Step | Hyp | Ref
| Expression |
1 | | eldif 3853 |
. . . . . . 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 5914 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ran (2^{nd}
‘(1^{st} ‘𝑇)) |
3 | | poimirlem12.2 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
4 | | elrabi 3582 |
. . . . . . . . . . . . . . . . 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 2851 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_{m} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
7 | | xp1st 7746 |
. . . . . . . . . . . . . . . 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 7747 |
. . . . . . . . . . . . . . 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 6687 |
. . . . . . . . . . . . . . 15
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
12 | | f1oeq1 6606 |
. . . . . . . . . . . . . . 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 3573 |
. . . . . . . . . . . . . 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 221 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
15 | | f1of 6618 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
16 | | frn 6511 |
. . . . . . . . . . . . 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 | sstrid 3888 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁)) |
19 | | poimirlem12.4 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
20 | | elrabi 3582 |
. . . . . . . . . . . . . . . 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 2851 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_{m} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
22 | | xp1st 7746 |
. . . . . . . . . . . . . . 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 7747 |
. . . . . . . . . . . . . 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 6687 |
. . . . . . . . . . . . . 14
⊢
(2^{nd} ‘(1^{st} ‘𝑈)) ∈ V |
27 | | f1oeq1 6606 |
. . . . . . . . . . . . . 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 3573 |
. . . . . . . . . . . . 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 221 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
30 | | f1ofo 6625 |
. . . . . . . . . . . 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 6597 |
. . . . . . . . . . . 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 | sseqtrrd 3918 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁))) |
34 | 33 | ssdifd 4031 |
. . . . . . . . 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 6624 |
. . . . . . . . . . . 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 500 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑈))) |
37 | | imadif 6423 |
. . . . . . . . . . 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 4370 |
. . . . . . . . . . . 12
⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)) |
40 | | poimirlem12.6 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1))) |
41 | | elfznn0 13091 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → 𝑀 ∈
ℕ_{0}) |
42 | | nn0p1nn 12015 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ_{0}
→ (𝑀 + 1) ∈
ℕ) |
43 | 40, 41, 42 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 + 1) ∈ ℕ) |
44 | | nnuz 12363 |
. . . . . . . . . . . . . . . 16
⊢ ℕ =
(ℤ_{≥}‘1) |
45 | 43, 44 | eleqtrdi 2843 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 + 1) ∈
(ℤ_{≥}‘1)) |
46 | | poimir.0 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℕ) |
47 | 46 | nncnd 11732 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℂ) |
48 | | npcan1 11143 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
49 | 47, 48 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
50 | | elfzuz3 12995 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ_{≥}‘𝑀)) |
51 | | peano2uz 12383 |
. . . . . . . . . . . . . . . . 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 2834 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘𝑀)) |
54 | | fzsplit2 13023 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 + 1) ∈
(ℤ_{≥}‘1) ∧ 𝑁 ∈ (ℤ_{≥}‘𝑀)) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
55 | 45, 53, 54 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
56 | | uncom 4043 |
. . . . . . . . . . . . . 14
⊢
((1...𝑀) ∪
((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)) |
57 | 55, 56 | eqtrdi 2789 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))) |
58 | 57 | difeq1d 4012 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀))) |
59 | | incom 4091 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) |
60 | 40, 41 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈
ℕ_{0}) |
61 | 60 | nn0red 12037 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
62 | 61 | ltp1d 11648 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
63 | | fzdisj 13025 |
. . . . . . . . . . . . . . 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 2785 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅) |
66 | | disj3 4343 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
67 | 65, 66 | sylib 221 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
68 | 39, 58, 67 | 3eqtr4a 2799 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁)) |
69 | 68 | imaeq2d 5903 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
70 | 38, 69 | eqtr3d 2775 |
. . . . . . . . 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 3917 |
