st 7198 |
. . . . . . . . . . . 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))) |
140 | 26, 139 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
141 | | elmapfn 7880 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
142 | 140, 141 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
143 | 142 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
144 | | 1ex 10035 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
145 | | fnconstg 6093 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) |
146 | 144, 145 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) |
147 | | c0ex 10034 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
148 | | fnconstg 6093 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
149 | 147, 148 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) |
150 | 146, 149 | pm3.2i 471 |
. . . . . . . . . . . 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))) |
151 | | imain 5974 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡(2nd ‘(1st
‘𝑈)) →
((2nd ‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
152 | 41, 151 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
153 | 57 | imaeq2d 5466 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑈)) “ ∅)) |
154 | | ima0 5481 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑈)) “ ∅) =
∅ |
155 | 153, 154 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
156 | 152, 155 | eqtr3d 2658 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) |
157 | | fnun 5997 |
. . . . . . . . . . . 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)))) |
158 | 150, 156,
157 | sylancr 695 |
. . . . . . . . . . 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)))) |
159 | | imaundi 5545 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
160 | 47 | imaeq2d 5466 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑈)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
161 | 160, 36 | eqtr3d 2658 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
162 | 159, 161 | syl5eqr 2670 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
163 | 162 | fneq2d 5982 |
. . . . . . . . . . 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))) |
164 | 158, 163 | mpbid 222 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
165 | 164 | adantr 481 |
. . . . . . . . 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)) |
166 | | ovexd 6680 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V) |
167 | | inidm 3822 |
. . . . . . . . 9
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
168 | | eqidd 2623 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
169 | | fvun2 6270 |
. . . . . . . . . . . . 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)) |
170 | 146, 149,
169 | mp3an12 1414 |
. . . . . . . . . . . 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)) |
171 | 156, 170 | sylan 488 |
. . . . . . . . . . 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)) |
172 | 147 | fvconst2 6469 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
173 | 172 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
174 | 171, 173 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
175 | 174 | adantr 481 |
. . . . . . . . 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) |
176 | 143, 165,
166, 166, 167, 168, 175 | ofval 6906 |
. . . . . . . 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)) |
177 | 138, 176 | mpdan 702 |
. . . . . . 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)) |
178 | | elmapi 7879 |
. . . . . . . . . . . . 13
⊢
((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
179 | 140, 178 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
180 | 179 | ffvelrnda 6359 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ (0..^𝐾)) |
181 | | elfzonn0 12512 |
. . . . . . . . . . 11
⊢
(((1st ‘(1st ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈
ℕ0) |
182 | 180, 181 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈
ℕ0) |
183 | 182 | nn0cnd 11353 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ ℂ) |
184 | 138, 183 | syldan 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ ℂ) |
185 | 184 | addid1d 10236 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st
‘(1st ‘𝑈))‘𝑦) + 0) = ((1st
‘(1st ‘𝑈))‘𝑦)) |
186 | 131, 177,
185 | 3eqtrd 2660 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
187 | 66, 186 | syldan 487 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
188 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
189 | 188 | breq2d 4665 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
190 | 189 | ifbid 4108 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
191 | 190 | csbeq1d 3540 |
. . . . . . . . . . . . . . . 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})))) |
192 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇)) |
193 | 192 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
194 | 192 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
195 | 194 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
196 | 195 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
197 | 194 | imaeq1d 5465 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
198 | 197 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
199 | 196, 198 | uneq12d 3768 |
. . . . . . . . . . . . . . . . . 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}))) |
200 | 193, 199 | oveq12d 6668 |
. . . . . . . . . . . . . . . . 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})))) |
201 | 200 | csbeq2dv 3992 |
. . . . . . . . . . . . . . . 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})))) |
202 | 191, 201 | eqtrd 2656 |
. . . . . . . . . . . . . . 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})))) |
203 | 202 | mpteq2dv 4745 |
. . . . . . . . . . . . . 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}))))) |
204 | 203 | eqeq2d 2632 |
. . . . . . . . . . . . 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})))))) |
205 | 204, 5 | elrab2 3366 |
. . . . . . . . . . . 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})))))) |
206 | 205 | simprbi 480 |
. . . . . . . . . . 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}))))) |
207 | 3, 206 | 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}))))) |
208 | | poimirlem11.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2nd
‘𝑇) =
0) |
209 | | breq12 4658 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 = (𝑀 − 1) ∧ (2nd
‘𝑇) = 0) →
(𝑦 < (2nd
‘𝑇) ↔ (𝑀 − 1) <
0)) |
210 | 208, 209 | sylan2 491 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < 0)) |
211 | 210 | ancoms 469 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2nd ‘𝑇) ↔ (𝑀 − 1) < 0)) |
212 | 211, 95 | ifbieq2d 4111 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀)) |
213 | 212, 112 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
214 | 213 | csbeq1d 3540 |
. . . . . . . . . . 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})))) |
215 | 115 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑀))) |
216 | 215 | xpeq1d 5138 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})) |
217 | 119 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
218 | 217 | xpeq1d 5138 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
