Step | Hyp | Ref
| Expression |
1 | | eldif 3550 |
. . . . . . 7
⊢ (𝑦 ∈ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ↔ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
2 | | imassrn 5396 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ran (2^{nd}
‘(1^{st} ‘𝑇)) |
3 | | poimirlem12.2 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
4 | | elrabi 3328 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
5 | | poimirlem22.s |
. . . . . . . . . . . . . . . . 17
⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
6 | 4, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
7 | | xp1st 7089 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
8 | 3, 6, 7 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1^{st}
‘𝑇) ∈
(((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
9 | | xp2nd 7090 |
. . . . . . . . . . . . . . 15
⊢
((1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
10 | 8, 9 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
11 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
12 | | f1oeq1 6040 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (2^{nd}
‘(1^{st} ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
13 | 11, 12 | elab 3319 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
14 | 10, 13 | sylib 207 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
15 | | f1of 6050 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
16 | | frn 5966 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁) → ran (2^{nd}
‘(1^{st} ‘𝑇)) ⊆ (1...𝑁)) |
17 | 14, 15, 16 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (2^{nd}
‘(1^{st} ‘𝑇)) ⊆ (1...𝑁)) |
18 | 2, 17 | syl5ss 3579 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁)) |
19 | | poimirlem12.4 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
20 | | elrabi 3328 |
. . . . . . . . . . . . . . . 16
⊢ (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
21 | 20, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
22 | | xp1st 7089 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑈) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
23 | 19, 21, 22 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1^{st}
‘𝑈) ∈
(((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
24 | | xp2nd 7090 |
. . . . . . . . . . . . . 14
⊢
((1^{st} ‘𝑈) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
26 | | fvex 6113 |
. . . . . . . . . . . . . 14
⊢
(2^{nd} ‘(1^{st} ‘𝑈)) ∈ V |
27 | | f1oeq1 6040 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = (2^{nd}
‘(1^{st} ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))) |
28 | 26, 27 | elab 3319 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
29 | 25, 28 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
30 | | f1ofo 6057 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁)) |
31 | | foima 6033 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
32 | 29, 30, 31 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
33 | 18, 32 | sseqtr4d 3605 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁))) |
34 | 33 | ssdifd 3708 |
. . . . . . . . 9
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ⊆ (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
35 | | dff1o3 6056 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑈)))) |
36 | 35 | simprbi 479 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑈))) |
37 | | imadif 5887 |
. . . . . . . . . . 11
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑈)) →
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
38 | 29, 36, 37 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
39 | | difun2 4000 |
. . . . . . . . . . . 12
⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)) |
40 | | poimirlem12.6 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1))) |
41 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → 𝑀 ∈
ℕ_{0}) |
42 | | nn0p1nn 11209 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ_{0}
→ (𝑀 + 1) ∈
ℕ) |
43 | 40, 41, 42 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 + 1) ∈ ℕ) |
44 | | nnuz 11599 |
. . . . . . . . . . . . . . . 16
⊢ ℕ =
(ℤ_{≥}‘1) |
45 | 43, 44 | syl6eleq 2698 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 + 1) ∈
(ℤ_{≥}‘1)) |
46 | | poimir.0 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℕ) |
47 | 46 | nncnd 10913 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℂ) |
48 | | npcan1 10334 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
49 | 47, 48 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
50 | | elfzuz3 12210 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ_{≥}‘𝑀)) |
51 | | peano2uz 11617 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑀) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑀)) |
52 | 40, 50, 51 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑀)) |
53 | 49, 52 | eqeltrrd 2689 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘𝑀)) |
54 | | fzsplit2 12237 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 + 1) ∈
(ℤ_{≥}‘1) ∧ 𝑁 ∈ (ℤ_{≥}‘𝑀)) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
55 | 45, 53, 54 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
56 | | uncom 3719 |
. . . . . . . . . . . . . 14
⊢
((1...𝑀) ∪
((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)) |
57 | 55, 56 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))) |
58 | 57 | difeq1d 3689 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀))) |
59 | | incom 3767 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) |
60 | 40, 41 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈
ℕ_{0}) |
61 | 60 | nn0red 11229 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
62 | 61 | ltp1d 10833 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
63 | | fzdisj 12239 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
65 | 59, 64 | syl5eq 2656 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅) |
66 | | disj3 3973 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
67 | 65, 66 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
68 | 39, 58, 67 | 3eqtr4a 2670 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁)) |
69 | 68 | imaeq2d 5385 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
70 | 38, 69 | eqtr3d 2646 |
. . . . . . . . 9
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
71 | 34, 70 | sseqtrd 3604 |
. . . . . . . 8
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
