| Step | Hyp | Ref
| Expression |
| 1 | | ccatws1f1o.1 |
. . . . . . . . . . 11
⊢ 𝑁 = (♯‘𝑇) |
| 2 | | ccatws1f1o.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁)) |
| 3 | | f1of 6848 |
. . . . . . . . . . . . 13
⊢ (𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁) → 𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 4 | 2, 3 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 5 | | iswrdi 14556 |
. . . . . . . . . . . 12
⊢ (𝑇:(0..^𝑁)⟶(0..^𝑁) → 𝑇 ∈ Word (0..^𝑁)) |
| 6 | | lencl 14571 |
. . . . . . . . . . . 12
⊢ (𝑇 ∈ Word (0..^𝑁) → (♯‘𝑇) ∈
ℕ0) |
| 7 | 4, 5, 6 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑇) ∈
ℕ0) |
| 8 | 1, 7 | eqeltrid 2845 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 9 | | fzossfzop1 13782 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ0
→ (0..^𝑁) ⊆
(0..^(𝑁 +
1))) |
| 10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(𝑁 + 1))) |
| 11 | | ccatws1f1o.2 |
. . . . . . . . 9
⊢ 𝐽 = (0..^(𝑁 + 1)) |
| 12 | 10, 11 | sseqtrrdi 4025 |
. . . . . . . 8
⊢ (𝜑 → (0..^𝑁) ⊆ 𝐽) |
| 13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (0..^𝑁) ⊆ 𝐽) |
| 14 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 15 | 1 | eqcomi 2746 |
. . . . . . . . . . . 12
⊢
(♯‘𝑇) =
𝑁 |
| 16 | 15 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑇) = 𝑁) |
| 17 | 16 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (0..^(♯‘𝑇)) = (0..^𝑁)) |
| 18 | 17 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝑇)) ↔ 𝑥 ∈ (0..^𝑁))) |
| 19 | 18 | biimpa 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^𝑁)) |
| 20 | 14, 19 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (𝑇‘𝑥) ∈ (0..^𝑁)) |
| 21 | 13, 20 | sseldd 3984 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (𝑇‘𝑥) ∈ 𝐽) |
| 22 | 21 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (𝑇‘𝑥) ∈ 𝐽) |
| 23 | 11 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐽 = (0..^(𝑁 + 1))) |
| 24 | | fzo0ssnn0 13785 |
. . . . . . . . . . . . 13
⊢
(0..^(𝑁 + 1))
⊆ ℕ0 |
| 25 | 23, 24 | eqsstrdi 4028 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ⊆
ℕ0) |
| 26 | 25 | sselda 3983 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ ℕ0) |
| 27 | 26 | nn0cnd 12589 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ ℂ) |
| 28 | 27 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ ℂ) |
| 29 | | nn0uz 12920 |
. . . . . . . . . . . . . 14
⊢
ℕ0 = (ℤ≥‘0) |
| 30 | 8, 29 | eleqtrdi 2851 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
| 31 | 30 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑁 ∈
(ℤ≥‘0)) |
| 32 | 23 | eleq2d 2827 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ (0..^(𝑁 + 1)))) |
| 33 | 32 | biimpa 476 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ (0..^(𝑁 + 1))) |
| 34 | 33 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(𝑁 + 1))) |
| 35 | | fzosplitsni 13817 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑥 ∈ (0..^(𝑁 + 1)) ↔ (𝑥 ∈ (0..^𝑁) ∨ 𝑥 = 𝑁))) |
| 36 | 35 | biimpa 476 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘0) ∧ 𝑥 ∈ (0..^(𝑁 + 1))) → (𝑥 ∈ (0..^𝑁) ∨ 𝑥 = 𝑁)) |
| 37 | 31, 34, 36 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → (𝑥 ∈ (0..^𝑁) ∨ 𝑥 = 𝑁)) |
| 38 | 18 | notbid 318 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (¬ 𝑥 ∈ (0..^(♯‘𝑇)) ↔ ¬ 𝑥 ∈ (0..^𝑁))) |
| 39 | 38 | biimpa 476 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → ¬ 𝑥 ∈ (0..^𝑁)) |
| 40 | 39 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → ¬ 𝑥 ∈ (0..^𝑁)) |
| 41 | 37, 40 | orcnd 879 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 = 𝑁) |
| 42 | 41, 1 | eqtrdi 2793 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 = (♯‘𝑇)) |
| 43 | 28, 42 | subeq0bd 11689 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → (𝑥 − (♯‘𝑇)) = 0) |
