Theorem clwwlkn1 16573

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 9294	. . 3 |- 1 e. NN
2		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2238	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlknx 16571	. . 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 1013	. . . 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 3626	. . . . . . . 8 |- A. i e. (/) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G )
8		oveq1 6082	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = ( 1 - 1 ) )
9		1m1e0 9352	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
10	8, 9	eqtrdi 2287	. . . . . . . . . . . 12 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = 0 )
11	10	oveq2d 6091	. . . . . . . . . . 11 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = ( 0..^ 0 ) )
12		fzo0 10555	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
13	11, 12	eqtrdi 2287	. . . . . . . . . 10 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = (/) )
14	13	raleqdv 2755	. . . . . . . . 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 11332	. . . . . . . . . 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 3790	. . . . . . . 8 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) } )
21		dfsn2 3719	. . . . . . . 8 |- { ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) }
22	20, 21	eqtr4di 2289	. . . . . . 7 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) } )
23	22	eleq1d 2307	. . . . . 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 456	. . . 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 459	. 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 1013	. . 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 1009   = wceq 1402   e. wcel 2209   A.wral 2528   (/)c0 3520   {csn 3705   {cpr 3706   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487   NNcn 9283  ..^cfzo 10527  ♯chash 11192  Word cword 11282  lastSclsw 11327  Vtxcvtx 16167  Edgcedg 16212   ClWWalksN cclwwlkn 16558

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-lsw 11328  df-ndx 13333  df-slot 13334  df-base 13336  df-vtx 16169  df-clwwlk 16547  df-clwwlkn 16559

This theorem is referenced by:  loopclwwlkn1b  16574  clwwlkn1loopb  16575