72 | 71 | sselda 3568 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∖ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
73 | 1, 72 | sylan2br 492 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
74 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑈)) |
75 | 74 | breq2d 4595 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑈))) |
76 | 75 | ifbid 4058 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1))) |
77 | 76 | csbeq1d 3506 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
78 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (1^{st} ‘𝑡) = (1^{st} ‘𝑈)) |
79 | 78 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑈))) |
80 | 78 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑈 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑈))) |
81 | 80 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑗))) |
82 | 81 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1})) |
83 | 80 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑗 + 1)...𝑁))) |
84 | 83 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) |
85 | 82, 84 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
86 | 79, 85 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → ((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
87 | 86 | csbeq2dv 3944 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
88 | 77, 87 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
89 | 88 | mpteq2dv 4673 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
90 | 89 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
91 | 90, 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 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
92 | 91 | simprbi 479 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
93 | 19, 92 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
94 | | breq1 4586 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (𝑦 < (2^{nd} ‘𝑈) ↔ 𝑀 < (2^{nd} ‘𝑈))) |
95 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) |
96 | 94, 95 | ifbieq1d 4059 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) = if(𝑀 < (2^{nd} ‘𝑈), 𝑀, (𝑦 + 1))) |
97 | 46 | nnred 10912 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℝ) |
98 | | peano2rem 10227 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈
ℝ) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
100 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) |
101 | 40, 100 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) |
102 | 97 | ltm1d 10835 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
103 | 61, 99, 97, 101, 102 | lelttrd 10074 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < 𝑁) |
104 | | poimirlem12.5 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘𝑈) = 𝑁) |
105 | 103, 104 | breqtrrd 4611 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2^{nd} ‘𝑈)) |
106 | 105 | iftrued 4044 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑀 < (2^{nd} ‘𝑈), 𝑀, (𝑦 + 1)) = 𝑀) |
107 | 96, 106 | sylan9eqr 2666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀) |
108 | 107 | csbeq1d 3506 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
109 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
110 | 109 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀))) |
111 | 110 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1})) |
112 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
113 | 112 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
114 | 113 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
115 | 114 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) |
116 | 111, 115 | uneq12d 3730 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
117 | 116 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
118 | 117 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
119 | 40, 118 | csbied 3526 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
120 | 119 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
121 | 108, 120 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2^{nd}
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
122 | | ovex 6577 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V |
123 | 122 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
124 | 93, 121, 40, 123 | fvmptd 6197 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑀) = ((1^{st} ‘(1^{st}
‘𝑈))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
125 | 124 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
126 | 125 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
127 | | imassrn 5396 |
. . . . . . . . . 10
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2^{nd}
‘(1^{st} ‘𝑈)) |
128 | | f1of 6050 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
129 | | frn 5966 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)⟶(1...𝑁) → ran (2^{nd}
‘(1^{st} ‘𝑈)) ⊆ (1...𝑁)) |
130 | 29, 128, 129 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ran (2^{nd}
‘(1^{st} ‘𝑈)) ⊆ (1...𝑁)) |
131 | 127, 130 | syl5ss 3579 |
. . . . . . . . 9
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁)) |
132 | 131 | sselda 3568 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁)) |
133 | | xp1st 7089 |
. . . . . . . . . . 11
⊢
((1^{st} ‘𝑈) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^{st}
‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
134 | | elmapfn 7766 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
135 | 23, 133, 134 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
136 | 135 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
137 | | 1ex 9914 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
138 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) |
139 | 137, 138 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) |
140 | | c0ex 9913 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
141 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
142 | 140, 141 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) |
143 | 139, 142 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
144 | | imain 5888 |
. . . . . . . . . . . . . 14
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑈)) →
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
145 | 29, 36, 144 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
146 | 64 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ∅)) |
147 | | ima0 5400 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ∅) =
