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Mirrors > Home > MPE Home > Th. List > erclwwlksnref | Structured version Visualization version GIF version |
Description: ∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlksn.w | ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) |
erclwwlksn.r | ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} |
Ref | Expression |
---|---|
erclwwlksnref | ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1039 | . . 3 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) ↔ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))) | |
2 | anidm 676 | . . . 4 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ↔ 𝑥 ∈ 𝑊) | |
3 | 2 | anbi1i 731 | . . 3 ⊢ (((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))) |
4 | 1, 3 | bitri 264 | . 2 ⊢ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))) |
5 | vex 3203 | . . 3 ⊢ 𝑥 ∈ V | |
6 | erclwwlksn.w | . . . 4 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) | |
7 | erclwwlksn.r | . . . 4 ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
8 | 6, 7 | erclwwlksneq 26944 | . . 3 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)))) |
9 | 5, 5, 8 | mp2an 708 | . 2 ⊢ (𝑥 ∼ 𝑥 ↔ (𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))) |
10 | eqid 2622 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
11 | 10 | clwwlknbp0 26884 | . . . . 5 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁))) |
12 | cshw0 13540 | . . . . . . . 8 ⊢ (𝑥 ∈ Word (Vtx‘𝐺) → (𝑥 cyclShift 0) = 𝑥) | |
13 | 12 | adantr 481 | . . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁) → (𝑥 cyclShift 0) = 𝑥) |
14 | 13 | adantl 482 | . . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) → (𝑥 cyclShift 0) = 𝑥) |
15 | nnnn0 11299 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
16 | 0elfz 12436 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
17 | 15, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 ∈ (0...𝑁)) |
18 | 17 | adantl 482 | . . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) → 0 ∈ (0...𝑁)) |
19 | 18 | adantr 481 | . . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) → 0 ∈ (0...𝑁)) |
20 | eqcom 2629 | . . . . . . . 8 ⊢ ((𝑥 cyclShift 0) = 𝑥 ↔ 𝑥 = (𝑥 cyclShift 0)) | |
21 | 20 | biimpi 206 | . . . . . . 7 ⊢ ((𝑥 cyclShift 0) = 𝑥 → 𝑥 = (𝑥 cyclShift 0)) |
22 | oveq2 6658 | . . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift 0)) | |
23 | 22 | eqeq2d 2632 | . . . . . . . 8 ⊢ (𝑛 = 0 → (𝑥 = (𝑥 cyclShift 𝑛) ↔ 𝑥 = (𝑥 cyclShift 0))) |
24 | 23 | rspcev 3309 | . . . . . . 7 ⊢ ((0 ∈ (0...𝑁) ∧ 𝑥 = (𝑥 cyclShift 0)) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) |
25 | 19, 21, 24 | syl2an 494 | . . . . . 6 ⊢ ((((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) ∧ (𝑥 cyclShift 0) = 𝑥) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) |
26 | 14, 25 | mpdan 702 | . . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = 𝑁)) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) |
27 | 11, 26 | syl 17 | . . . 4 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) |
28 | 27, 6 | eleq2s 2719 | . . 3 ⊢ (𝑥 ∈ 𝑊 → ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛)) |
29 | 28 | pm4.71i 664 | . 2 ⊢ (𝑥 ∈ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑥 = (𝑥 cyclShift 𝑛))) |
30 | 4, 9, 29 | 3bitr4ri 293 | 1 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 class class class wbr 4653 {copab 4712 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕcn 11020 ℕ0cn0 11292 ...cfz 12326 #chash 13117 Word cword 13291 cyclShift ccsh 13534 Vtxcvtx 25874 ClWWalksN cclwwlksn 26876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-csh 13535 df-clwwlks 26877 df-clwwlksn 26878 |
This theorem is referenced by: erclwwlksn 26949 |
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