Proof of Theorem poimirlem12
| Step | Hyp | Ref
| Expression |
| 1 | | eldif 3961 |
. . . . . . 7
⊢ (𝑦 ∈ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ↔ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 2 | | imassrn 6089 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ran (2nd
‘(1st ‘𝑇)) |
| 3 | | poimirlem12.2 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
| 4 | | elrabi 3687 |
. . . . . . . . . . . . . . . . 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 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 5 | | poimirlem22.s |
. . . . . . . . . . . . . . . . 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 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
| 6 | 4, 5 | eleq2s 2859 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 7 | | xp1st 8046 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 8 | 3, 6, 7 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1st
‘𝑇) ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 9 | | xp2nd 8047 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 10 | 8, 9 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 11 | | fvex 6919 |
. . . . . . . . . . . . . . 15
⊢
(2nd ‘(1st ‘𝑇)) ∈ V |
| 12 | | f1oeq1 6836 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (2nd
‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
| 13 | 11, 12 | elab 3679 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 14 | 10, 13 | sylib 218 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2nd
‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 15 | | f1of 6848 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁)) |
| 16 | | frn 6743 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)⟶(1...𝑁) → ran (2nd
‘(1st ‘𝑇)) ⊆ (1...𝑁)) |
| 17 | 14, 15, 16 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (2nd
‘(1st ‘𝑇)) ⊆ (1...𝑁)) |
| 18 | 2, 17 | sstrid 3995 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁)) |
| 19 | | poimirlem12.4 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
| 20 | | elrabi 3687 |
. . . . . . . . . . . . . . . 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 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 21 | 20, 5 | eleq2s 2859 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 22 | | xp1st 8046 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 23 | 19, 21, 22 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1st
‘𝑈) ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 24 | | xp2nd 8047 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 25 | 23, 24 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 26 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘(1st ‘𝑈)) ∈ V |
| 27 | | f1oeq1 6836 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = (2nd
‘(1st ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))) |
| 28 | 26, 27 | elab 3679 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 29 | 25, 28 | sylib 218 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2nd
‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 30 | | f1ofo 6855 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁)) |
| 31 | | foima 6825 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
| 32 | 29, 30, 31 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = (1...𝑁)) |
| 33 | 18, 32 | sseqtrrd 4021 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd
‘(1st ‘𝑈)) “ (1...𝑁))) |
| 34 | 33 | ssdifd 4145 |
. . . . . . . . 9
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ⊆ (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 35 | | dff1o3 6854 |
. . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑈)))) |
| 36 | 35 | simprbi 496 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑈))) |
| 37 | | imadif 6650 |
. . . . . . . . . . 11
⊢ (Fun
◡(2nd ‘(1st
‘𝑈)) →
((2nd ‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 38 | 29, 36, 37 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 39 | | difun2 4481 |
. . . . . . . . . . . 12
⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)) |
| 40 | | poimirlem12.6 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1))) |
| 41 | | elfznn0 13660 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → 𝑀 ∈
ℕ0) |
| 42 | | nn0p1nn 12565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ0
→ (𝑀 + 1) ∈
ℕ) |
| 43 | 40, 41, 42 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 + 1) ∈ ℕ) |
| 44 | | nnuz 12921 |
. . . . . . . . . . . . . . . 16
⊢ ℕ =
(ℤ≥‘1) |
| 45 | 43, 44 | eleqtrdi 2851 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 + 1) ∈
(ℤ≥‘1)) |
| 46 | | poimir.0 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 47 | 46 | nncnd 12282 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 48 | | npcan1 11688 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
| 49 | 47, 48 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
| 50 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
| 51 | | peano2uz 12943 |
. . . . . . . . . . . . . . . . 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 2842 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 54 | | fzsplit2 13589 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
| 55 | 45, 53, 54 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) |
| 56 | | uncom 4158 |
. . . . . . . . . . . . . 14
⊢
((1...𝑀) ∪
((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)) |
| 57 | 55, 56 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))) |
| 58 | 57 | difeq1d 4125 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀))) |
| 59 | | incom 4209 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) |