. . . . . . . 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 3877 |
. . . . . . 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 598 |
. . . . . 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 6674 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑈)) |
75 | 74 | breq2d 5042 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑈))) |
76 | 75 | ifbid 4437 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1))) |
77 | 76 | csbeq1d 3794 |
. . . . . . . . . . . . . . . 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 | | 2fveq3 6679 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑈))) |
79 | | 2fveq3 6679 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑈 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑈))) |
80 | 79 | imaeq1d 5902 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑗))) |
81 | 80 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1})) |
82 | 79 | imaeq1d 5902 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑗 + 1)...𝑁))) |
83 | 82 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) |
84 | 81, 83 | uneq12d 4054 |
. . . . . . . . . . . . . . . . . 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}))) |
85 | 78, 84 | oveq12d 7188 |
. . . . . . . . . . . . . . . . 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})))) |
86 | 85 | csbeq2dv 3797 |
. . . . . . . . . . . . . . . 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})))) |
87 | 77, 86 | eqtrd 2773 |
. . . . . . . . . . . . . . 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})))) |
88 | 87 | mpteq2dv 5126 |
. . . . . . . . . . . . . 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}))))) |
89 | 88 | eqeq2d 2749 |
. . . . . . . . . . . . 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})))))) |
90 | 89, 5 | elrab2 3591 |
. . . . . . . . . . . 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})))))) |
91 | 90 | simprbi 500 |
. . . . . . . . . . 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}))))) |
92 | 19, 91 | 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}))))) |
93 | | breq1 5033 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (𝑦 < (2^{nd} ‘𝑈) ↔ 𝑀 < (2^{nd} ‘𝑈))) |
94 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) |
95 | 93, 94 | ifbieq1d 4438 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) = if(𝑀 < (2^{nd} ‘𝑈), 𝑀, (𝑦 + 1))) |
96 | 46 | nnred 11731 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℝ) |
97 | | peano2rem 11031 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈
ℝ) |
98 | 96, 97 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
99 | | elfzle2 13002 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) |
100 | 40, 99 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) |
101 | 96 | ltm1d 11650 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
102 | 61, 98, 96, 100, 101 | lelttrd 10876 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < 𝑁) |
103 | | poimirlem12.5 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘𝑈) = 𝑁) |
104 | 102, 103 | breqtrrd 5058 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2^{nd} ‘𝑈)) |
105 | 104 | iftrued 4422 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑀 < (2^{nd} ‘𝑈), 𝑀, (𝑦 + 1)) = 𝑀) |
106 | 95, 105 | sylan9eqr 2795 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2^{nd} ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀) |
107 | 106 | csbeq1d 3794 |
. . . . . . . . . . 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})))) |
108 | | oveq2 7178 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
109 | 108 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
(1...𝑀))) |
110 | 109 | xpeq1d 5554 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1})) |
111 | | oveq1 7177 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
112 | 111 | oveq1d 7185 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
113 | 112 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
114 | 113 | xpeq1d 5554 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) |
115 | 110, 114 | uneq12d 4054 |
. . . . . . . . . . . . . . 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}))) |
116 | 115 | oveq2d 7186 |
. . . . . . . . . . . . . 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})))) |
117 | 116 | adantl 485 |
. . . . . . . . . . . . 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})))) |
118 | 40, 117 | csbied 3826 |
. . . . . . . . . . . 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})))) |
119 | 118 | adantr 484 |
. . . . . . . . . . 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})))) |
120 | 107, 119 | eqtrd 2773 |
. . . . . . . . . 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})))) |
121 | | ovexd 7205 |
. . . . . . . . . 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) |
122 | 92, 120, 40, 121 | fvmptd 6782 |
. . . . . . . . 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})))) |
123 | 122 | fveq1d 6676 |
. . . . . . . 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)) |
124 | 123 | adantr 484 |
. . . . . . 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)) |
125 | | imassrn 5914 |
. . . . . . . . . 10
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2^{nd}
‘(1^{st} ‘𝑈)) |
126 | | f1of 6618 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
