Theorem clwwlknonex2e 16561

Description: Extending a closed walk W on vertex X by an additional edge (forth and back) results in a closed walk on vertex X. (Contributed by AV, 17-Apr-2022.)

Hypotheses

Ref	Expression
clwwlknonex2.v	|- V = (Vtx ` G )
clwwlknonex2.e	|- E = (Edg ` G )

Assertion

Ref	Expression
clwwlknonex2e	|- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X (ClWWalksNOn ` G ) N ) )

Proof of Theorem clwwlknonex2e

Step	Hyp	Ref	Expression
1		clwwlknonex2.v	. . 3 |- V = (Vtx ` G )
2		clwwlknonex2.e	. . 3 |- E = (Edg ` G )
3	1, 2	clwwlknonex2 16560	. 2 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )
4		isclwwlknon 16551	. . . . 5 |- ( W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) <-> ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) )
5		isclwwlkn 16534	. . . . . . . . . 10 |- ( W e. ( ( N - 2 ) ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = ( N - 2 ) ) )
6	1	clwwlkbp 16516	. . . . . . . . . . . . 13 |- ( W e. (ClWWalks ` G ) -> ( G e. _V /\ W e. Word V /\ W =/= (/) ) )
7	6	simp2d 1037	. . . . . . . . . . . 12 |- ( W e. (ClWWalks ` G ) -> W e. Word V )
8		clwwlkgt0 16517	. . . . . . . . . . . 12 |- ( W e. (ClWWalks ` G ) -> 0 < (♯ ` W ) )
9	7, 8	jca 306	. . . . . . . . . . 11 |- ( W e. (ClWWalks ` G ) -> ( W e. Word V /\ 0 < (♯ ` W ) ) )
10	9	adantr 276	. . . . . . . . . 10 |- ( ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( W e. Word V /\ 0 < (♯ ` W ) ) )
11	5, 10	sylbi 121	. . . . . . . . 9 |- ( W e. ( ( N - 2 ) ClWWalksN G ) -> ( W e. Word V /\ 0 < (♯ ` W ) ) )
12	11	ad2antrl 490	. . . . . . . 8 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) ) -> ( W e. Word V /\ 0 < (♯ ` W ) ) )
13		simpl1 1027	. . . . . . . 8 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) ) -> X e. V )
14		simpl2 1028	. . . . . . . 8 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) ) -> Y e. V )
15		ccat2s1fstg 11361	. . . . . . . 8 |- ( ( ( W e. Word V /\ 0 < (♯ ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
16	12, 13, 14, 15	syl12anc 1272	. . . . . . 7 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
17		simprr 533	. . . . . . 7 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) ) -> ( W ` 0 ) = X )
18	16, 17	eqtrd 2267	. . . . . 6 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = X )
19	18	ex 115	. . . . 5 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = X ) )
20	4, 19	biimtrid 152	. . . 4 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = X ) )
21	20	a1d 22	. . 3 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> ( W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = X ) ) )
22	21	3imp 1220	. 2 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = X )
23		isclwwlknon 16551	. 2 |- ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X (ClWWalksNOn ` G ) N ) <-> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = X ) )
24	3, 22, 23	sylanbrc 417	1 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X (ClWWalksNOn ` G ) N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   (/)c0 3512   {cpr 3695   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   < clt 8324   - cmin 8460   2c2 9305   3c3 9306   ZZ>=cuz 9871  ♯chash 11163  Word cword 11249   ++ cconcat 11303   <"cs1 11328  Vtxcvtx 16133  Edgcedg 16178  ClWWalkscclwwlk 16512   ClWWalksN cclwwlkn 16524  ClWWalksNOncclwwlknon 16547

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-2 9313  df-3 9314  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-concat 11304  df-s1 11329  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlk 16513  df-clwwlkn 16525  df-clwwlknon 16548

This theorem is referenced by: (None)