| 44 | 43 | fveq2d 6910 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) = (〈“𝑁”〉‘0)) |
| 45 | | s1fv 14648 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ (〈“𝑁”〉‘0) = 𝑁) |
| 46 | 8, 45 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (〈“𝑁”〉‘0) = 𝑁) |
| 47 | 46 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → (〈“𝑁”〉‘0) = 𝑁) |
| 48 | 44, 47 | eqtrd 2777 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) = 𝑁) |
| 49 | | fzonn0p1 13781 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈ (0..^(𝑁 + 1))) |
| 50 | 8, 49 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ (0..^(𝑁 + 1))) |
| 51 | 50, 11 | eleqtrrdi 2852 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ 𝐽) |
| 52 | 51 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑁 ∈ 𝐽) |
| 53 | 48, 52 | eqeltrd 2841 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐽) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑇))) → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) ∈ 𝐽) |
| 54 | 22, 53 | ifclda 4561 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) ∈ 𝐽) |
| 55 | 54 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐽 if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) ∈ 𝐽) |
| 56 | 12 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ 𝐽) |
| 57 | | f1ocnv 6860 |
. . . . . . . . . 10
⊢ (𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁) → ◡𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁)) |
| 58 | | f1of 6848 |
. . . . . . . . . 10
⊢ (◡𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁) → ◡𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 59 | 2, 57, 58 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 60 | 59 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → ◡𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 61 | 60 | ffvelcdmda 7104 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → (◡𝑇‘𝑦) ∈ (0..^𝑁)) |
| 62 | 56, 61 | sseldd 3984 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → (◡𝑇‘𝑦) ∈ 𝐽) |
| 63 | 1 | oveq2i 7442 |
. . . . . . . . 9
⊢
(0..^𝑁) =
(0..^(♯‘𝑇)) |
| 64 | 61, 63 | eleqtrdi 2851 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → (◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇))) |
| 65 | 64 | iftrued 4533 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → if((◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇)), (𝑇‘(◡𝑇‘𝑦)), (〈“𝑁”〉‘((◡𝑇‘𝑦) − (♯‘𝑇)))) = (𝑇‘(◡𝑇‘𝑦))) |
| 66 | 2 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁)) |
| 67 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 ∈ (0..^𝑁)) |
| 68 | | f1ocnvfv2 7297 |
. . . . . . . 8
⊢ ((𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑇‘(◡𝑇‘𝑦)) = 𝑦) |
| 69 | 66, 67, 68 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → (𝑇‘(◡𝑇‘𝑦)) = 𝑦) |
| 70 | 65, 69 | eqtr2d 2778 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → 𝑦 = if((◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇)), (𝑇‘(◡𝑇‘𝑦)), (〈“𝑁”〉‘((◡𝑇‘𝑦) − (♯‘𝑇))))) |
| 71 | | simpr 484 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 72 | 30 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑁 ∈
(ℤ≥‘0)) |
| 73 | 72, 33, 36 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ (0..^𝑁) ∨ 𝑥 = 𝑁)) |
| 74 | 73 | ad5ant14 758 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → (𝑥 ∈ (0..^𝑁) ∨ 𝑥 = 𝑁)) |
| 75 | 67 | ad3antrrr 730 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑦 ∈ (0..^𝑁)) |
| 76 | 71 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 77 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑥 = 𝑁) |
| 78 | | fzonel 13713 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ¬
𝑁 ∈ (0..^𝑁) |
| 79 | 78 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → ¬ 𝑁 ∈ (0..^𝑁)) |
