Proof of Theorem repswcshw
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 0csh0 14831 | . . . . 5
⊢ (∅
cyclShift 𝐼) =
∅ | 
| 2 |  | repsw0 14815 | . . . . . 6
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 0) = ∅) | 
| 3 | 2 | oveq1d 7446 | . . . . 5
⊢ (𝑆 ∈ 𝑉 → ((𝑆 repeatS 0) cyclShift 𝐼) = (∅ cyclShift 𝐼)) | 
| 4 | 1, 3, 2 | 3eqtr4a 2803 | . . . 4
⊢ (𝑆 ∈ 𝑉 → ((𝑆 repeatS 0) cyclShift 𝐼) = (𝑆 repeatS 0)) | 
| 5 | 4 | 3ad2ant1 1134 | . . 3
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 0) cyclShift 𝐼) = (𝑆 repeatS 0)) | 
| 6 |  | oveq2 7439 | . . . . 5
⊢ (𝑁 = 0 → (𝑆 repeatS 𝑁) = (𝑆 repeatS 0)) | 
| 7 | 6 | oveq1d 7446 | . . . 4
⊢ (𝑁 = 0 → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = ((𝑆 repeatS 0) cyclShift 𝐼)) | 
| 8 | 7, 6 | eqeq12d 2753 | . . 3
⊢ (𝑁 = 0 → (((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁) ↔ ((𝑆 repeatS 0) cyclShift 𝐼) = (𝑆 repeatS 0))) | 
| 9 | 5, 8 | imbitrrid 246 | . 2
⊢ (𝑁 = 0 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁))) | 
| 10 |  | idd 24 | . . . 4
⊢ (¬
𝑁 = 0 → (𝑆 ∈ 𝑉 → 𝑆 ∈ 𝑉)) | 
| 11 |  | df-ne 2941 | . . . . 5
⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0) | 
| 12 |  | elnnne0 12540 | . . . . . 6
⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0
∧ 𝑁 ≠
0)) | 
| 13 | 12 | simplbi2com 502 | . . . . 5
⊢ (𝑁 ≠ 0 → (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℕ)) | 
| 14 | 11, 13 | sylbir 235 | . . . 4
⊢ (¬
𝑁 = 0 → (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℕ)) | 
| 15 |  | idd 24 | . . . 4
⊢ (¬
𝑁 = 0 → (𝐼 ∈ ℤ → 𝐼 ∈
ℤ)) | 
| 16 | 10, 14, 15 | 3anim123d 1445 | . . 3
⊢ (¬
𝑁 = 0 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → (𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ))) | 
| 17 |  | nnnn0 12533 | . . . . . . 7
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) | 
| 18 | 17 | anim2i 617 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ 𝑉 ∧ 𝑁 ∈
ℕ0)) | 
| 19 |  | repsw 14813 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) ∈ Word 𝑉) | 
| 20 | 18, 19 | syl 17 | . . . . 5
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑆 repeatS 𝑁) ∈ Word 𝑉) | 
| 21 |  | cshword 14829 | . . . . 5
⊢ (((𝑆 repeatS 𝑁) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod (♯‘(𝑆 repeatS 𝑁))), (♯‘(𝑆 repeatS 𝑁))〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod (♯‘(𝑆 repeatS 𝑁)))))) | 
| 22 | 20, 21 | stoic3 1776 | . . . 4
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod (♯‘(𝑆 repeatS 𝑁))), (♯‘(𝑆 repeatS 𝑁))〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod (♯‘(𝑆 repeatS 𝑁)))))) | 
| 23 |  | repswlen 14814 | . . . . . . . . . 10
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) →
(♯‘(𝑆 repeatS
𝑁)) = 𝑁) | 
| 24 | 18, 23 | syl 17 | . . . . . . . . 9
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) →
(♯‘(𝑆 repeatS
𝑁)) = 𝑁) | 
| 25 | 24 | oveq2d 7447 | . . . . . . . 8
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝐼 mod (♯‘(𝑆 repeatS 𝑁))) = (𝐼 mod 𝑁)) | 
| 26 | 25, 24 | opeq12d 4881 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 〈(𝐼 mod (♯‘(𝑆 repeatS 𝑁))), (♯‘(𝑆 repeatS 𝑁))〉 = 〈(𝐼 mod 𝑁), 𝑁〉) | 