127 | | frn 6511 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑈)):(1...𝑁)⟶(1...𝑁) → ran (2^{nd}
‘(1^{st} ‘𝑈)) ⊆ (1...𝑁)) |
128 | 29, 126, 127 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ran (2^{nd}
‘(1^{st} ‘𝑈)) ⊆ (1...𝑁)) |
129 | 125, 128 | sstrid 3888 |
. . . . . . . . 9
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁)) |
130 | 129 | sselda 3877 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁)) |
131 | | xp1st 7746 |
. . . . . . . . . . 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))) |
132 | | elmapfn 8475 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{m} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
133 | 23, 131, 132 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
134 | 133 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1^{st}
‘(1^{st} ‘𝑈)) Fn (1...𝑁)) |
135 | | 1ex 10715 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
136 | | fnconstg 6566 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) |
137 | 135, 136 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) |
138 | | c0ex 10713 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
139 | | fnconstg 6566 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
140 | 138, 139 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) |
141 | 137, 140 | pm3.2i 474 |
. . . . . . . . . . . 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))) |
142 | | imain 6424 |
. . . . . . . . . . . . . 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)))) |
143 | 29, 36, 142 | 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)))) |
144 | 64 | imaeq2d 5903 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑈)) “ ∅)) |
145 | | ima0 5919 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑈)) “ ∅) =
∅ |
146 | 144, 145 | eqtrdi 2789 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
147 | 143, 146 | eqtr3d 2775 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) |
148 | | fnun 6449 |
. . . . . . . . . . . 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)))) |
149 | 141, 147,
148 | sylancr 590 |
. . . . . . . . . . 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)))) |
150 | | imaundi 5982 |
. . . . . . . . . . . . 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))) |
151 | 55 | imaeq2d 5903 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑈)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
152 | 151, 32 | eqtr3d 2775 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
153 | 150, 152 | eqtr3id 2787 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
154 | 153 | fneq2d 6432 |
. . . . . . . . . . 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))) |
155 | 149, 154 | mpbid 235 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
156 | 155 | adantr 484 |
. . . . . . . . 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)) |
157 | | ovexd 7205 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V) |
158 | | inidm 4109 |
. . . . . . . . 9
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
159 | | eqidd 2739 |
. . . . . . . . 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)) |
160 | | fvun2 6760 |
. . . . . . . . . . . . 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)) |
161 | 137, 140,
160 | mp3an12 1452 |
. . . . . . . . . . . 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)) |
162 | 147, 161 | sylan 583 |
. . . . . . . . . . 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)) |
163 | 138 | fvconst2 6976 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
164 | 163 | adantl 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
165 | 162, 164 | eqtrd 2773 |
. . . . . . . . . 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) |
166 | 165 | adantr 484 |
. . . . . . . . 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) |
167 | 134, 156,
157, 157, 158, 159, 166 | ofval 7435 |
. . . . . . . 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)) |
168 | 130, 167 | mpdan 687 |
. . . . . . 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)) |
169 | | elmapi 8459 |
. . . . . . . . . . . . 13
⊢
((1^{st} ‘(1^{st} ‘𝑈)) ∈ ((0..^𝐾) ↑_{m} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
170 | 23, 131, 169 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
171 | 170 | ffvelrnda 6861 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ (0..^𝐾)) |
172 | | elfzonn0 13173 |
. . . . . . . . . . 11
⊢
(((1^{st} ‘(1^{st} ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈
ℕ_{0}) |
173 | 171, 172 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈
ℕ_{0}) |
174 | 173 | nn0cnd 12038 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) ∈ ℂ) |
175 | 174 | addid1d 10918 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) + 0) = ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦)) |
176 | 130, 175 | syldan 594 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) + 0) = ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦)) |
177 | 124, 168,
176 | 3eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀)‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑈))‘𝑦)) |
178 | 73, 177 | syldan 594 |
. . . . 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)) |
179 | | fveq2 6674 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
180 | 179 | breq2d 5042 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
