| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | poimir.0 | . . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 2 | 1 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
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 |  | poimirlem22.1 | . . . . . . . . 9
⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) | 
| 5 | 4 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) | 
| 6 |  | simplrl 777 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
𝑧 ∈ 𝑆) | 
| 7 |  | simprl 771 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(2nd ‘𝑧) =
0) | 
| 8 | 2, 3, 5, 6, 7 | poimirlem10 37637 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
((𝐹‘(𝑁 − 1)) ∘f
− ((1...𝑁) ×
{1})) = (1st ‘(1st ‘𝑧))) | 
| 9 |  | simplrr 778 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
𝑘 ∈ 𝑆) | 
| 10 |  | simprr 773 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(2nd ‘𝑘) =
0) | 
| 11 | 2, 3, 5, 9, 10 | poimirlem10 37637 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
((𝐹‘(𝑁 − 1)) ∘f
− ((1...𝑁) ×
{1})) = (1st ‘(1st ‘𝑘))) | 
| 12 | 8, 11 | eqtr3d 2779 | . . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(1st ‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘))) | 
| 13 |  | elrabi 3687 | . . . . . . . . . . . . . 14
⊢ (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))) | 
| 14 | 13, 3 | eleq2s 2859 | . . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) | 
| 15 |  | xp1st 8046 | . . . . . . . . . . . . 13
⊢ (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)})) | 
| 16 | 14, 15 | syl 17 | . . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) | 
| 17 |  | 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)}) | 
| 18 | 16, 17 | syl 17 | . . . . . . . . . . 11
⊢ (𝑧 ∈ 𝑆 → (2nd
‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) | 
| 19 |  | fvex 6919 | . . . . . . . . . . . 12
⊢
(2nd ‘(1st ‘𝑧)) ∈ V | 
| 20 |  | 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))) | 
| 21 | 19, 20 | 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)) | 
| 22 | 18, 21 | sylib 218 | . . . . . . . . . 10
⊢ (𝑧 ∈ 𝑆 → (2nd
‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)) | 
| 23 |  | f1ofn 6849 | . . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) | 
| 24 | 22, 23 | syl 17 | . . . . . . . . 9
⊢ (𝑧 ∈ 𝑆 → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) | 
| 25 | 24 | adantr 480 | . . . . . . . 8
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd
‘(1st ‘𝑧)) Fn (1...𝑁)) | 
| 26 | 25 | ad2antlr 727 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(2nd ‘(1st ‘𝑧)) Fn (1...𝑁)) | 
| 27 |  | elrabi 3687 | . . . . . . . . . . . . . 14
⊢ (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))) | 
| 28 | 27, 3 | eleq2s 2859 | . . . . . . . . . . . . 13
⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) | 
| 29 |  | xp1st 8046 | . . . . . . . . . . . . 13
⊢ (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)})) | 
| 30 | 28, 29 | syl 17 | . . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝑆 → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) | 
| 31 |  | 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)}) | 
| 32 | 30, 31 | syl 17 | . . . . . . . . . . 11
⊢ (𝑘 ∈ 𝑆 → (2nd
‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) | 
| 33 |  | fvex 6919 | . . . . . . . . . . . 12
⊢
(2nd ‘(1st ‘𝑘)) ∈ V | 
| 34 |  | 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))) | 
| 35 | 33, 34 | 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)) | 
| 36 | 32, 35 | sylib 218 | . . . . . . . . . 10
⊢ (𝑘 ∈ 𝑆 → (2nd
‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)) | 
| 37 |  | f1ofn 6849 | . . . . . . . . . 10
⊢
((2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) | 
| 38 | 36, 37 | syl 17 | . . . . . . . . 9
⊢ (𝑘 ∈ 𝑆 → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) | 
| 39 | 38 | adantl 481 | . . . . . . . 8
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd
‘(1st ‘𝑘)) Fn (1...𝑁)) | 
| 40 | 39 | ad2antlr 727 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(2nd ‘(1st ‘𝑘)) Fn (1...𝑁)) | 
| 41 |  | eleq1 2829 | . . . . . . . . . . . . 13
⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁))) | 
| 42 | 41 | anbi2d 630 | . . . . . . . . . . . 12
⊢ (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)))) | 
| 43 |  | oveq2 7439 | . . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛)) | 
| 44 | 43 | imaeq2d 6078 | . . . . . . . . . . . . 13