| 80 | 63 | eleq2i 2833 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ (0..^𝑁) ↔ 𝑁 ∈ (0..^(♯‘𝑇))) |
| 81 | 79, 80 | sylnib 328 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → ¬ 𝑁 ∈ (0..^(♯‘𝑇))) |
| 82 | 77, 81 | eqneltrd 2861 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → ¬ 𝑥 ∈ (0..^(♯‘𝑇))) |
| 83 | 82 | iffalsed 4536 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) |
| 84 | 11, 24 | eqsstri 4030 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐽 ⊆
ℕ0 |
| 85 | | simpllr 776 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑥 ∈ 𝐽) |
| 86 | 84, 85 | sselid 3981 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑥 ∈ ℕ0) |
| 87 | 86 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑥 ∈ ℂ) |
| 88 | 77, 1 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑥 = (♯‘𝑇)) |
| 89 | 87, 88 | subeq0bd 11689 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → (𝑥 − (♯‘𝑇)) = 0) |
| 90 | 89 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) = (〈“𝑁”〉‘0)) |
| 91 | 46 | ad5antr 734 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → (〈“𝑁”〉‘0) = 𝑁) |
| 92 | 90, 91 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) = 𝑁) |
| 93 | 76, 83, 92 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → 𝑦 = 𝑁) |
| 94 | 93, 79 | eqneltrd 2861 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 = 𝑁) → ¬ 𝑦 ∈ (0..^𝑁)) |
| 95 | 75, 94 | pm2.65da 817 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → ¬ 𝑥 = 𝑁) |
| 96 | 74, 95 | olcnd 878 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑥 ∈ (0..^𝑁)) |
| 97 | 96, 63 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑥 ∈ (0..^(♯‘𝑇))) |
| 98 | 97 | iftrued 4533 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = (𝑇‘𝑥)) |
| 99 | 71, 98 | eqtrd 2777 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑦 = (𝑇‘𝑥)) |
| 100 | 99 | fveq2d 6910 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → (◡𝑇‘𝑦) = (◡𝑇‘(𝑇‘𝑥))) |
| 101 | 66 | ad2antrr 726 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁)) |
| 102 | | f1ocnvfv1 7296 |
. . . . . . . . . 10
⊢ ((𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁) ∧ 𝑥 ∈ (0..^𝑁)) → (◡𝑇‘(𝑇‘𝑥)) = 𝑥) |
| 103 | 101, 96, 102 | syl2anc 584 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → (◡𝑇‘(𝑇‘𝑥)) = 𝑥) |
| 104 | 100, 103 | eqtr2d 2778 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑥 = (◡𝑇‘𝑦)) |
| 105 | 104 | ex 412 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) ∧ 𝑥 ∈ 𝐽) → (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) → 𝑥 = (◡𝑇‘𝑦))) |
| 106 | 105 | ralrimiva 3146 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → ∀𝑥 ∈ 𝐽 (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) → 𝑥 = (◡𝑇‘𝑦))) |
| 107 | | eleq1 2829 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝑇‘𝑦) → (𝑥 ∈ (0..^(♯‘𝑇)) ↔ (◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇)))) |
| 108 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝑇‘𝑦) → (𝑇‘𝑥) = (𝑇‘(◡𝑇‘𝑦))) |
| 109 | | fvoveq1 7454 |
. . . . . . . . 9
⊢ (𝑥 = (◡𝑇‘𝑦) → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) = (〈“𝑁”〉‘((◡𝑇‘𝑦) − (♯‘𝑇)))) |
| 110 | 107, 108,
109 | ifbieq12d 4554 |
. . . . . . . 8
⊢ (𝑥 = (◡𝑇‘𝑦) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = if((◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇)), (𝑇‘(◡𝑇‘𝑦)), (〈“𝑁”〉‘((◡𝑇‘𝑦) − (♯‘𝑇))))) |
| 111 | 110 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑥 = (◡𝑇‘𝑦) → (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) ↔ 𝑦 = if((◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇)), (𝑇‘(◡𝑇‘𝑦)), (〈“𝑁”〉‘((◡𝑇‘𝑦) − (♯‘𝑇)))))) |
| 112 | 111 | eqreu 3735 |
. . . . . 6
