Theorem clwwlkn1 16160

Description: A closed walk of length 1 represented as word is a word consisting of 1 symbol representing a vertex connected to itself by (at least) one edge, that is, a loop. (Contributed by AV, 24-Apr-2021.) (Revised by AV, 11-Feb-2022.)

Assertion

Ref	Expression
clwwlkn1	|- ( W e. ( 1 ClWWalksN G ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )

Proof of Theorem clwwlkn1

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn 9132	. . 3 |- 1 e. NN
2		eqid 2229	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2229	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlknx 16158	. . 3 |- ( 1 e. NN -> ( W e. ( 1 ClWWalksN G ) <-> ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ (♯ ` W ) = 1 ) ) )
5	1, 4	ax-mp 5	. 2 |- ( W e. ( 1 ClWWalksN G ) <-> ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ (♯ ` W ) = 1 ) )
6		3anass 1006	. . . 4 |- ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( W e. Word (Vtx ` G ) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) ) )
7		ral0 3593	. . . . . . . 8 |- A. i e. (/) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G )
8		oveq1 6014	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = ( 1 - 1 ) )
9		1m1e0 9190	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
10	8, 9	eqtrdi 2278	. . . . . . . . . . . 12 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = 0 )
11	10	oveq2d 6023	. . . . . . . . . . 11 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = ( 0..^ 0 ) )
12		fzo0 10378	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
13	11, 12	eqtrdi 2278	. . . . . . . . . 10 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = (/) )
14	13	raleqdv 2734	. . . . . . . . 9 |- ( (♯ ` W ) = 1 -> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. (/) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) ) )
15	14	adantr 276	. . . . . . . 8 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> A. i e. (/) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) ) )
16	7, 15	mpbiri 168	. . . . . . 7 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) )
17	16	biantrurd 305	. . . . . 6 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> ( { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) <-> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) ) )
18		lsw1 11134	. . . . . . . . . 10 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = 1 ) -> (lastS ` W ) = ( W ` 0 ) )
19	18	ancoms 268	. . . . . . . . 9 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> (lastS ` W ) = ( W ` 0 ) )
20	19	preq1d 3749	. . . . . . . 8 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) } )
21		dfsn2 3680	. . . . . . . 8 |- { ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) }
22	20, 21	eqtr4di 2280	. . . . . . 7 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) } )
23	22	eleq1d 2298	. . . . . 6 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> ( { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) <-> { ( W ` 0 ) } e. (Edg ` G ) ) )
24	17, 23	bitr3d 190	. . . . 5 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> ( ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> { ( W ` 0 ) } e. (Edg ` G ) ) )
25	24	pm5.32da 452	. . . 4 |- ( (♯ ` W ) = 1 -> ( ( W e. Word (Vtx ` G ) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) ) <-> ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) ) )
26	6, 25	bitrid 192	. . 3 |- ( (♯ ` W ) = 1 -> ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) ) )
27	26	pm5.32ri 455	. 2 |- ( ( ( W e. Word (Vtx ` G ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ (♯ ` W ) = 1 ) <-> ( ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) /\ (♯ ` W ) = 1 ) )
28		3anass 1006	. . 3 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) <-> ( (♯ ` W ) = 1 /\ ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) ) )
29		ancom 266	. . 3 |- ( ( (♯ ` W ) = 1 /\ ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) ) <-> ( ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) /\ (♯ ` W ) = 1 ) )
30	28, 29	bitr2i 185	. 2 |- ( ( ( W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) /\ (♯ ` W ) = 1 ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )
31	5, 27, 30	3bitri 206	1 |- ( W e. ( 1 ClWWalksN G ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   (/)c0 3491   {csn 3666   {cpr 3667   ` cfv 5318  (class class class)co 6007   0cc0 8010   1c1 8011   + caddc 8013   - cmin 8328   NNcn 9121  ..^cfzo 10350  ♯chash 11009  Word cword 11084  lastSclsw 11129  Vtxcvtx 15829  Edgcedg 15874   ClWWalksN cclwwlkn 16146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-lsw 11130  df-ndx 13051  df-slot 13052  df-base 13054  df-vtx 15831  df-clwwlk 16135  df-clwwlkn 16147

This theorem is referenced by:  loopclwwlkn1b  16161  clwwlkn1loopb  16162