⊢ (𝑚 = 𝑛 → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑧)) “
(1...𝑛))) | 
| 45 | 43 | imaeq2d 6078 | . . . . . . . . . . . . 13
⊢ (𝑚 = 𝑛 → ((2nd
‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) | 
| 46 | 44, 45 | eqeq12d 2753 | . . . . . . . . . . . 12
⊢ (𝑚 = 𝑛 → (((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚)) ↔
((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛)))) | 
| 47 | 42, 46 | imbi12d 344 | . . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))))) | 
| 48 | 1 | ad3antrrr 730 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ) | 
| 49 | 4 | ad3antrrr 730 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁))) | 
| 50 |  | simpl 482 | . . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆) | 
| 51 | 50 | ad3antlr 731 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧 ∈ 𝑆) | 
| 52 |  | simplrl 777 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd ‘𝑧) = 0) | 
| 53 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆) | 
| 54 | 53 | ad3antlr 731 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ 𝑆) | 
| 55 |  | simplrr 778 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd ‘𝑘) = 0) | 
| 56 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁)) | 
| 57 | 48, 3, 49, 51, 52, 54, 55, 56 | poimirlem11 37638 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) ⊆ ((2nd
‘(1st ‘𝑘)) “ (1...𝑚))) | 
| 58 | 48, 3, 49, 54, 55, 51, 52, 56 | poimirlem11 37638 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑚)) ⊆ ((2nd
‘(1st ‘𝑧)) “ (1...𝑚))) | 
| 59 | 57, 58 | eqssd 4001 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚))) | 
| 60 | 47, 59 | chvarvv 1998 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) | 
| 61 |  | simpll 767 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
𝜑) | 
| 62 |  | elfznn 13593 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ) | 
| 63 |  | nnm1nn0 12567 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
ℕ0) | 
| 64 | 62, 63 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈
ℕ0) | 
| 65 | 64 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈
ℕ0) | 
| 66 | 62 | nncnd 12282 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ) | 
| 67 |  | ax-1cn 11213 | . . . . . . . . . . . . . . . . . . . 20
⊢ 1 ∈
ℂ | 
| 68 |  | subeq0 11535 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑛 −
1) = 0 ↔ 𝑛 =
1)) | 
| 69 | 66, 67, 68 | sylancl 586 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1)) | 
| 70 | 69 | necon3abid 2977 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1)) | 
| 71 | 70 | biimpar 477 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0) | 
| 72 |  | elnnne0 12540 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑛 − 1) ∈ ℕ
↔ ((𝑛 − 1)
∈ ℕ0 ∧ (𝑛 − 1) ≠ 0)) | 
| 73 | 65, 71, 72 | sylanbrc 583 | . . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ) | 
| 74 | 73 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ) | 
| 75 | 64 | nn0red 12588 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ) | 
| 76 | 75 | adantl 481 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ) | 
| 77 | 62 | nnred 12281 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ) | 
| 78 | 77 | adantl 481 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ) | 
| 79 | 1 | nnred 12281 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℝ) | 
| 80 | 79 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ) | 
| 81 | 77 | lem1d 12201 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛) | 
| 82 | 81 | adantl 481 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛) | 
| 83 |  | elfzle2 13568 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ≤ 𝑁) | 
| 84 | 83 | adantl 481 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≤ 𝑁) | 
| 85 | 76, 78, 80, 82, 84 | letrd 11418 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁) | 
| 86 | 85 | adantrr 717 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁) | 
| 87 | 1 | nnzd 12640 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 88 |  | fznn 13632 | . . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℤ → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁))) | 
| 89 | 87, 88 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁))) | 
| 90 | 89 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁))) | 
| 91 | 74, 86, 90 | mpbir2and 713 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁)) | 
| 92 | 61, 91 | sylan 580 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧
(𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁)) | 