| 60 | 40, 41 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 61 | 60 | nn0red 12588 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 62 | 61 | ltp1d 12198 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
| 63 | | fzdisj 13591 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
| 64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅) |
| 65 | 59, 64 | eqtrid 2789 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅) |
| 66 | | disj3 4454 |
. . . . . . . . . . . . 13
⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
| 67 | 65, 66 | sylib 218 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))) |
| 68 | 39, 58, 67 | 3eqtr4a 2803 |
. . . . . . . . . . 11
⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁)) |
| 69 | 68 | imaeq2d 6078 |
. . . . . . . . . 10
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 70 | 38, 69 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) = ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 71 | 34, 70 | sseqtrd 4020 |
. . . . . . . 8
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ⊆ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 72 | 71 | sselda 3983 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∖ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 73 | 1, 72 | sylan2br 595 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 74 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (2nd ‘𝑡) = (2nd ‘𝑈)) |
| 75 | 74 | breq2d 5155 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑈))) |
| 76 | 75 | ifbid 4549 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1))) |
| 77 | 76 | csbeq1d 3903 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 78 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑈))) |
| 79 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑈 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑈))) |
| 80 | 79 | imaeq1d 6077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑈)) “
(1...𝑗))) |
| 81 | 80 | xpeq1d 5714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1})) |
| 82 | 79 | imaeq1d 6077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑈 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑈)) “ ((𝑗 + 1)...𝑁))) |
| 83 | 82 | xpeq1d 5714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑈 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) |
| 84 | 81, 83 | uneq12d 4169 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑈 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
| 85 | 78, 84 | oveq12d 7449 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑈 → ((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 86 | 85 | csbeq2dv 3906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 87 | 77, 86 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 88 | 87 | mpteq2dv 5244 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 89 | 88 | eqeq2d 2748 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
| 90 | 89, 5 | elrab2 3695 |
. . . . . . . . . . . 12
⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
| 91 | 90 | simprbi 496 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 92 | 19, 91 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 93 | | breq1 5146 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (𝑦 < (2nd ‘𝑈) ↔ 𝑀 < (2nd ‘𝑈))) |
| 94 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → 𝑦 = 𝑀) |
| 95 | 93, 94 | ifbieq1d 4550 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) = if(𝑀 < (2nd ‘𝑈), 𝑀, (𝑦 + 1))) |
| 96 | 46 | nnred 12281 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 97 | | peano2rem 11576 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈
ℝ) |
| 98 | 96, 97 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
| 99 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ (0...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) |
| 100 | 40, 99 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) |
| 101 | 96 | ltm1d 12200 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
| 102 | 61, 98, 96, 100, 101 | lelttrd 11419 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 < 𝑁) |
| 103 | | poimirlem12.5 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘𝑈) = 𝑁) |
| 104 | 102, 103 | breqtrrd 5171 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2nd ‘𝑈)) |
| 105 | 104 | iftrued 4533 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑀 < (2nd ‘𝑈), 𝑀, (𝑦 + 1)) = 𝑀) |
| 106 | 95, 105 | sylan9eqr 2799 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2nd ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀) |
| 107 | 106 | csbeq1d 3903 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 108 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀)) |
| 109 | 108 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑈)) “
(1...𝑀))) |
| 110 | 109 | xpeq1d 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1})) |
| 111 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1)) |
| 112 | 111 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁)) |
| 113 | 112 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 114 | 113 | xpeq1d 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) |
| 115 | 110, 114 | uneq12d 4169 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 116 | 115 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 117 | 116 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 118 | 40, 117 | csbied 3935 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 119 | 118 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 120 | 107, 119 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2nd
‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 121 | | ovexd 7466 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
