Theorem clwwlkn1 16268

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 9153	. . 3 |- 1 e. NN
2		eqid 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2231	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlknx 16266	. . 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 1008	. . . 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 3596	. . . . . . . 8 |- A. i e. (/) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G )
8		oveq1 6024	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = ( 1 - 1 ) )
9		1m1e0 9211	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
10	8, 9	eqtrdi 2280	. . . . . . . . . . . 12 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = 0 )
11	10	oveq2d 6033	. . . . . . . . . . 11 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = ( 0..^ 0 ) )
12		fzo0 10404	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
13	11, 12	eqtrdi 2280	. . . . . . . . . 10 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = (/) )
14	13	raleqdv 2736	. . . . . . . . 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 11162	. . . . . . . . . 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 3754	. . . . . . . 8 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) } )
21		dfsn2 3683	. . . . . . . 8 |- { ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) }
22	20, 21	eqtr4di 2282	. . . . . . 7 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) } )
23	22	eleq1d 2300	. . . . . 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 1008	. . 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   (/)c0 3494   {csn 3669   {cpr 3670   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   - cmin 8349   NNcn 9142  ..^cfzo 10376  ♯chash 11036  Word cword 11112  lastSclsw 11157  Vtxcvtx 15862  Edgcedg 15907   ClWWalksN cclwwlkn 16253

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-lsw 11158  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864  df-clwwlk 16242  df-clwwlkn 16254

This theorem is referenced by:  loopclwwlkn1b  16269  clwwlkn1loopb  16270