⊢ (((◡𝑇‘𝑦) ∈ 𝐽 ∧ 𝑦 = if((◡𝑇‘𝑦) ∈ (0..^(♯‘𝑇)), (𝑇‘(◡𝑇‘𝑦)), (〈“𝑁”〉‘((◡𝑇‘𝑦) − (♯‘𝑇)))) ∧ ∀𝑥 ∈ 𝐽 (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) → 𝑥 = (◡𝑇‘𝑦))) → ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 113 | 62, 70, 106, 112 | syl3anc 1373 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 ∈ (0..^𝑁)) → ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 114 | 51 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → 𝑁 ∈ 𝐽) |
| 115 | 8 | nn0cnd 12589 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 116 | 115 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → 𝑁 ∈ ℂ) |
| 117 | 1 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → 𝑁 = (♯‘𝑇)) |
| 118 | 116, 117 | subeq0bd 11689 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → (𝑁 − (♯‘𝑇)) = 0) |
| 119 | 118 | fveq2d 6910 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → (〈“𝑁”〉‘(𝑁 − (♯‘𝑇))) = (〈“𝑁”〉‘0)) |
| 120 | 46 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → (〈“𝑁”〉‘0) = 𝑁) |
| 121 | 119, 120 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → (〈“𝑁”〉‘(𝑁 − (♯‘𝑇))) = 𝑁) |
| 122 | 78 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → ¬ 𝑁 ∈ (0..^𝑁)) |
| 123 | 122, 80 | sylnib 328 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → ¬ 𝑁 ∈ (0..^(♯‘𝑇))) |
| 124 | 123 | iffalsed 4536 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → if(𝑁 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑁), (〈“𝑁”〉‘(𝑁 − (♯‘𝑇)))) = (〈“𝑁”〉‘(𝑁 − (♯‘𝑇)))) |
| 125 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁) |
| 126 | 121, 124,
125 | 3eqtr4rd 2788 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → 𝑦 = if(𝑁 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑁), (〈“𝑁”〉‘(𝑁 − (♯‘𝑇))))) |
| 127 | 30 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑁 ∈
(ℤ≥‘0)) |
| 128 | 127 | ad3antrrr 730 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑁 ∈
(ℤ≥‘0)) |
| 129 | 33 | ad5ant14 758 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑥 ∈ (0..^(𝑁 + 1))) |
| 130 | 128, 129,
36 | syl2anc 584 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → (𝑥 ∈ (0..^𝑁) ∨ 𝑥 = 𝑁)) |
| 131 | | simpr 484 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 132 | | simpllr 776 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑦 = 𝑁) |
| 133 | 131, 132 | eqtr3d 2779 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = 𝑁) |
| 134 | 133 | adantr 480 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = 𝑁) |
| 135 | 63 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → (0..^𝑁) = (0..^(♯‘𝑇))) |
| 136 | 135 | eleq2d 2827 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → (𝑥 ∈ (0..^𝑁) ↔ 𝑥 ∈ (0..^(♯‘𝑇)))) |
| 137 | 136 | biimpa 476 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^(♯‘𝑇))) |
| 138 | 137 | iftrued 4533 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = (𝑇‘𝑥)) |
| 139 | 4 | ad4antr 732 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑇:(0..^𝑁)⟶(0..^𝑁)) |
| 140 | 139 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑇‘𝑥) ∈ (0..^𝑁)) |
| 141 | 138, 140 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) ∈ (0..^𝑁)) |
| 142 | 134, 141 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑁 ∈ (0..^𝑁)) |
| 143 | 78 | a1i 11 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) ∧ 𝑥 ∈ (0..^𝑁)) → ¬ 𝑁 ∈ (0..^𝑁)) |
| 144 | 142, 143 | pm2.65da 817 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → ¬ 𝑥 ∈ (0..^𝑁)) |
| 145 | 130, 144 | orcnd 879 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) → 𝑥 = 𝑁) |
| 146 | 145 | ex 412 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) ∧ 𝑥 ∈ 𝐽) → (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) → 𝑥 = 𝑁)) |
| 147 | 146 | ralrimiva 3146 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → ∀𝑥 ∈ 𝐽 (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) → 𝑥 = 𝑁)) |