| 122 | 92, 120, 40, 121 | fvmptd 7023 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑀) = ((1st ‘(1st
‘𝑈))
∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 123 | 122 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑀)‘𝑦) = (((1st ‘(1st
‘𝑈))
∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
| 124 | 123 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀)‘𝑦) = (((1st ‘(1st
‘𝑈))
∘f + ((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
| 125 | | imassrn 6089 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2nd
‘(1st ‘𝑈)) |
| 126 | | f1of 6848 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁)) |
| 127 | | frn 6743 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑈)):(1...𝑁)⟶(1...𝑁) → ran (2nd
‘(1st ‘𝑈)) ⊆ (1...𝑁)) |
| 128 | 29, 126, 127 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ran (2nd
‘(1st ‘𝑈)) ⊆ (1...𝑁)) |
| 129 | 125, 128 | sstrid 3995 |
. . . . . . . . 9
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁)) |
| 130 | 129 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁)) |
| 131 | | xp1st 8046 |
. . . . . . . . . . 11
⊢
((1st ‘𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
| 132 | | elmapfn 8905 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
| 133 | 23, 131, 132 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
| 134 | 133 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1st
‘(1st ‘𝑈)) Fn (1...𝑁)) |
| 135 | | 1ex 11257 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
| 136 | | fnconstg 6796 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) |
| 137 | 135, 136 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) |
| 138 | | c0ex 11255 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
| 139 | | fnconstg 6796 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 140 | 138, 139 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) |
| 141 | 137, 140 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢
((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 142 | | imain 6651 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡(2nd ‘(1st
‘𝑈)) →
((2nd ‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
| 143 | 29, 36, 142 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
| 144 | 64 | imaeq2d 6078 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑈)) “ ∅)) |
| 145 | | ima0 6095 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑈)) “ ∅) =
∅ |
| 146 | 144, 145 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
| 147 | 143, 146 | eqtr3d 2779 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) |
| 148 | | fnun 6682 |
. . . . . . . . . . . 12
⊢
((((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
| 149 | 141, 147,
148 | sylancr 587 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) |
| 150 | | imaundi 6169 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) |
| 151 | 55 | imaeq2d 6078 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑈)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
| 152 | 151, 32 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
| 153 | 150, 152 | eqtr3id 2791 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
| 154 | 153 | fneq2d 6662 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
| 155 | 149, 154 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
| 156 | 155 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
| 157 | | ovexd 7466 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V) |
| 158 | | inidm 4227 |
. . . . . . . . 9
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
| 159 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
| 160 | | fvun2 7001 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
| 161 | 137, 140,
160 | mp3an12 1453 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑈)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
| 162 | 147, 161 | sylan 580 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦)) |
| 163 | 138 | fvconst2 7224 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
| 164 | 163 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0) |
| 165 | 162, 164 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
| 166 | 165 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0) |
| 167 | 134, 156,
157, 157, 158, 159, 166 | ofval 7708 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑈))‘𝑦) + 0)) |
| 168 | 130, 167 | mpdan 687 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st
‘(1st ‘𝑈)) ∘f + ((((2nd
‘(1st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑈))‘𝑦) + 0)) |
| 169 | | elmapi 8889 |
. . . . . . . . . . . . 13
⊢
((1st ‘(1st ‘𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
| 170 | 23, 131, 169 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st
‘(1st ‘𝑈)):(1...𝑁)⟶(0..^𝐾)) |
| 171 | 170 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ (0..^𝐾)) |
| 172 | | elfzonn0 13747 |
. . . . . . . . . . 11
⊢
(((1st ‘(1st ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈
ℕ0) |
| 173 | 171, 172 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈
ℕ0) |
| 174 | 173 | nn0cnd 12589 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑈))‘𝑦) ∈ ℂ) |
| 175 | 174 | addridd 11461 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑈))‘𝑦) + 0) = ((1st
‘(1st ‘𝑈))‘𝑦)) |
| 176 | 130, 175 | syldan 591 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st
‘(1st ‘𝑈))‘𝑦) + 0) = ((1st
‘(1st ‘𝑈))‘𝑦)) |
| 177 | 124, 168,
176 | 3eqtrd 2781 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘𝑀)‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
| 178 | 73, 177 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘𝑀)‘𝑦) = ((1st ‘(1st
‘𝑈))‘𝑦)) |
