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Mirrors > Home > MPE Home > Th. List > clwwlknon1loop | Structured version Visualization version GIF version |
Description: If there is a loop at vertex π, the set of (closed) walks on π of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of π. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon1.v | β’ π = (VtxβπΊ) |
clwwlknon1.c | β’ πΆ = (ClWWalksNOnβπΊ) |
clwwlknon1.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
clwwlknon1loop | β’ ((π β π β§ {π} β πΈ) β (ππΆ1) = {β¨βπββ©}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 770 | . . . 4 β’ ((π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ)) β π€ = β¨βπββ©) | |
2 | s1cl 14499 | . . . . . . . . 9 β’ (π β π β β¨βπββ© β Word π) | |
3 | 2 | adantr 482 | . . . . . . . 8 β’ ((π β π β§ {π} β πΈ) β β¨βπββ© β Word π) |
4 | 3 | adantr 482 | . . . . . . 7 β’ (((π β π β§ {π} β πΈ) β§ π€ = β¨βπββ©) β β¨βπββ© β Word π) |
5 | eleq1 2822 | . . . . . . . 8 β’ (π€ = β¨βπββ© β (π€ β Word π β β¨βπββ© β Word π)) | |
6 | 5 | adantl 483 | . . . . . . 7 β’ (((π β π β§ {π} β πΈ) β§ π€ = β¨βπββ©) β (π€ β Word π β β¨βπββ© β Word π)) |
7 | 4, 6 | mpbird 257 | . . . . . 6 β’ (((π β π β§ {π} β πΈ) β§ π€ = β¨βπββ©) β π€ β Word π) |
8 | simpr 486 | . . . . . . 7 β’ ((π β π β§ {π} β πΈ) β {π} β πΈ) | |
9 | 8 | anim1ci 617 | . . . . . 6 β’ (((π β π β§ {π} β πΈ) β§ π€ = β¨βπββ©) β (π€ = β¨βπββ© β§ {π} β πΈ)) |
10 | 7, 9 | jca 513 | . . . . 5 β’ (((π β π β§ {π} β πΈ) β§ π€ = β¨βπββ©) β (π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ))) |
11 | 10 | ex 414 | . . . 4 β’ ((π β π β§ {π} β πΈ) β (π€ = β¨βπββ© β (π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ)))) |
12 | 1, 11 | impbid2 225 | . . 3 β’ ((π β π β§ {π} β πΈ) β ((π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ)) β π€ = β¨βπββ©)) |
13 | 12 | alrimiv 1931 | . 2 β’ ((π β π β§ {π} β πΈ) β βπ€((π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ)) β π€ = β¨βπββ©)) |
14 | clwwlknon1.v | . . . . . 6 β’ π = (VtxβπΊ) | |
15 | clwwlknon1.c | . . . . . 6 β’ πΆ = (ClWWalksNOnβπΊ) | |
16 | clwwlknon1.e | . . . . . 6 β’ πΈ = (EdgβπΊ) | |
17 | 14, 15, 16 | clwwlknon1 29090 | . . . . 5 β’ (π β π β (ππΆ1) = {π€ β Word π β£ (π€ = β¨βπββ© β§ {π} β πΈ)}) |
18 | 17 | eqeq1d 2735 | . . . 4 β’ (π β π β ((ππΆ1) = {β¨βπββ©} β {π€ β Word π β£ (π€ = β¨βπββ© β§ {π} β πΈ)} = {β¨βπββ©})) |
19 | 18 | adantr 482 | . . 3 β’ ((π β π β§ {π} β πΈ) β ((ππΆ1) = {β¨βπββ©} β {π€ β Word π β£ (π€ = β¨βπββ© β§ {π} β πΈ)} = {β¨βπββ©})) |
20 | rabeqsn 4631 | . . 3 β’ ({π€ β Word π β£ (π€ = β¨βπββ© β§ {π} β πΈ)} = {β¨βπββ©} β βπ€((π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ)) β π€ = β¨βπββ©)) | |
21 | 19, 20 | bitrdi 287 | . 2 β’ ((π β π β§ {π} β πΈ) β ((ππΆ1) = {β¨βπββ©} β βπ€((π€ β Word π β§ (π€ = β¨βπββ© β§ {π} β πΈ)) β π€ = β¨βπββ©))) |
22 | 13, 21 | mpbird 257 | 1 β’ ((π β π β§ {π} β πΈ) β (ππΆ1) = {β¨βπββ©}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 βwal 1540 = wceq 1542 β wcel 2107 {crab 3406 {csn 4590 βcfv 6500 (class class class)co 7361 1c1 11060 Word cword 14411 β¨βcs1 14492 Vtxcvtx 27996 Edgcedg 28047 ClWWalksNOncclwwlknon 29080 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-lsw 14460 df-s1 14493 df-clwwlk 28975 df-clwwlkn 29018 df-clwwlknon 29081 |
This theorem is referenced by: clwwlknon1sn 29093 clwwlknon1le1 29094 |
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