∅ |
148 | 146, 147 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
149 | 145, 148 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) |
150 | | fnun 5911 |
. . . . . . . . . . . 12
⊢
((((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
151 | 143, 149,
150 | sylancr 694 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
152 | | imaundi 5464 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
153 | 55 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
154 | 153, 32 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
155 | 152, 154 | syl5eqr 2658 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
156 | 155 | fneq2d 5896 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
157 | 151, 156 | mpbid 221 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
158 | 157 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
159 | | ovex 6577 |
. . . . . . . . . 10
⊢
(1...𝑁) ∈
V |
160 | 159 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V) |
161 | | inidm 3784 |
. . . . . . . . 9
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
162 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
163 | | fvun2 6180 |
. . . . . . . . . . . . 13
⊢
(((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
164 | 139, 142,
163 | mp3an12 1406 |
. . . . . . . . . . . 12
⊢
(((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
165 | 149, 164 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
166 | 140 | fvconst2 6374 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
167 | 166 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
168 | 165, 167 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
169 | 168 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
170 | 136, 158,
160, 160, 161, 162, 169 | ofval 6804 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))‘𝑦) + 0)) |
171 | 132, 170 | mpdan 699 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^{st}
‘(1^{st} ‘𝑈)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑈))‘𝑦) + 0)) |
172 | | elmapi 7765 |
. . . . . . . . . . . . 13
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
173 | 23, 133, 172 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
174 | 173 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ (0..^𝐾)) |
175 | | elfzonn0 12380 |
. . . . . . . . . . 11
⊢
(((1^{st} ‘(1^{st} ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈
ℕ_{0}) |
176 | 174, 175 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈
ℕ_{0}) |
177 | 176 | nn0cnd 11230 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ ℂ) |
178 | 177 | addid1d 10115 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) + 0) = ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦)) |
179 | 132, 178 | syldan 486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) + 0) = ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦)) |
180 | 126, 171,
179 | 3eqtrd 2648 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀)‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
181 | 73, 180 | syldan 486 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘𝑀)‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
182 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
183 | 182 | breq2d 4595 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
184 | 183 | ifbid 4058 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
185 | 184 | csbeq1d 3506 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
186 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (1^{st} ‘𝑡) = (1^{st} ‘𝑇)) |
187 | 186 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
188 | 186 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
189 | 188 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
190 | 189 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
191 | 188 | imaeq1d 5384 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
192 | 191 | xpeq1d 5062 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
193 | 190, 192 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
194 | 187, 193 | oveq12d 6567 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → ((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
195 | 194 | csbeq2dv 3944 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
196 | 185, 195 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
197 | 196 | mpteq2dv 4673 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
198 | 197 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
199 | 198, 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 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
200 | 199 | simprbi 479 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
201 | 3, 200 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
202 | | breq1 4586 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (𝑦 < (2^{nd} ‘𝑇) ↔ 𝑀 < (2^{nd} ‘𝑇))) |
203 | 202, 95 | ifbieq1d 4059 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2^{nd} ‘𝑇), 𝑀, (𝑦 + 1))) |
204 | | poimirlem12.3 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘𝑇) = 𝑁) |
205 | 103, 204 | breqtrrd 4611 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2^{nd} ‘𝑇)) |
206 | 205 | iftrued 4044 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑀 < (2^{nd} ‘𝑇), 𝑀, (𝑦 + 1)) = 𝑀) |
207 | 203, 206 | sylan9eqr 2666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
208 | 207 | csbeq1d 3506 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
209 | 109 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀))) |
210 | 209 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})) |
211 | 113 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
212 | 211 | xpeq1d 5062 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
213 | 210, 212 | uneq12d 3730 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
214 | 213 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
215 | 214 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
216 | 40, 215 | csbied 3526 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
217 | 216 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋𝑀 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
218 | 208, 217 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
219 | | ovex 6577 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V |
220 | 219 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
221 | 201, 218,
