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Mirrors > Home > MPE Home > Th. List > erclwwlkneqlen | Structured version Visualization version GIF version |
Description: If two classes are equivalent regarding ∼, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-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 |
---|---|
erclwwlkneqlen | ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 → (♯‘𝑇) = (♯‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erclwwlkn.w | . . 3 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) | |
2 | erclwwlkn.r | . . 3 ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
3 | 1, 2 | erclwwlkneq 28459 | . 2 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))) |
4 | fveq2 6792 | . . . . 5 ⊢ (𝑇 = (𝑈 cyclShift 𝑛) → (♯‘𝑇) = (♯‘(𝑈 cyclShift 𝑛))) | |
5 | eqid 2733 | . . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
6 | 5 | clwwlknwrd 28426 | . . . . . . . 8 ⊢ (𝑈 ∈ (𝑁 ClWWalksN 𝐺) → 𝑈 ∈ Word (Vtx‘𝐺)) |
7 | 6, 1 | eleq2s 2852 | . . . . . . 7 ⊢ (𝑈 ∈ 𝑊 → 𝑈 ∈ Word (Vtx‘𝐺)) |
8 | 7 | adantl 481 | . . . . . 6 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) → 𝑈 ∈ Word (Vtx‘𝐺)) |
9 | elfzelz 13284 | . . . . . 6 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ∈ ℤ) | |
10 | cshwlen 14540 | . . . . . 6 ⊢ ((𝑈 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ ℤ) → (♯‘(𝑈 cyclShift 𝑛)) = (♯‘𝑈)) | |
11 | 8, 9, 10 | syl2an 595 | . . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ 𝑛 ∈ (0...𝑁)) → (♯‘(𝑈 cyclShift 𝑛)) = (♯‘𝑈)) |
12 | 4, 11 | sylan9eqr 2795 | . . . 4 ⊢ ((((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑇 = (𝑈 cyclShift 𝑛)) → (♯‘𝑇) = (♯‘𝑈)) |
13 | 12 | rexlimdva2 3148 | . . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → (♯‘𝑇) = (♯‘𝑈))) |
14 | 13 | 3impia 1115 | . 2 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)) → (♯‘𝑇) = (♯‘𝑈)) |
15 | 3, 14 | syl6bi 252 | 1 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 → (♯‘𝑇) = (♯‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1537 ∈ wcel 2101 ∃wrex 3068 class class class wbr 5077 {copab 5139 ‘cfv 6447 (class class class)co 7295 0cc0 10899 ℤcz 12347 ...cfz 13267 ♯chash 14072 Word cword 14245 cyclShift ccsh 14529 Vtxcvtx 27394 ClWWalksN cclwwlkn 28416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-inf 9230 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-n0 12262 df-z 12348 df-uz 12611 df-rp 12759 df-fz 13268 df-fzo 13411 df-fl 13540 df-mod 13618 df-hash 14073 df-word 14246 df-concat 14302 df-substr 14382 df-pfx 14412 df-csh 14530 df-clwwlk 28374 df-clwwlkn 28417 |
This theorem is referenced by: erclwwlknsym 28462 erclwwlkntr 28463 |
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