| 148 | | eleq1 2829 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑥 ∈ (0..^(♯‘𝑇)) ↔ 𝑁 ∈ (0..^(♯‘𝑇)))) |
| 149 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (𝑇‘𝑥) = (𝑇‘𝑁)) |
| 150 | | fvoveq1 7454 |
. . . . . . . . 9
⊢ (𝑥 = 𝑁 → (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))) = (〈“𝑁”〉‘(𝑁 − (♯‘𝑇)))) |
| 151 | 148, 149,
150 | ifbieq12d 4554 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) = if(𝑁 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑁), (〈“𝑁”〉‘(𝑁 − (♯‘𝑇))))) |
| 152 | 151 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) ↔ 𝑦 = if(𝑁 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑁), (〈“𝑁”〉‘(𝑁 − (♯‘𝑇)))))) |
| 153 | 152 | eqreu 3735 |
. . . . . 6
⊢ ((𝑁 ∈ 𝐽 ∧ 𝑦 = if(𝑁 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑁), (〈“𝑁”〉‘(𝑁 − (♯‘𝑇)))) ∧ ∀𝑥 ∈ 𝐽 (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) → 𝑥 = 𝑁)) → ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 154 | 114, 126,
147, 153 | syl3anc 1373 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐽) ∧ 𝑦 = 𝑁) → ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 155 | 23 | eleq2d 2827 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐽 ↔ 𝑦 ∈ (0..^(𝑁 + 1)))) |
| 156 | 155 | biimpa 476 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ (0..^(𝑁 + 1))) |
| 157 | | fzosplitsni 13817 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑦 ∈ (0..^(𝑁 + 1)) ↔ (𝑦 ∈ (0..^𝑁) ∨ 𝑦 = 𝑁))) |
| 158 | 157 | biimpa 476 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘0) ∧ 𝑦 ∈ (0..^(𝑁 + 1))) → (𝑦 ∈ (0..^𝑁) ∨ 𝑦 = 𝑁)) |
| 159 | 127, 156,
158 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑦 ∈ (0..^𝑁) ∨ 𝑦 = 𝑁)) |
| 160 | 113, 154,
159 | mpjaodan 961 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 161 | 160 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ 𝐽 ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 162 | | s1len 14644 |
. . . . . . . 8
⊢
(♯‘〈“𝑁”〉) = 1 |
| 163 | 15, 162 | oveq12i 7443 |
. . . . . . 7
⊢
((♯‘𝑇) +
(♯‘〈“𝑁”〉)) = (𝑁 + 1) |
| 164 | 163 | oveq2i 7442 |
. . . . . 6
⊢
(0..^((♯‘𝑇) + (♯‘〈“𝑁”〉))) = (0..^(𝑁 + 1)) |
| 165 | 164, 11 | eqtr4i 2768 |
. . . . 5
⊢
(0..^((♯‘𝑇) + (♯‘〈“𝑁”〉))) = 𝐽 |
| 166 | 165 | mpteq1i 5238 |
. . . 4
⊢ (𝑥 ∈
(0..^((♯‘𝑇) +
(♯‘〈“𝑁”〉))) ↦ if(𝑥 ∈
(0..^(♯‘𝑇)),
(𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) = (𝑥 ∈ 𝐽 ↦ if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))) |
| 167 | 166 | f1ompt 7131 |
. . 3
⊢ ((𝑥 ∈
(0..^((♯‘𝑇) +
(♯‘〈“𝑁”〉))) ↦ if(𝑥 ∈
(0..^(♯‘𝑇)),
(𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))):𝐽–1-1-onto→𝐽 ↔ (∀𝑥 ∈ 𝐽 if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))) ∈ 𝐽 ∧ ∀𝑦 ∈ 𝐽 ∃!𝑥 ∈ 𝐽 𝑦 = if(𝑥 ∈ (0..^(♯‘𝑇)), (𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))))) |
| 168 | 55, 161, 167 | sylanbrc 583 |
. 2
⊢ (𝜑 → (𝑥 ∈ (0..^((♯‘𝑇) +
(♯‘〈“𝑁”〉))) ↦ if(𝑥 ∈
(0..^(♯‘𝑇)),
(𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))):𝐽–1-1-onto→𝐽) |
| 169 | | ovex 7464 |
. . . . 5
⊢
(0..^𝑁) ∈
V |
| 170 | | fex 7246 |
. . . . 5
⊢ ((𝑇:(0..^𝑁)⟶(0..^𝑁) ∧ (0..^𝑁) ∈ V) → 𝑇 ∈ V) |
| 171 | 4, 169, 170 | sylancl 586 |
. . . 4
⊢ (𝜑 → 𝑇 ∈ V) |
| 172 | | s1cli 14643 |
. . . 4
⊢
〈“𝑁”〉 ∈ Word V |
| 173 | | ccatfval 14611 |
. . . 4
⊢ ((𝑇 ∈ V ∧
〈“𝑁”〉
∈ Word V) → (𝑇 ++
〈“𝑁”〉) = (𝑥 ∈ (0..^((♯‘𝑇) +
(♯‘〈“𝑁”〉))) ↦ if(𝑥 ∈
(0..^(♯‘𝑇)),
(𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))))) |
| 174 | 171, 172,
173 | sylancl 586 |
. . 3
⊢ (𝜑 → (𝑇 ++ 〈“𝑁”〉) = (𝑥 ∈ (0..^((♯‘𝑇) +
(♯‘〈“𝑁”〉))) ↦ if(𝑥 ∈
(0..^(♯‘𝑇)),
(𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇)))))) |
| 175 | 174 | f1oeq1d 6843 |
. 2
⊢ (𝜑 → ((𝑇 ++ 〈“𝑁”〉):𝐽–1-1-onto→𝐽 ↔ (𝑥 ∈ (0..^((♯‘𝑇) +
(♯‘〈“𝑁”〉))) ↦ if(𝑥 ∈
(0..^(♯‘𝑇)),
(𝑇‘𝑥), (〈“𝑁”〉‘(𝑥 − (♯‘𝑇))))):𝐽–1-1-onto→𝐽)) |
| 176 | 168, 175 | mpbird 257 |
1
⊢ (𝜑 → (𝑇 ++ 〈“𝑁”〉):𝐽–1-1-onto→𝐽) |