181 | 180 | ifbid 4437 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
182 | 181 | csbeq1d 3794 |
. . . . . . . . . . . . . . . 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})))) |
183 | | 2fveq3 6679 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
184 | | 2fveq3 6679 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
185 | 184 | imaeq1d 5902 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
186 | 185 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
187 | 184 | imaeq1d 5902 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
188 | 187 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
189 | 186, 188 | uneq12d 4054 |
. . . . . . . . . . . . . . . . . 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}))) |
190 | 183, 189 | oveq12d 7188 |
. . . . . . . . . . . . . . . . 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})))) |
191 | 190 | csbeq2dv 3797 |
. . . . . . . . . . . . . . . 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})))) |
192 | 182, 191 | eqtrd 2773 |
. . . . . . . . . . . . . . 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})))) |
193 | 192 | mpteq2dv 5126 |
. . . . . . . . . . . . . 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}))))) |
194 | 193 | eqeq2d 2749 |
. . . . . . . . . . . . 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})))))) |
195 | 194, 5 | elrab2 3591 |
. . . . . . . . . . . 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})))))) |
196 | 195 | simprbi 500 |
. . . . . . . . . . 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}))))) |
197 | 3, 196 | 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}))))) |
198 | | breq1 5033 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (𝑦 < (2^{nd} ‘𝑇) ↔ 𝑀 < (2^{nd} ‘𝑇))) |
199 | 198, 94 | ifbieq1d 4438 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2^{nd} ‘𝑇), 𝑀, (𝑦 + 1))) |
200 | | poimirlem12.3 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2^{nd}
‘𝑇) = 𝑁) |
201 | 102, 200 | breqtrrd 5058 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2^{nd} ‘𝑇)) |
202 | 201 | iftrued 4422 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑀 < (2^{nd} ‘𝑇), 𝑀, (𝑦 + 1)) = 𝑀) |
203 | 199, 202 | sylan9eqr 2795 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
204 | 203 | csbeq1d 3794 |
. . . . . . . . . . 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})))) |
205 | 108 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑀))) |
206 | 205 | xpeq1d 5554 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})) |
207 | 112 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
208 | 207 | xpeq1d 5554 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
209 | 206, 208 | uneq12d 4054 |
. . . . . . . . . . . . . . 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}))) |
210 | 209 | oveq2d 7186 |
. . . . . . . . . . . . . 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})))) |
211 | 210 | adantl 485 |
. . . . . . . . . . . . 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})))) |
212 | 40, 211 | csbied 3826 |
. . . . . . . . . . . 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})))) |
213 | 212 | adantr 484 |
. . . . . . . . . . 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})))) |
214 | 204, 213 | eqtrd 2773 |
. . . . . . . . . 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})))) |
215 | | ovexd 7205 |
. . . . . . . . . 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) |
216 | 197, 214,
40, 215 | fvmptd 6782 |
. . . . . . . . 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})))) |
217 | 216 | fveq1d 6676 |
. . . . . . . 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)) |
218 | 217 | adantr 484 |
. . . . . . 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)) |
219 | 18 | sselda 3877 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁)) |
220 | | xp1st 7746 |
. . . . . . . . . . 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))) |
221 | | elmapfn 8475 |
. . . . . . . . . . 11
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{m} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
222 | 8, 220, 221 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
223 | 222 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (1^{st}
‘(1^{st} ‘𝑇)) Fn (1...𝑁)) |
224 | | fnconstg 6566 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) |
225 | 135, 224 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) |
226 | | fnconstg 6566 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
227 | 138, 226 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
228 | 225, 227 | pm3.2i 474 |
. . . . . . . . . . . 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))) |
229 | | dff1o3 6624 |
. . . . . . . . . . . . . . 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)))) |
230 | 229 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^{◡}(2^{nd} ‘(1^{st}
‘𝑇))) |
231 | | imain 6424 |
. . . . . . . . . . . . . 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)))) |
232 | 14, 230, 231 | 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)))) |
233 | 64 | imaeq2d 5903 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ∅)) |
234 | | ima0 5919 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) “ ∅) =
∅ |
