Theorem clwwlknonex2e 16847

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 16846	. 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 16837	. . . . 5 |- ( W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) <-> ( W e. ( ( N - 2 ) ClWWalksN G ) /\ ( W ` 0 ) = X ) )
5		isclwwlkn 16820	. . . . . . . . . 10 |- ( W e. ( ( N - 2 ) ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = ( N - 2 ) ) )
6	1	clwwlkbp 16802	. . . . . . . . . . . . 13 |- ( W e. (ClWWalks ` G ) -> ( G e. _V /\ W e. Word V /\ W =/= (/) ) )
7	6	simp2d 1041	. . . . . . . . . . . 12 |- ( W e. (ClWWalks ` G ) -> W e. Word V )
8		clwwlkgt0 16803	. . . . . . . . . . . 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 494	. . . . . . . 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 1031	. . . . . . . 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 1032	. . . . . . . 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 11432	. . . . . . . 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 1276	. . . . . . 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 537	. . . . . . 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 2271	. . . . . 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 1224	. 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 16837	. 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 421	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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   {cpr 3710   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   0cc0 8180   < clt 8361   - cmin 8499   2c2 9358   3c3 9359   ZZ>=cuz 9931  ♯chash 11230  Word cword 11320   ++ cconcat 11374   <"cs1 11399  Vtxcvtx 16419  Edgcedg 16464  ClWWalkscclwwlk 16798   ClWWalksN cclwwlkn 16810  ClWWalksNOncclwwlknon 16833

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-inn 9308  df-2 9366  df-3 9367  df-n0 9569  df-z 9650  df-uz 9932  df-fz 10423  df-fzo 10561  df-ihash 11231  df-word 11321  df-lsw 11366  df-concat 11375  df-s1 11400  df-ndx 13407  df-slot 13408  df-base 13410  df-vtx 16421  df-clwwlk 16799  df-clwwlkn 16811  df-clwwlknon 16834

This theorem is used by: (None)