Theorem clwwlkn1 16359

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 9213	. . 3 |- 1 e. NN
2		eqid 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
3		eqid 2231	. . . 4 |- (Edg ` G ) = (Edg ` G )
4	2, 3	isclwwlknx 16357	. . 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 1009	. . . 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 3598	. . . . . . . 8 |- A. i e. (/) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G )
8		oveq1 6035	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = ( 1 - 1 ) )
9		1m1e0 9271	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
10	8, 9	eqtrdi 2280	. . . . . . . . . . . 12 |- ( (♯ ` W ) = 1 -> ( (♯ ` W ) - 1 ) = 0 )
11	10	oveq2d 6044	. . . . . . . . . . 11 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = ( 0..^ 0 ) )
12		fzo0 10467	. . . . . . . . . . 11 |- ( 0..^ 0 ) = (/)
13	11, 12	eqtrdi 2280	. . . . . . . . . 10 |- ( (♯ ` W ) = 1 -> ( 0..^ ( (♯ ` W ) - 1 ) ) = (/) )
14	13	raleqdv 2737	. . . . . . . . 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 11229	. . . . . . . . . 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 3758	. . . . . . . 8 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` 0 ) , ( W ` 0 ) } )
21		dfsn2 3687	. . . . . . . 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 1009	. . 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 1005   = wceq 1398   e. wcel 2202   A.wral 2511   (/)c0 3496   {csn 3673   {cpr 3674   ` cfv 5333  (class class class)co 6028   0cc0 8092   1c1 8093   + caddc 8095   - cmin 8409   NNcn 9202  ..^cfzo 10439  ♯chash 11100  Word cword 11179  lastSclsw 11224  Vtxcvtx 15953  Edgcedg 15998   ClWWalksN cclwwlkn 16344

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-lsw 11225  df-ndx 13165  df-slot 13166  df-base 13168  df-vtx 15955  df-clwwlk 16333  df-clwwlkn 16345

This theorem is referenced by:  loopclwwlkn1b  16360  clwwlkn1loopb  16361