40, 220 | fvmptd 6197 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑀) = ((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
222 | 221 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
223 | 222 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))
∘_{𝑓} + ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) × {1}) ∪
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
224 | 18 | sselda 3568 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁)) |
225 | | xp1st 7089 |
. . . . . . . . . . 11
⊢
((1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
226 | | elmapfn 7766 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
227 | 8, 225, 226 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
228 | 227 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
229 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) |
230 | 137, 229 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) |
231 | | fnconstg 6006 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
232 | 140, 231 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
233 | 230, 232 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
234 | | dff1o3 6056 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇)))) |
235 | 234 | simprbi 479 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇))) |
236 | | imain 5888 |
. . . . . . . . . . . . . 14
⊢ (Fun
^{◡}(2^{nd} ‘(1^{st}
‘𝑇)) →
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
237 | 14, 235, 236 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
238 | 64 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ∅)) |
239 | | ima0 5400 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ∅) =
∅ |
240 | 238, 239 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
241 | 237, 240 | eqtr3d 2646 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
242 | | fnun 5911 |
. . . . . . . . . . . 12
⊢
((((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
243 | 233, 241,
242 | sylancr 694 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
244 | | imaundi 5464 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
245 | 55 | imaeq2d 5385 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
246 | | f1ofo 6057 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
247 | | foima 6033 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
248 | 14, 246, 247 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
249 | 245, 248 | eqtr3d 2646 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
250 | 244, 249 | syl5eqr 2658 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
251 | 250 | fneq2d 5896 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
252 | 243, 251 | mpbid 221 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
253 | 252 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
254 | 159 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V) |
255 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
256 | | fvun1 6179 |
. . . . . . . . . . . . 13
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
257 | 230, 232,
256 | mp3an12 1406 |
. . . . . . . . . . . 12
⊢
(((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
258 | 241, 257 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
259 | 137 | fvconst2 6374 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
260 | 259 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
261 | 258, 260 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
262 | 261 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
263 | 228, 253,
254, 254, 161, 255, 262 | ofval 6804 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
264 | 224, 263 | mpdan 699 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
265 | 223, 264 | eqtrd 2644 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
266 | 265 | adantrr 749 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
267 | 46 | nngt0d 10941 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝑁) |
268 | 267, 104 | breqtrrd 4611 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (2^{nd}
‘𝑈)) |
269 | 46, 5, 19, 268 | poimirlem5 32584 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘0) = (1^{st}
‘(1^{st} ‘𝑈))) |
270 | 267, 204 | breqtrrd 4611 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (2^{nd}
‘𝑇)) |
271 | 46, 5, 3, 270 | poimirlem5 32584 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘0) = (1^{st}
‘(1^{st} ‘𝑇))) |
272 | 269, 271 | eqtr3d 2646 |
. . . . . . 7
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) = (1^{st} ‘(1^{st}
‘𝑇))) |
273 | 272 | fveq1d 6105 |
. . . . . 6
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
274 | 273 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
275 | 181, 266,
274 | 3eqtr3d 2652 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
276 | | elmapi 7765 |
. . . . . . . . . . . 12
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
277 | 8, 225, 276 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
278 | 277 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈ (0..^𝐾)) |
279 | | elfzonn0 12380 |
. . . . . . . . . 10
⊢
(((1^{st} ‘(1^{st} ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈
ℕ_{0}) |
280 | 278, 279 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈
ℕ_{0}) |
281 | 280 | nn0red 11229 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈ ℝ) |
282 | 281 | ltp1d 10833 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) < (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1)) |
283 | 281, 282 | gtned 10051 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) ≠ ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
284 | 224, 283 | syldan 486 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) ≠ ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
285 | 284 | neneqd 2787 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ¬ (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
286 | 285 | adantrr 749 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) → ¬ (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
287 | 275, 286 | pm2.65da 598 |
. . 3
⊢ (𝜑 → ¬ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
288 | | iman 439 |
. . 3
⊢ ((𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) ↔ ¬ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
289 | 287, 288 | sylibr 223 |
. 2
⊢ (𝜑 → (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
290 | 289 | ssrdv 3574 |
1
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) |