| 179 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇)) |
| 180 | 179 | breq2d 5155 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇))) |
| 181 | 180 | ifbid 4549 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1))) |
| 182 | 181 | csbeq1d 3903 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 183 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑇))) |
| 184 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑇 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑇))) |
| 185 | 184 | imaeq1d 6077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑗))) |
| 186 | 185 | xpeq1d 5714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1})) |
| 187 | 184 | imaeq1d 6077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑇 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
| 188 | 187 | xpeq1d 5714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑇 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
| 189 | 186, 188 | uneq12d 4169 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑇 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
| 190 | 183, 189 | oveq12d 7449 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑇 → ((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 191 | 190 | csbeq2dv 3906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 192 | 182, 191 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 193 | 192 | mpteq2dv 5244 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 194 | 193 | eqeq2d 2748 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘f + ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
| 195 | 194, 5 | elrab2 3695 |
. . . . . . . . . . . 12
⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
| 196 | 195 | simprbi 496 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 197 | 3, 196 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
| 198 | | breq1 5146 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → (𝑦 < (2nd ‘𝑇) ↔ 𝑀 < (2nd ‘𝑇))) |
| 199 | 198, 94 | ifbieq1d 4550 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑀 → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = if(𝑀 < (2nd ‘𝑇), 𝑀, (𝑦 + 1))) |
| 200 | | poimirlem12.3 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd
‘𝑇) = 𝑁) |
| 201 | 102, 200 | breqtrrd 5171 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 < (2nd ‘𝑇)) |
| 202 | 201 | iftrued 4533 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if(𝑀 < (2nd ‘𝑇), 𝑀, (𝑦 + 1)) = 𝑀) |
| 203 | 199, 202 | sylan9eqr 2799 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀) |
| 204 | 203 | csbeq1d 3903 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
| 205 | 108 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑇)) “
(1...𝑀))) |
| 206 | 205 | xpeq1d 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})) |
| 207 | 112 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑀 → ((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 208 | 207 | xpeq1d 5714 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑀 → (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) |
| 209 | 206, 208 | uneq12d 4169 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑀 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) |
| 210 | 209 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑀 → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 211 | 210 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 212 | 40, 211 | csbied 3935 |
. . . . . . . . . . . 12
⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 213 | 212 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋𝑀 / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 214 | 204, 213 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < (2nd
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 215 | | ovexd 7466 |
. . . . . . . . . 10
⊢ (𝜑 → ((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V) |
| 216 | 197, 214,
40, 215 | fvmptd 7023 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑀) = ((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))) |
| 217 | 216 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑀)‘𝑦) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
| 218 | 217 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘𝑀)‘𝑦) = (((1st ‘(1st
‘𝑇))
∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦)) |
| 219 | 18 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁)) |
| 220 | | xp1st 8046 |
. . . . . . . . . . 11
⊢
((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁))) |
| 221 | | elmapfn 8905 |
. . . . . . . . . . 11
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 222 | 8, 220, 221 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 223 | 222 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (1st
‘(1st ‘𝑇)) Fn (1...𝑁)) |
| 224 | | fnconstg 6796 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
V → (((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) |
| 225 | 135, 224 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) |
| 226 | | fnconstg 6796 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
V → (((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 227 | 138, 226 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) |
| 228 | 225, 227 | pm3.2i 470 |
. . . . . . . . . . . 12
⊢
((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 229 | | dff1o3 6854 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑇)))) |
| 230 | 229 | simprbi 496 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑇))) |
| 231 | | imain 6651 |
. . . . . . . . . . . . . 14
⊢ (Fun
◡(2nd ‘(1st
‘𝑇)) →
