| Step | Hyp | Ref
| Expression |
| 1 | | poimir.0 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 2 | 1 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑁 ∈ ℕ) |
| 3 | | poimirlem22.s |
. . . . . . . 8
⊢ 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}))))} |
| 4 | | simplrl 777 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑧 ∈ 𝑆) |
| 5 | 1 | nngt0d 12315 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝑁) |
| 6 | | breq2 5147 |
. . . . . . . . . . 11
⊢
((2nd ‘𝑧) = 𝑁 → (0 < (2nd ‘𝑧) ↔ 0 < 𝑁)) |
| 7 | 6 | biimparc 479 |
. . . . . . . . . 10
⊢ ((0 <
𝑁 ∧ (2nd
‘𝑧) = 𝑁) → 0 < (2nd
‘𝑧)) |
| 8 | 5, 7 | sylan 580 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (2nd
‘𝑧) = 𝑁) → 0 < (2nd
‘𝑧)) |
| 9 | 8 | ad2ant2r 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 0 < (2nd
‘𝑧)) |
| 10 | 2, 3, 4, 9 | poimirlem5 37632 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (𝐹‘0) = (1st
‘(1st ‘𝑧))) |
| 11 | | simplrr 778 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑘 ∈ 𝑆) |
| 12 | | breq2 5147 |
. . . . . . . . . . 11
⊢
((2nd ‘𝑘) = 𝑁 → (0 < (2nd ‘𝑘) ↔ 0 < 𝑁)) |
| 13 | 12 | biimparc 479 |
. . . . . . . . . 10
⊢ ((0 <
𝑁 ∧ (2nd
‘𝑘) = 𝑁) → 0 < (2nd
‘𝑘)) |
| 14 | 5, 13 | sylan 580 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (2nd
‘𝑘) = 𝑁) → 0 < (2nd
‘𝑘)) |
| 15 | 14 | ad2ant2rl 749 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 0 < (2nd
‘𝑘)) |
| 16 | 2, 3, 11, 15 | poimirlem5 37632 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (𝐹‘0) = (1st
‘(1st ‘𝑘))) |
| 17 | 10, 16 | eqtr3d 2779 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (1st
‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘))) |
| 18 | | elrabi 3687 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ {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}))))} → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 19 | 18, 3 | eleq2s 2859 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 20 | | xp1st 8046 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 21 | | xp2nd 8047 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 22 | 19, 20, 21 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝑆 → (2nd
‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 23 | | fvex 6919 |
. . . . . . . . . . . 12
⊢
(2nd ‘(1st ‘𝑧)) ∈ V |
| 24 | | f1oeq1 6836 |
. . . . . . . . . . . 12
⊢ (𝑓 = (2nd
‘(1st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))) |
| 25 | 23, 24 | elab 3679 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 26 | 22, 25 | sylib 218 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑆 → (2nd
‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 27 | | f1ofn 6849 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑆 → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) |
| 29 | 28 | adantr 480 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) |
| 30 | 29 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) |
| 31 | | elrabi 3687 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ {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}))))} → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 32 | 31, 3 | eleq2s 2859 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
| 33 | | xp1st 8046 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 34 | | xp2nd 8047 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 35 | 32, 33, 34 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝑆 → (2nd
‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
| 36 | | fvex 6919 |
. . . . . . . . . . . 12
⊢
(2nd ‘(1st ‘𝑘)) ∈ V |
| 37 | | f1oeq1 6836 |
. . . . . . . . . . . 12
⊢ (𝑓 = (2nd
‘(1st ‘𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))) |
| 38 | 36, 37 | elab 3679 |
. . . . . . . . . . 11
⊢
((2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 39 | 35, 38 | sylib 218 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝑆 → (2nd
‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)) |
| 40 | | f1ofn 6849 |
. . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) |
| 41 | 39, 40 | syl 17 |
. . . . . . . . 9
⊢ (𝑘 ∈ 𝑆 → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) |
| 42 | 41 | adantl 481 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) |
| 43 | 42 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) |
| 44 | | simpllr 776 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) |
| 45 | | oveq2 7439 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) |
| 46 | 45 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑁 → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑧)) “
(1...𝑁))) |
