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Mirrors > Home > MPE Home > Th. List > erclwwlkeq | Structured version Visualization version GIF version |
Description: Two classes are equivalent regarding ∼ if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlk.r | ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} |
Ref | Expression |
---|---|
erclwwlkeq | ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2902 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (ClWWalks‘𝐺) ↔ 𝑈 ∈ (ClWWalks‘𝐺))) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑢 ∈ (ClWWalks‘𝐺) ↔ 𝑈 ∈ (ClWWalks‘𝐺))) |
3 | eleq1 2902 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ (ClWWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺))) | |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ 𝑊 ∈ (ClWWalks‘𝐺))) |
5 | fveq2 6672 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | |
6 | 5 | oveq2d 7174 | . . . . 5 ⊢ (𝑤 = 𝑊 → (0...(♯‘𝑤)) = (0...(♯‘𝑊))) |
7 | 6 | adantl 484 | . . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (0...(♯‘𝑤)) = (0...(♯‘𝑊))) |
8 | simpl 485 | . . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → 𝑢 = 𝑈) | |
9 | oveq1 7165 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛)) | |
10 | 9 | adantl 484 | . . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛)) |
11 | 8, 10 | eqeq12d 2839 | . . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛))) |
12 | 7, 11 | rexeqbidv 3404 | . . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))) |
13 | 2, 4, 12 | 3anbi123d 1432 | . 2 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → ((𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛)) ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))) |
14 | erclwwlk.r | . 2 ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} | |
15 | 13, 14 | brabga 5423 | 1 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 {copab 5130 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ...cfz 12895 ♯chash 13693 cyclShift ccsh 14152 ClWWalkscclwwlk 27761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: erclwwlkeqlen 27799 erclwwlkref 27800 erclwwlksym 27801 erclwwlktr 27802 |
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