235 | 233, 234 | eqtrdi 2789 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
236 | 232, 235 | eqtr3d 2775 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∩ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
237 | | fnun 6449 |
. . . . . . . . . . . 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)))) |
238 | 228, 236,
237 | sylancr 590 |
. . . . . . . . . . 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)))) |
239 | | imaundi 5982 |
. . . . . . . . . . . . 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))) |
240 | 55 | imaeq2d 5903 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
241 | | f1ofo 6625 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
242 | | foima 6597 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
243 | 14, 241, 242 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
244 | 240, 243 | eqtr3d 2775 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
245 | 239, 244 | eqtr3id 2787 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∪ ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
246 | 245 | fneq2d 6432 |
. . . . . . . . . . 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))) |
247 | 238, 246 | mpbid 235 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
248 | 247 | adantr 484 |
. . . . . . . . 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)) |
249 | | ovexd 7205 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V) |
250 | | eqidd 2739 |
. . . . . . . . 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)) |
251 | | fvun1 6759 |
. . . . . . . . . . . . 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)) |
252 | 225, 227,
251 | mp3an12 1452 |
. . . . . . . . . . . 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)) |
253 | 236, 252 | sylan 583 |
. . . . . . . . . . 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)) |
254 | 135 | fvconst2 6976 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
255 | 254 | adantl 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
256 | 253, 255 | eqtrd 2773 |
. . . . . . . . . 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) |
257 | 256 | adantr 484 |
. . . . . . . . 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) |
258 | 223, 248,
249, 249, 158, 250, 257 | ofval 7435 |
. . . . . . . 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)) |
259 | 219, 258 | mpdan 687 |
. . . . . . 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)) |
260 | 218, 259 | eqtrd 2773 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ((𝐹‘𝑀)‘𝑦) = (((1^{st} ‘(1^{st}
‘𝑇))‘𝑦) + 1)) |
261 | 260 | adantrr 717 |
. . . . 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)) |
262 | 46 | nngt0d 11765 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝑁) |
263 | 262, 103 | breqtrrd 5058 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (2^{nd}
‘𝑈)) |
264 | 46, 5, 19, 263 | poimirlem5 35405 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘0) = (1^{st}
‘(1^{st} ‘𝑈))) |
265 | 262, 200 | breqtrrd 5058 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (2^{nd}
‘𝑇)) |
266 | 46, 5, 3, 265 | poimirlem5 35405 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘0) = (1^{st}
‘(1^{st} ‘𝑇))) |
267 | 264, 266 | eqtr3d 2775 |
. . . . . . 7
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑈)) = (1^{st} ‘(1^{st}
‘𝑇))) |
268 | 267 | fveq1d 6676 |
. . . . . 6
⊢ (𝜑 → ((1^{st}
‘(1^{st} ‘𝑈))‘𝑦) = ((1^{st} ‘(1^{st}
‘𝑇))‘𝑦)) |
269 | 268 | adantr 484 |
. . . . 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)) |
270 | 178, 261,
269 | 3eqtr3d 2781 |
. . . 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)) |
271 | | elmapi 8459 |
. . . . . . . . . . . 12
⊢
((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{m} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
272 | 8, 220, 271 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
273 | 272 | ffvelrnda 6861 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈ (0..^𝐾)) |
274 | | elfzonn0 13173 |
. . . . . . . . . 10
⊢
(((1^{st} ‘(1^{st} ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈
ℕ_{0}) |
275 | 273, 274 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈
ℕ_{0}) |
276 | 275 | nn0red 12037 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) ∈ ℝ) |
277 | 276 | ltp1d 11648 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) < (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1)) |
278 | 276, 277 | gtned 10853 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) ≠ ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
279 | 219, 278 | syldan 594 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) ≠ ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
280 | 279 | neneqd 2939 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀))) → ¬ (((1^{st}
‘(1^{st} ‘𝑇))‘𝑦) + 1) = ((1^{st}
‘(1^{st} ‘𝑇))‘𝑦)) |
281 | 280 | adantrr 717 |
. . . 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)) |
282 | 270, 281 | pm2.65da 817 |
. . 3
⊢ (𝜑 → ¬ (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
283 | | iman 405 |
. . 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)))) |
284 | 282, 283 | sylibr 237 |
. 2
⊢ (𝜑 → (𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀)))) |
285 | 284 | ssrdv 3883 |
1
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑀)) ⊆ ((2^{nd}
‘(1^{st} ‘𝑈)) “ (1...𝑀))) |