| 47 | | f1ofo 6855 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁)) |
| 48 | | foima 6825 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑁)) = (1...𝑁)) |
| 49 | 26, 47, 48 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝑆 → ((2nd
‘(1st ‘𝑧)) “ (1...𝑁)) = (1...𝑁)) |
| 50 | 46, 49 | sylan9eqr 2799 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 = 𝑁) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = (1...𝑁)) |
| 51 | 50 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ 𝑛 = 𝑁) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = (1...𝑁)) |
| 52 | 45 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑁 → ((2nd
‘(1st ‘𝑘)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑁))) |
| 53 | | f1ofo 6855 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑘)):(1...𝑁)–onto→(1...𝑁)) |
| 54 | | foima 6825 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘(1st ‘𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑁)) = (1...𝑁)) |
| 55 | 39, 53, 54 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝑆 → ((2nd
‘(1st ‘𝑘)) “ (1...𝑁)) = (1...𝑁)) |
| 56 | 52, 55 | sylan9eqr 2799 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 = 𝑁) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑛)) = (1...𝑁)) |
| 57 | 56 | adantll 714 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ 𝑛 = 𝑁) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑛)) = (1...𝑁)) |
| 58 | 51, 57 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ 𝑛 = 𝑁) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 59 | 44, 58 | sylan 580 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = 𝑁) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 60 | | simpll 767 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝜑) |
| 61 | | elnnuz 12922 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈
(ℤ≥‘1)) |
| 62 | 1, 61 | sylib 218 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
| 63 | | fzm1 13647 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈
(ℤ≥‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
| 64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
| 65 | 64 | anbi1d 631 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁))) |
| 66 | 65 | biimpa 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁)) |
| 67 | | df-ne 2941 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ≠ 𝑁 ↔ ¬ 𝑛 = 𝑁) |
| 68 | 67 | anbi2i 623 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁)) |
| 69 | | pm5.61 1003 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁)) |
| 70 | 68, 69 | bitri 275 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁)) |
| 71 | 66, 70 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁)) |
| 72 | | fz1ssfz0 13663 |
. . . . . . . . . . . . . . . . 17
⊢
(1...(𝑁 − 1))
⊆ (0...(𝑁 −
1)) |
| 73 | 72 | sseli 3979 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ (0...(𝑁 − 1))) |
| 74 | 73 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁) → 𝑛 ∈ (0...(𝑁 − 1))) |
| 75 | 71, 74 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → 𝑛 ∈ (0...(𝑁 − 1))) |
| 76 | 60, 75 | sylan 580 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → 𝑛 ∈ (0...(𝑁 − 1))) |
| 77 | | eleq1 2829 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑛 → (𝑚 ∈ (0...(𝑁 − 1)) ↔ 𝑛 ∈ (0...(𝑁 − 1)))) |
| 78 | 77 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))))) |
| 79 | | oveq2 7439 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛)) |
| 80 | 79 | imaeq2d 6078 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑛 → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑧)) “
(1...𝑛))) |
| 81 | 79 | imaeq2d 6078 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑛 → ((2nd
‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 82 | 80, 81 | eqeq12d 2753 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑛 → (((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚)) ↔
((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛)))) |
| 83 | 78, 82 | imbi12d 344 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))))) |
| 84 | 1 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ) |
| 85 | | poimirlem22.1 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) |
| 86 | 85 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) |
| 87 | | simpl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆) |
| 88 | 87 | ad3antlr 731 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑧 ∈ 𝑆) |
| 89 | | simplrl 777 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑧) = 𝑁) |
| 90 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆) |
| 91 | 90 | ad3antlr 731 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ 𝑆) |
| 92 | | simplrr 778 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd
‘𝑘) = 𝑁) |
| 93 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...(𝑁 − 1))) |
| 94 | 84, 3, 86, 88, 89, 91, 92, 93 | poimirlem12 37639 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) ⊆ ((2nd
‘(1st ‘𝑘)) “ (1...𝑚))) |
| 95 | 84, 3, 86, 91, 92, 88, 89, 93 | poimirlem12 37639 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑚)) ⊆ ((2nd
‘(1st ‘𝑧)) “ (1...𝑚))) |
| 96 | 94, 95 | eqssd 4001 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚))) |
| 97 | 83, 96 | chvarvv 1998 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 98 | 76, 97 | syldan 591 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 99 | 98 | anassrs 467 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ 𝑁) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 100 | 59, 99 | pm2.61dane 3029 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 101 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁)) |
| 102 | | elfzelz 13564 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ) |
| 103 | 1 | nnzd 12640 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 104 | | elfzm1b 13642 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1)))) |
| 105 | 102, 103,
104 | syl2anr 597 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1)))) |
| 106 | 101, 105 | mpbid 232 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1))) |
| 107 | 60, 106 | sylan 580 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1))) |
| 108 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (𝑛 − 1) ∈
V |
| 109 | | eleq1 2829 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (0...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1)))) |
| 110 | 109 | anbi2d 630 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))))) |
| 111 | | oveq2 7439 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1))) |
| 112 | 111 | imaeq2d 6078 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑛 − 1) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑧)) “
(1...(𝑛 −
1)))) |
| 113 | 111 | imaeq2d 6078 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑛 − 1) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...(𝑛 −
1)))) |
| 114 | 112, 113 | eqeq12d 2753 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑛 − 1) → (((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚)) ↔
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) |
| 115 | 110, 114 | imbi12d 344 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))) |
| 116 | 108, 115,
96 | vtocl 3558 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) |
| 117 | 107, 116 | syldan 591 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) |
| 118 | 100, 117 | difeq12d 4127 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))) = (((2nd
‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) |
| 119 | | fnsnfv 6988 |
. . . . . . . . . . . 12
⊢
(((2nd ‘(1st ‘𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st
‘𝑧)) “ {𝑛})) |
| 120 | 28, 119 | sylan 580 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st
‘𝑧)) “ {𝑛})) |
| 121 | | elfznn 13593 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ) |
| 122 | | uncom 4158 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑛 − 1))
∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1))) |
| 123 | 122 | difeq1i 4122 |
. . . . . . . . . . . . . . . 16
⊢
(((1...(𝑛 −
1)) ∪ {𝑛}) ∖
(1...(𝑛 − 1))) =
(({𝑛} ∪ (1...(𝑛 − 1))) ∖
(1...(𝑛 −
1))) |
| 124 | | difun2 4481 |
. . . . . . . . . . . . . . . 16
⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1))) |
| 125 | 123, 124 | eqtri 2765 |
. . . . . . . . . . . . . . 15
⊢
(((1...(𝑛 −
1)) ∪ {𝑛}) ∖
(1...(𝑛 − 1))) =
({𝑛} ∖ (1...(𝑛 − 1))) |
| 126 | | nncn 12274 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
| 127 | | npcan1 11688 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛) |
| 128 | 126, 127 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛) |
| 129 | | elnnuz 12922 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈
(ℤ≥‘1)) |
| 130 | 129 | biimpi 216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
(ℤ≥‘1)) |
| 131 | 128, 130 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈
(ℤ≥‘1)) |
| 132 | | nnm1nn0 12567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
ℕ0) |
| 133 | 132 | nn0zd 12639 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
ℤ) |
| 134 | | uzid 12893 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 − 1) ∈ ℤ
→ (𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1))) |