| 93 |  | ovex 7464 | . . . . . . . . . . . . . 14
⊢ (𝑛 − 1) ∈
V | 
| 94 |  | eleq1 2829 | . . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁))) | 
| 95 | 94 | anbi2d 630 | . . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧
(𝑛 − 1) ∈
(1...𝑁)))) | 
| 96 |  | oveq2 7439 | . . . . . . . . . . . . . . . . 17
⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1))) | 
| 97 | 96 | imaeq2d 6078 | . . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝑛 − 1) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑧)) “
(1...(𝑛 −
1)))) | 
| 98 | 96 | imaeq2d 6078 | . . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝑛 − 1) → ((2nd
‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...(𝑛 −
1)))) | 
| 99 | 97, 98 | eqeq12d 2753 | . . . . . . . . . . . . . . 15
⊢ (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))))) | 
| 100 | 95, 99 | imbi12d 344 | . . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧
(𝑛 − 1) ∈
(1...𝑁)) →
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))) | 
| 101 | 93, 100, 59 | vtocl 3558 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧
(𝑛 − 1) ∈
(1...𝑁)) →
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) | 
| 102 | 92, 101 | syldan 591 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧
(𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) | 
| 103 | 102 | expr 456 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) | 
| 104 |  | ima0 6095 | . . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑧)) “ ∅) =
∅ | 
| 105 |  | ima0 6095 | . . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑘)) “ ∅) =
∅ | 
| 106 | 104, 105 | eqtr4i 2768 | . . . . . . . . . . . 12
⊢
((2nd ‘(1st ‘𝑧)) “ ∅) = ((2nd
‘(1st ‘𝑘)) “ ∅) | 
| 107 |  | oveq1 7438 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1)) | 
| 108 |  | 1m1e0 12338 | . . . . . . . . . . . . . . . 16
⊢ (1
− 1) = 0 | 
| 109 | 107, 108 | eqtrdi 2793 | . . . . . . . . . . . . . . 15
⊢ (𝑛 = 1 → (𝑛 − 1) = 0) | 
| 110 | 109 | oveq2d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑛 = 1 → (1...(𝑛 − 1)) =
(1...0)) | 
| 111 |  | fz10 13585 | . . . . . . . . . . . . . 14
⊢ (1...0) =
∅ | 
| 112 | 110, 111 | eqtrdi 2793 | . . . . . . . . . . . . 13
⊢ (𝑛 = 1 → (1...(𝑛 − 1)) =
∅) | 
| 113 | 112 | imaeq2d 6078 | . . . . . . . . . . . 12
⊢ (𝑛 = 1 → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑧)) “ ∅)) | 
| 114 | 112 | imaeq2d 6078 | . . . . . . . . . . . 12
⊢ (𝑛 = 1 → ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ ∅)) | 
| 115 | 106, 113,
114 | 3eqtr4a 2803 | . . . . . . . . . . 11
⊢ (𝑛 = 1 → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) | 
| 116 | 103, 115 | pm2.61d2 181 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) | 
| 117 | 60, 116 | difeq12d 4127 | . . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 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))))) | 
| 118 |  | fnsnfv 6988 | . . . . . . . . . . . 12
⊢
(((2nd ‘(1st ‘𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st
‘𝑧)) “ {𝑛})) | 
| 119 | 24, 118 | sylan 580 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st
‘𝑧)) “ {𝑛})) | 
| 120 | 62 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ) | 
| 121 |  | uncom 4158 | . . . . . . . . . . . . . . . 16
⊢
((1...(𝑛 − 1))
∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1))) | 
| 122 | 121 | difeq1i 4122 | . . . . . . . . . . . . . . 15
⊢
(((1...(𝑛 −
1)) ∪ {𝑛}) ∖
(1...(𝑛 − 1))) =
(({𝑛} ∪ (1...(𝑛 − 1))) ∖
(1...(𝑛 −
1))) | 
| 123 |  | difun2 4481 | . . . . . . . . . . . . . . 15
⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1))) | 
| 124 | 122, 123 | eqtri 2765 | . . . . . . . . . . . . . 14
⊢
(((1...(𝑛 −
1)) ∪ {𝑛}) ∖
(1...(𝑛 − 1))) =
({𝑛} ∖ (1...(𝑛 − 1))) | 
| 125 |  | nncn 12274 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) | 
| 126 |  | npcan1 11688 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛) | 
| 127 | 125, 126 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛) | 
| 128 |  | elnnuz 12922 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈
(ℤ≥‘1)) | 
| 129 | 128 | biimpi 216 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
(ℤ≥‘1)) | 
| 130 | 127, 129 | eqeltrd 2841 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈
(ℤ≥‘1)) | 
| 131 | 63 | nn0zd 12639 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
ℤ) | 
| 132 |  | uzid 12893 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 − 1) ∈ ℤ