219 | 216, 218 | uneq12d 3768 |
. . . . . . . . . . . . . . 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}))) |
220 | 219 | oveq2d 6666 |
. . . . . . . . . . . . . 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})))) |
221 | 220 | adantl 482 |
. . . . . . . . . . . . 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})))) |
222 | 45, 221 | csbied 3560 |
. . . . . . . . . . . 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})))) |
223 | 222 | adantr 481 |
. . . . . . . . . . 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})))) |
224 | 214, 223 | eqtrd 2656 |
. . . . . . . . . 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})))) |
225 | | ovexd 6680 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
226 | 207, 224,
102, 225 | fvmptd 6288 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st
‘(1st ‘𝑇)) ∘𝑓 +
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
227 | 226 | fveq1d 6193 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))
∘𝑓 + ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
228 | 227 | adantr 481 |
. . . . . . 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)) |
229 | 20 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁)) |
230 | | xp1st 7198 |
. . . . . . . . . . . 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))) |
231 | 9, 230 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
232 | | elmapfn 7880 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
233 | 231, 232 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
234 | 233 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
235 | | fnconstg 6093 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
236 | 144, 235 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) |
237 | | fnconstg 6093 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
238 | 147, 237 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
239 | 236, 238 | pm3.2i 471 |
. . . . . . . . . . . 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))) |
240 | | dff1o3 6143 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
241 | 240 | simprbi 480 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
242 | 15, 241 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Fun ◡(2nd ‘(1st
‘𝑇))) |
243 | | imain 5974 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
244 | 242, 243 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
245 | 57 | imaeq2d 5466 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
246 | | ima0 5481 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
247 | 245, 246 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
248 | 244, 247 | eqtr3d 2658 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
249 | | fnun 5997 |
. . . . . . . . . . . 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)))) |
250 | 239, 248,
249 | sylancr 695 |
. . . . . . . . . . 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)))) |
251 | | imaundi 5545 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
252 | 47 | imaeq2d 5466 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
253 | | f1ofo 6144 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
254 | 15, 253 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
255 | | foima 6120 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
256 | 254, 255 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
257 | 252, 256 | eqtr3d 2658 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
258 | 251, 257 | syl5eqr 2670 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
259 | 258 | fneq2d 5982 |
. . . . . . . . . . 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))) |
260 | 250, 259 | mpbid 222 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
261 | 260 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
262 | | ovexd 6680 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V) |
263 | | eqidd 2623 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
264 | | fvun1 6269 |
. . . . . . . . . . . . 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)) |
265 | 236, 238,
264 | mp3an12 1414 |
. . . . . . . . . . . 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)) |
266 | 248, 265 | sylan 488 |
. . . . . . . . . . 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)) |
267 | 144 | fvconst2 6469 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
268 | 267 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
269 | 266, 268 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
270 | 269 | adantr 481 |
. . . . . . . . 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) |
271 | 234, 261,
262, 262, 167, 263, 270 | ofval 6906 |
. . . . . . . 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)) |
272 | 229, 271 | mpdan 702 |
. . . . . . 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)) |
273 | 228, 272 | eqtrd 2656 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
274 | 273 | adantrr 753 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
275 | | poimirlem22.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁))) |
276 | 98, 5, 275, 21, 87 | poimirlem10 33419 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓
− ((1...𝑁) ×
{1})) = (1st ‘(1st ‘𝑈))) |
277 | 98, 5, 275, 3, 208 | poimirlem10 33419 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘𝑓
− ((1...𝑁) ×
{1})) = (1st ‘(1st ‘𝑇))) |
278 | 276, 277 | eqtr3d 2658 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) = (1st ‘(1st
‘𝑇))) |
279 | 278 | fveq1d 6193 |
. . . . . 6
⊢ (𝜑 → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
280 | 279 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
281 | 187, 274,
280 | 3eqtr3d 2664 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
282 | | elmapi 7879 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
283 | 231, 282 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
284 | 283 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈ (0..^𝐾)) |
285 | | elfzonn0 12512 |
. . . . . . . . . 10
⊢
(((1st ‘(1st ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈
ℕ0) |
286 | 284, 285 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈
ℕ0) |
287 | 286 | nn0red 11352 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈ ℝ) |
288 | 287 | ltp1d 10954 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) < (((1st
‘(1st ‘𝑇))‘𝑦) + 1)) |
289 | 287, 288 | gtned 10172 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) ≠ ((1st
‘(1st ‘𝑇))‘𝑦)) |
290 | 229, 289 | syldan 487 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) ≠ ((1st
‘(1st ‘𝑇))‘𝑦)) |
291 | 290 | neneqd 2799 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ¬ (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
292 | 291 | adantrr 753 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ¬ (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
293 | 281, 292 | pm2.65da 600 |
. . 3
⊢ (𝜑 → ¬ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
294 | | iman 440 |
. . 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)))) |
295 | 293, 294 | sylibr 224 |
. 2
⊢ (𝜑 → (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
296 | 295 | ssrdv 3609 |
1
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) |