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 232 | 14, 230, 231 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 233 | 64 | imaeq2d 6078 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd
‘(1st ‘𝑇)) “ ∅)) |
| 234 | | ima0 6095 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑇)) “ ∅) =
∅ |
| 235 | 233, 234 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅) |
| 236 | 232, 235 | eqtr3d 2779 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) |
| 237 | | fnun 6682 |
. . . . . . . . . . . 12
⊢
((((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 238 | 228, 236,
237 | sylancr 587 |
. . . . . . . . . . 11
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))) |
| 239 | | imaundi 6169 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) |
| 240 | 55 | imaeq2d 6078 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = ((2nd ‘(1st
‘𝑇)) “
((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))) |
| 241 | | f1ofo 6855 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁)) |
| 242 | | foima 6825 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
| 243 | 14, 241, 242 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑁)) = (1...𝑁)) |
| 244 | 240, 243 | eqtr3d 2779 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
| 245 | 239, 244 | eqtr3id 2791 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁)) |
| 246 | 245 | fneq2d 6662 |
. . . . . . . . . . 11
⊢ (𝜑 → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∪ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
| 247 | 238, 246 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
| 248 | 247 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
| 249 | | ovexd 7466 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V) |
| 250 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
| 251 | | fvun1 7000 |
. . . . . . . . . . . . 13
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
| 252 | 225, 227,
251 | mp3an12 1453 |
. . . . . . . . . . . 12
⊢
(((((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ∩ ((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
| 253 | 236, 252 | sylan 580 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st
‘𝑇)) “
(1...𝑀)) ×
{1})‘𝑦)) |
| 254 | 135 | fvconst2 7224 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
| 255 | 254 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1) |
| 256 | 253, 255 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
| 257 | 256 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1) |
| 258 | 223, 248,
249, 249, 158, 250, 257 | ofval 7708 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
| 259 | 219, 258 | mpdan 687 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((1st
‘(1st ‘𝑇)) ∘f + ((((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd
‘(1st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
| 260 | 218, 259 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘𝑀)‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
| 261 | 260 | adantrr 717 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘𝑀)‘𝑦) = (((1st ‘(1st
‘𝑇))‘𝑦) + 1)) |
| 262 | 46 | nngt0d 12315 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝑁) |
| 263 | 262, 103 | breqtrrd 5171 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (2nd
‘𝑈)) |
| 264 | 46, 5, 19, 263 | poimirlem5 37632 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘0) = (1st
‘(1st ‘𝑈))) |
| 265 | 262, 200 | breqtrrd 5171 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (2nd
‘𝑇)) |
| 266 | 46, 5, 3, 265 | poimirlem5 37632 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘0) = (1st
‘(1st ‘𝑇))) |
| 267 | 264, 266 | eqtr3d 2779 |
. . . . . . 7
⊢ (𝜑 → (1st
‘(1st ‘𝑈)) = (1st ‘(1st
‘𝑇))) |
| 268 | 267 | fveq1d 6908 |
. . . . . 6
⊢ (𝜑 → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
| 269 | 268 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ((1st
‘(1st ‘𝑈))‘𝑦) = ((1st ‘(1st
‘𝑇))‘𝑦)) |
| 270 | 178, 261,
269 | 3eqtr3d 2785 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
| 271 | | elmapi 8889 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
| 272 | 8, 220, 271 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (1st
‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾)) |
| 273 | 272 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈ (0..^𝐾)) |
| 274 | | elfzonn0 13747 |
. . . . . . . . . 10
⊢
(((1st ‘(1st ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈
ℕ0) |
| 275 | 273, 274 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈
ℕ0) |
| 276 | 275 | nn0red 12588 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) ∈ ℝ) |
| 277 | 276 | ltp1d 12198 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1st
‘(1st ‘𝑇))‘𝑦) < (((1st
‘(1st ‘𝑇))‘𝑦) + 1)) |
| 278 | 276, 277 | gtned 11396 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) ≠ ((1st
‘(1st ‘𝑇))‘𝑦)) |
| 279 | 219, 278 | syldan 591 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → (((1st
‘(1st ‘𝑇))‘𝑦) + 1) ≠ ((1st
‘(1st ‘𝑇))‘𝑦)) |
| 280 | 279 | neneqd 2945 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀))) → ¬ (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
| 281 | 280 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) → ¬ (((1st
‘(1st ‘𝑇))‘𝑦) + 1) = ((1st
‘(1st ‘𝑇))‘𝑦)) |
| 282 | 270, 281 | pm2.65da 817 |
. . 3
⊢ (𝜑 → ¬ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 283 | | iman 401 |
. . 3
⊢ ((𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) ↔ ¬ (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 284 | 282, 283 | sylibr 234 |
. 2
⊢ (𝜑 → (𝑦 ∈ ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀)))) |
| 285 | 284 | ssrdv 3989 |
1
⊢ (𝜑 → ((2nd
‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd
‘(1st ‘𝑈)) “ (1...𝑀))) |