→ (𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1))) | 
| 133 | 131, 132 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1))) | 
| 134 |  | peano2uz 12943 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 − 1) ∈
(ℤ≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈
(ℤ≥‘(𝑛 − 1))) | 
| 135 | 133, 134 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈
(ℤ≥‘(𝑛 − 1))) | 
| 136 | 127, 135 | eqeltrrd 2842 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
(ℤ≥‘(𝑛 − 1))) | 
| 137 |  | fzsplit2 13589 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝑛 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑛 ∈ (ℤ≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛))) | 
| 138 | 130, 136,
137 | syl2anc 584 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ →
(1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛))) | 
| 139 | 127 | oveq1d 7446 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛)) | 
| 140 |  | nnz 12634 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) | 
| 141 |  | fzsn 13606 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛}) | 
| 142 | 140, 141 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛}) | 
| 143 | 139, 142 | eqtrd 2777 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛}) | 
| 144 | 143 | uneq2d 4168 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ →
((1...(𝑛 − 1)) ∪
(((𝑛 − 1) +
1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛})) | 
| 145 | 138, 144 | eqtrd 2777 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ →
(1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛})) | 
| 146 | 145 | difeq1d 4125 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ →
((1...𝑛) ∖
(1...(𝑛 − 1))) =
(((1...(𝑛 − 1)) ∪
{𝑛}) ∖ (1...(𝑛 − 1)))) | 
| 147 |  | nnre 12273 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) | 
| 148 |  | ltm1 12109 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛) | 
| 149 |  | peano2rem 11576 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈
ℝ) | 
| 150 |  | ltnle 11340 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝑛 − 1) ∈ ℝ ∧
𝑛 ∈ ℝ) →
((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1))) | 
| 151 | 149, 150 | mpancom 688 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1))) | 
| 152 | 148, 151 | mpbid 232 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℝ → ¬
𝑛 ≤ (𝑛 − 1)) | 
| 153 |  | elfzle2 13568 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1)) | 
| 154 | 152, 153 | nsyl 140 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℝ → ¬
𝑛 ∈ (1...(𝑛 − 1))) | 
| 155 | 147, 154 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → ¬
𝑛 ∈ (1...(𝑛 − 1))) | 
| 156 |  | incom 4209 | . . . . . . . . . . . . . . . . 17
⊢
((1...(𝑛 − 1))
∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1))) | 
| 157 | 156 | eqeq1i 2742 | . . . . . . . . . . . . . . . 16
⊢
(((1...(𝑛 −
1)) ∩ {𝑛}) = ∅
↔ ({𝑛} ∩
(1...(𝑛 − 1))) =
∅) | 
| 158 |  | disjsn 4711 | . . . . . . . . . . . . . . . 16
⊢
(((1...(𝑛 −
1)) ∩ {𝑛}) = ∅
↔ ¬ 𝑛 ∈
(1...(𝑛 −
1))) | 
| 159 |  | disj3 4454 | . . . . . . . . . . . . . . . 16
⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1)))) | 
| 160 | 157, 158,
159 | 3bitr3i 301 | . . . . . . . . . . . . . . 15
⊢ (¬
𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1)))) | 
| 161 | 155, 160 | sylib 218 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1)))) | 
| 162 | 124, 146,
161 | 3eqtr4a 2803 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ →
((1...𝑛) ∖
(1...(𝑛 − 1))) =
{𝑛}) | 
| 163 | 120, 162 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛}) | 
| 164 | 163 | imaeq2d 6078 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd
‘(1st ‘𝑧)) “ {𝑛})) | 
| 165 |  | dff1o3 6854 | . . . . . . . . . . . . . . 15
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd
‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st
‘𝑧)))) | 
| 166 | 165 | simprbi 496 | . . . . . . . . . . . . . 14
⊢
((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st
‘𝑧))) | 
| 167 | 22, 166 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑆 → Fun ◡(2nd ‘(1st
‘𝑧))) | 
| 168 |  | 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))))) | 
| 169 | 167, 168 | syl 17 | . . . . . . . . . . . 12
⊢ (𝑧 ∈ 𝑆 → ((2nd
‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) | 
| 170 | 169 | 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))))) | 
| 171 | 119, 164,
170 | 3eqtr2d 2783 | . . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st