| 135 | | peano2uz 12943 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈
(ℤ≥‘(𝑛 − 1))) |
| 136 | 133, 134,
135 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈
(ℤ≥‘(𝑛 − 1))) |
| 137 | 128, 136 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
(ℤ≥‘(𝑛 − 1))) |
| 138 | | fzsplit2 13589 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑛 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑛 ∈ (ℤ≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛))) |
| 139 | 131, 137,
138 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ →
(1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛))) |
| 140 | 128 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛)) |
| 141 | | nnz 12634 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
| 142 | | fzsn 13606 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛}) |
| 143 | 141, 142 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛}) |
| 144 | 140, 143 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛}) |
| 145 | 144 | uneq2d 4168 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ →
((1...(𝑛 − 1)) ∪
(((𝑛 − 1) +
1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛})) |
| 146 | 139, 145 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ →
(1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛})) |
| 147 | 146 | difeq1d 4125 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ →
((1...𝑛) ∖
(1...(𝑛 − 1))) =
(((1...(𝑛 − 1)) ∪
{𝑛}) ∖ (1...(𝑛 − 1)))) |
| 148 | | nnre 12273 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
| 149 | | ltm1 12109 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛) |
| 150 | | peano2rem 11576 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈
ℝ) |
| 151 | | ltnle 11340 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑛 − 1) ∈ ℝ ∧
𝑛 ∈ ℝ) →
((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1))) |
| 152 | 150, 151 | mpancom 688 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1))) |
| 153 | 149, 152 | mpbid 232 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℝ → ¬
𝑛 ≤ (𝑛 − 1)) |
| 154 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1)) |
| 155 | 153, 154 | nsyl 140 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℝ → ¬
𝑛 ∈ (1...(𝑛 − 1))) |
| 156 | 148, 155 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ¬
𝑛 ∈ (1...(𝑛 − 1))) |
| 157 | | incom 4209 |
. . . . . . . . . . . . . . . . . 18
⊢
((1...(𝑛 − 1))
∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1))) |
| 158 | 157 | eqeq1i 2742 |
. . . . . . . . . . . . . . . . 17
⊢
(((1...(𝑛 −
1)) ∩ {𝑛}) = ∅
↔ ({𝑛} ∩
(1...(𝑛 − 1))) =
∅) |
| 159 | | disjsn 4711 |
. . . . . . . . . . . . . . . . 17
⊢
(((1...(𝑛 −
1)) ∩ {𝑛}) = ∅
↔ ¬ 𝑛 ∈
(1...(𝑛 −
1))) |
| 160 | | disj3 4454 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1)))) |
| 161 | 158, 159,
160 | 3bitr3i 301 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1)))) |
| 162 | 156, 161 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1)))) |
| 163 | 125, 147,
162 | 3eqtr4a 2803 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ →
((1...𝑛) ∖
(1...(𝑛 − 1))) =
{𝑛}) |
| 164 | 121, 163 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (1...𝑁) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛}) |
| 165 | 164 | imaeq2d 6078 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (1...𝑁) → ((2nd
‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd
‘(1st ‘𝑧)) “ {𝑛})) |
| 166 | 165 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd
‘(1st ‘𝑧)) “ {𝑛})) |
| 167 | | dff1o3 6854 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑧)))) |
| 168 | 167 | simprbi 496 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑧))) |
| 169 | | imadif 6650 |
. . . . . . . . . . . . 13
⊢ (Fun
◡(2nd ‘(1st
‘𝑧)) →
((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) |
| 170 | 26, 168, 169 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝑆 → ((2nd
‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) |
| 171 | 170 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) |
| 172 | 120, 166,
171 | 3eqtr2d 2783 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st
‘𝑧)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) |
| 173 | 4, 172 | sylan 580 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st
‘𝑧)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) |
| 174 | | eleq1 2829 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆)) |
| 175 | 174 | anbi1d 631 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)))) |
| 176 | | 2fveq3 6911 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑘 → (2nd
‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) |
