![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > erclwwlkneq | Structured version Visualization version GIF version |
Description: Two classes are equivalent regarding ∼ if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlkn.w | ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) |
erclwwlkn.r | ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} |
Ref | Expression |
---|---|
erclwwlkneq | ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2822 | . . . 4 ⊢ (𝑡 = 𝑇 → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊)) | |
2 | 1 | adantr 482 | . . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊)) |
3 | eleq1 2822 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊)) | |
4 | 3 | adantl 483 | . . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊)) |
5 | simpl 484 | . . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → 𝑡 = 𝑇) | |
6 | oveq1 7368 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛)) | |
7 | 6 | adantl 483 | . . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛)) |
8 | 5, 7 | eqeq12d 2749 | . . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛))) |
9 | 8 | rexbidv 3172 | . . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))) |
10 | 2, 4, 9 | 3anbi123d 1437 | . 2 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛)) ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))) |
11 | erclwwlkn.r | . 2 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
12 | 10, 11 | brabga 5495 | 1 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 class class class wbr 5109 {copab 5171 (class class class)co 7361 0cc0 11059 ...cfz 13433 cyclShift ccsh 14685 ClWWalksN cclwwlkn 29017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: erclwwlkneqlen 29061 erclwwlknref 29062 erclwwlknsym 29063 erclwwlkntr 29064 eclclwwlkn1 29068 |
Copyright terms: Public domain | W3C validator |