‘𝑧)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) | 
| 172 | 6, 171 | sylan 580 | . . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st
‘𝑧)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) | 
| 173 |  | eleq1 2829 | . . . . . . . . . . . . 13
⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆)) | 
| 174 | 173 | anbi1d 631 | . . . . . . . . . . . 12
⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)))) | 
| 175 |  | 2fveq3 6911 | . . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑘 → (2nd
‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) | 
| 176 | 175 | fveq1d 6908 | . . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑘 → ((2nd
‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st
‘𝑘))‘𝑛)) | 
| 177 | 176 | sneqd 4638 | . . . . . . . . . . . . 13
⊢ (𝑧 = 𝑘 → {((2nd
‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st
‘𝑘))‘𝑛)}) | 
| 178 | 175 | imaeq1d 6077 | . . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑘 → ((2nd
‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st
‘𝑘)) “
(1...𝑛))) | 
| 179 | 175 | imaeq1d 6077 | . . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑘 → ((2nd
‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd
‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))) | 
| 180 | 178, 179 | 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))))) | 
| 181 | 177, 180 | 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)))))) | 
| 182 | 174, 181 | 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))))))) | 
| 183 | 182, 171 | chvarvv 1998 | . . . . . . . . . 10
⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st
‘𝑘)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) | 
| 184 | 9, 183 | sylan 580 | . . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st
‘𝑘)) “
(1...𝑛)) ∖
((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))) | 
| 185 | 117, 172,
184 | 3eqtr4d 2787 | . . . . . . . 8
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd
‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st
‘𝑘))‘𝑛)}) | 
| 186 |  | fvex 6919 | . . . . . . . . 9
⊢
((2nd ‘(1st ‘𝑧))‘𝑛) ∈ V | 
| 187 | 186 | sneqr 4840 | . . . . . . . 8
⊢
({((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st
‘𝑘))‘𝑛)} → ((2nd
‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st
‘𝑘))‘𝑛)) | 
| 188 | 185, 187 | syl 17 | . . . . . . 7
⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd
‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st
‘𝑘))‘𝑛)) | 
| 189 | 26, 40, 188 | eqfnfvd 7054 | . . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(2nd ‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) | 
| 190 |  | 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))) | 
| 191 | 16, 30, 190 | syl2an 596 | . . . . . . 7
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st
‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘)) ∧
(2nd ‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) ↔
(1st ‘𝑧) =
(1st ‘𝑘))) | 
| 192 | 191 | ad2antlr 727 | . . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(((1st ‘(1st ‘𝑧)) = (1st ‘(1st
‘𝑘)) ∧
(2nd ‘(1st ‘𝑧)) = (2nd ‘(1st
‘𝑘))) ↔
(1st ‘𝑧) =
(1st ‘𝑘))) | 
| 193 | 12, 189, 192 | mpbi2and 712 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(1st ‘𝑧) =
(1st ‘𝑘)) | 
| 194 |  | eqtr3 2763 | . . . . . 6
⊢
(((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → (2nd
‘𝑧) = (2nd
‘𝑘)) | 
| 195 | 194 | adantl 481 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(2nd ‘𝑧) =
(2nd ‘𝑘)) | 
| 196 |  | 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)) | 
| 197 | 14, 28, 196 | syl2an 596 | . . . . . 6
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd
‘𝑧) = (2nd
‘𝑘)) ↔ 𝑧 = 𝑘)) | 
| 198 | 197 | ad2antlr 727 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
(((1st ‘𝑧)
= (1st ‘𝑘)
∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘)) | 
| 199 | 193, 195,
198 | mpbi2and 712 | . . . 4
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0)) →
𝑧 = 𝑘) | 
| 200 | 199 | ex 412 | . . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2nd ‘𝑧) = 0 ∧ (2nd
‘𝑘) = 0) → 𝑧 = 𝑘)) | 
| 201 | 200 | ralrimivva 3202 | . 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → 𝑧 = 𝑘)) | 
| 202 |  | fveqeq2 6915 | . . 3
⊢ (𝑧 = 𝑘 → ((2nd ‘𝑧) = 0 ↔ (2nd
‘𝑘) =
0)) | 
| 203 | 202 | rmo4 3736 | . 2
⊢
(∃*𝑧 ∈
𝑆 (2nd
‘𝑧) = 0 ↔
∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → 𝑧 = 𝑘)) | 
| 204 | 201, 203 | sylibr 234 | 1
⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0) |