| 27 | 26 | oveq2d 7447 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁) substr 〈(𝐼 mod (♯‘(𝑆 repeatS 𝑁))), (♯‘(𝑆 repeatS 𝑁))〉) = ((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉)) | 
| 28 | 25 | oveq2d 7447 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁) prefix (𝐼 mod (♯‘(𝑆 repeatS 𝑁)))) = ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁))) | 
| 29 | 27, 28 | oveq12d 7449 | . . . . 5
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod (♯‘(𝑆 repeatS 𝑁))), (♯‘(𝑆 repeatS 𝑁))〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod (♯‘(𝑆 repeatS 𝑁))))) = (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁)))) | 
| 30 | 29 | 3adant3 1133 | . . . 4
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod (♯‘(𝑆 repeatS 𝑁))), (♯‘(𝑆 repeatS 𝑁))〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod (♯‘(𝑆 repeatS 𝑁))))) = (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁)))) | 
| 31 | 18 | 3adant3 1133 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝑆 ∈ 𝑉 ∧ 𝑁 ∈
ℕ0)) | 
| 32 |  | zmodcl 13931 | . . . . . . . . . 10
⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐼 mod 𝑁) ∈
ℕ0) | 
| 33 | 32 | ancoms 458 | . . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ∈
ℕ0) | 
| 34 | 17 | adantr 480 | . . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈
ℕ0) | 
| 35 | 33, 34 | jca 511 | . . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝐼 mod 𝑁) ∈ ℕ0 ∧ 𝑁 ∈
ℕ0)) | 
| 36 | 35 | 3adant1 1131 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝐼 mod 𝑁) ∈ ℕ0 ∧ 𝑁 ∈
ℕ0)) | 
| 37 |  | nnre 12273 | . . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ) | 
| 38 | 37 | leidd 11829 | . . . . . . . 8
⊢ (𝑁 ∈ ℕ → 𝑁 ≤ 𝑁) | 
| 39 | 38 | 3ad2ant2 1135 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → 𝑁 ≤ 𝑁) | 
| 40 |  | repswswrd 14822 | . . . . . . 7
⊢ (((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ((𝐼 mod 𝑁) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
∧ 𝑁 ≤ 𝑁) → ((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉) = (𝑆 repeatS (𝑁 − (𝐼 mod 𝑁)))) | 
| 41 | 31, 36, 39, 40 | syl3anc 1373 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉) = (𝑆 repeatS (𝑁 − (𝐼 mod 𝑁)))) | 
| 42 |  | simp1 1137 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → 𝑆 ∈ 𝑉) | 
| 43 | 17 | 3ad2ant2 1135 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈
ℕ0) | 
| 44 |  | zmodfzp1 13935 | . . . . . . . . 9
⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐼 mod 𝑁) ∈ (0...𝑁)) | 
| 45 | 44 | ancoms 458 | . . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ∈ (0...𝑁)) | 
| 46 | 45 | 3adant1 1131 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ∈ (0...𝑁)) | 
| 47 |  | repswpfx 14823 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ (𝐼 mod 𝑁) ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁)) = (𝑆 repeatS (𝐼 mod 𝑁))) | 
| 48 | 42, 43, 46, 47 | syl3anc 1373 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁)) = (𝑆 repeatS (𝐼 mod 𝑁))) | 
| 49 | 41, 48 | oveq12d 7449 | . . . . 5
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁))) = ((𝑆 repeatS (𝑁 − (𝐼 mod 𝑁))) ++ (𝑆 repeatS (𝐼 mod 𝑁)))) | 
| 50 | 32 | nn0red 12588 | . . . . . . . . . 10
⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐼 mod 𝑁) ∈ ℝ) | 
| 51 | 50 | ancoms 458 | . . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ∈ ℝ) | 
| 52 | 37 | adantr 480 | . . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → 𝑁 ∈
ℝ) | 
| 53 |  | zre 12617 | . . . . . . . . . 10
⊢ (𝐼 ∈ ℤ → 𝐼 ∈
ℝ) | 
| 54 |  | nnrp 13046 | . . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ+) | 
| 55 |  | modlt 13920 | . . . . . . . . . 10
⊢ ((𝐼 ∈ ℝ ∧ 𝑁 ∈ ℝ+)
→ (𝐼 mod 𝑁) < 𝑁) | 
| 56 | 53, 54, 55 | syl2anr 597 | . . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) < 𝑁) | 
| 57 | 51, 52, 56 | ltled 11409 | . . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ≤ 𝑁) | 
| 58 | 57 | 3adant1 1131 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ≤ 𝑁) | 
| 59 | 33 | 3adant1 1131 | . . . . . . . 8
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝐼 mod 𝑁) ∈
ℕ0) | 
| 60 |  | nn0sub 12576 | . . . . . . . 8
⊢ (((𝐼 mod 𝑁) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)
→ ((𝐼 mod 𝑁) ≤ 𝑁 ↔ (𝑁 − (𝐼 mod 𝑁)) ∈
ℕ0)) | 
| 61 | 59, 43, 60 | syl2anc 584 | . . . . . . 7
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝐼 mod 𝑁) ≤ 𝑁 ↔ (𝑁 − (𝐼 mod 𝑁)) ∈
ℕ0)) | 
| 62 | 58, 61 | mpbid 232 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝑁 − (𝐼 mod 𝑁)) ∈
ℕ0) | 
| 63 |  | repswccat 14824 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑁 − (𝐼 mod 𝑁)) ∈ ℕ0 ∧ (𝐼 mod 𝑁) ∈ ℕ0) → ((𝑆 repeatS (𝑁 − (𝐼 mod 𝑁))) ++ (𝑆 repeatS (𝐼 mod 𝑁))) = (𝑆 repeatS ((𝑁 − (𝐼 mod 𝑁)) + (𝐼 mod 𝑁)))) | 
| 64 | 42, 62, 59, 63 | syl3anc 1373 | . . . . 5
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS (𝑁 − (𝐼 mod 𝑁))) ++ (𝑆 repeatS (𝐼 mod 𝑁))) = (𝑆 repeatS ((𝑁 − (𝐼 mod 𝑁)) + (𝐼 mod 𝑁)))) | 
| 65 |  | nncn 12274 | . . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) | 
| 66 | 65 | adantl 481 | . . . . . . . . 9
⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈
ℂ) | 
| 67 | 32 | nn0cnd 12589 | . . . . . . . . 9
⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐼 mod 𝑁) ∈ ℂ) | 
| 68 | 66, 67 | npcand 11624 | . . . . . . . 8
⊢ ((𝐼 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑁 − (𝐼 mod 𝑁)) + (𝐼 mod 𝑁)) = 𝑁) | 
| 69 | 68 | ancoms 458 | . . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑁 − (𝐼 mod 𝑁)) + (𝐼 mod 𝑁)) = 𝑁) | 
| 70 | 69 | 3adant1 1131 | . . . . . 6
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑁 − (𝐼 mod 𝑁)) + (𝐼 mod 𝑁)) = 𝑁) | 
| 71 | 70 | oveq2d 7447 | . . . . 5
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (𝑆 repeatS ((𝑁 − (𝐼 mod 𝑁)) + (𝐼 mod 𝑁))) = (𝑆 repeatS 𝑁)) | 
| 72 | 49, 64, 71 | 3eqtrd 2781 | . . . 4
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → (((𝑆 repeatS 𝑁) substr 〈(𝐼 mod 𝑁), 𝑁〉) ++ ((𝑆 repeatS 𝑁) prefix (𝐼 mod 𝑁))) = (𝑆 repeatS 𝑁)) | 
| 73 | 22, 30, 72 | 3eqtrd 2781 | . . 3
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁)) | 
| 74 | 16, 73 | syl6 35 | . 2
⊢ (¬
𝑁 = 0 → ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁))) | 
| 75 | 9, 74 | pm2.61i 182 | 1
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁)) |