| 177 | 176 | fveq1d 6908 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑘 → ((2nd
‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st
‘𝑘))‘𝑛)) |
| 178 | 177 | sneqd 4638 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑘 → {((2nd
‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st
‘𝑘))‘𝑛)}) |
| 179 | 176 | imaeq1d 6077 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑘 → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) |
| 180 | 176 | imaeq1d 6077 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑘 → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) |
| 181 | 179, 180 | difeq12d 4127 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑘 → (((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))) = (((2nd
‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) |
| 182 | 178, 181 | eqeq12d 2753 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑘 → ({((2nd
‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st
‘𝑧)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))) ↔ {((2nd
‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st
‘𝑘)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))) |
| 183 | 175, 182 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑘 → (((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st
‘𝑧)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) ↔ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st
‘𝑘)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))))) |
| 184 | 183, 172 | chvarvv 1998 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st
‘𝑘)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) |
| 185 | 11, 184 | sylan 580 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st
‘𝑘)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) |
| 186 | 118, 173,
185 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st
‘𝑘))‘𝑛)}) |
| 187 | | fvex 6919 |
. . . . . . . . 9
⊢
((2nd ‘(1st ‘𝑧))‘𝑛) ∈ V |
| 188 | 187 | sneqr 4840 |
. . . . . . . 8
⊢
({((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st
‘𝑘))‘𝑛)} → ((2nd
‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st
‘𝑘))‘𝑛)) |
| 189 | 186, 188 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st
‘𝑘))‘𝑛)) |
| 190 | 30, 43, 189 | eqfnfvd 7054 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd
‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) |
| 191 | 19, 20 | syl 17 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 192 | 32, 33 | syl 17 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑆 → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
| 193 | | xpopth 8055 |
. . . . . . . 8
⊢
(((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (((1st
‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘)) ∧
(2nd ‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) ↔
(1st ‘𝑧) =
(1st ‘𝑘))) |
| 194 | 191, 192,
193 | syl2an 596 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st
‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘)) ∧
(2nd ‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) ↔
(1st ‘𝑧) =
(1st ‘𝑘))) |
| 195 | 194 | ad2antlr 727 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (((1st
‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘)) ∧
(2nd ‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) ↔
(1st ‘𝑧) =
(1st ‘𝑘))) |
| 196 | 17, 190, 195 | mpbi2and 712 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (1st ‘𝑧) = (1st ‘𝑘)) |
| 197 | | eqtr3 2763 |
. . . . . 6
⊢
(((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → (2nd ‘𝑧) = (2nd ‘𝑘)) |
| 198 | 197 | adantl 481 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd ‘𝑧) = (2nd ‘𝑘)) |
| 199 | | xpopth 8055 |
. . . . . . 7
⊢ ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd
‘𝑧) = (2nd
‘𝑘)) ↔ 𝑧 = 𝑘)) |
| 200 | 19, 32, 199 | syl2an 596 |
. . . . . 6
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd
‘𝑧) = (2nd
‘𝑘)) ↔ 𝑧 = 𝑘)) |
| 201 | 200 | ad2antlr 727 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd
‘𝑧) = (2nd
‘𝑘)) ↔ 𝑧 = 𝑘)) |
| 202 | 196, 198,
201 | mpbi2and 712 |
. . . 4
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑧 = 𝑘) |
| 203 | 202 | ex 412 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 𝑧 = 𝑘)) |
| 204 | 203 | ralrimivva 3202 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 𝑧 = 𝑘)) |
| 205 | | fveqeq2 6915 |
. . 3
⊢ (𝑧 = 𝑘 → ((2nd ‘𝑧) = 𝑁 ↔ (2nd ‘𝑘) = 𝑁)) |
| 206 | 205 | rmo4 3736 |
. 2
⊢
(∃*𝑧 ∈
𝑆 (2nd
‘𝑧) = 𝑁 ↔ ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 𝑧 = 𝑘)) |
| 207 | 204, 206 | sylibr 234 |
1
⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁) |