Theorem wlklenvclwlk 16494

Description: The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)

Assertion

Ref	Expression
wlklenvclwlk	|- ( W e. Word (Vtx ` G ) -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) -> (♯ ` F ) = (♯ ` W ) ) )

Proof of Theorem wlklenvclwlk

Step	Hyp	Ref	Expression
1		df-br 4115	. . 3 |- ( F (Walks ` G ) ( W ++ <" ( W ` 0 ) "> ) <-> <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) )
2		wlkcl 16453	. . . 4 |- ( F (Walks ` G ) ( W ++ <" ( W ` 0 ) "> ) -> (♯ ` F ) e. NN0 )
3		wlklenvp1 16458	. . . 4 |- ( F (Walks ` G ) ( W ++ <" ( W ` 0 ) "> ) -> (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) )
4	2, 3	jca 306	. . 3 |- ( F (Walks ` G ) ( W ++ <" ( W ` 0 ) "> ) -> ( (♯ ` F ) e. NN0 /\ (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) ) )
5	1, 4	sylbir 135	. 2 |- ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) -> ( (♯ ` F ) e. NN0 /\ (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) ) )
6		0z 9605	. . . . . . . . 9 |- 0 e. ZZ
7		fvexg 5694	. . . . . . . . 9 |- ( ( W e. Word (Vtx ` G ) /\ 0 e. ZZ ) -> ( W ` 0 ) e. _V )
8	6, 7	mpan2 425	. . . . . . . 8 |- ( W e. Word (Vtx ` G ) -> ( W ` 0 ) e. _V )
9		ccatws1leng 11347	. . . . . . . 8 |- ( ( W e. Word (Vtx ` G ) /\ ( W ` 0 ) e. _V ) -> (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` W ) + 1 ) )
10	8, 9	mpdan 421	. . . . . . 7 |- ( W e. Word (Vtx ` G ) -> (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` W ) + 1 ) )
11	10	eqeq1d 2243	. . . . . 6 |- ( W e. Word (Vtx ` G ) -> ( (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) <-> ( (♯ ` W ) + 1 ) = ( (♯ ` F ) + 1 ) ) )
12		eqcom 2236	. . . . . 6 |- ( ( (♯ ` W ) + 1 ) = ( (♯ ` F ) + 1 ) <-> ( (♯ ` F ) + 1 ) = ( (♯ ` W ) + 1 ) )
13	11, 12	bitrdi 196	. . . . 5 |- ( W e. Word (Vtx ` G ) -> ( (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) <-> ( (♯ ` F ) + 1 ) = ( (♯ ` W ) + 1 ) ) )
14	13	adantr 276	. . . 4 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> ( (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) <-> ( (♯ ` F ) + 1 ) = ( (♯ ` W ) + 1 ) ) )
15		nn0cn 9523	. . . . . . 7 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) e. CC )
16	15	adantl 277	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> (♯ ` F ) e. CC )
17		lencl 11253	. . . . . . . 8 |- ( W e. Word (Vtx ` G ) -> (♯ ` W ) e. NN0 )
18	17	nn0cnd 9572	. . . . . . 7 |- ( W e. Word (Vtx ` G ) -> (♯ ` W ) e. CC )
19	18	adantr 276	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> (♯ ` W ) e. CC )
20		1cnd 8306	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> 1 e. CC )
21	16, 19, 20	addcan2d 8474	. . . . 5 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> ( ( (♯ ` F ) + 1 ) = ( (♯ ` W ) + 1 ) <-> (♯ ` F ) = (♯ ` W ) ) )
22	21	biimpd 144	. . . 4 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> ( ( (♯ ` F ) + 1 ) = ( (♯ ` W ) + 1 ) -> (♯ ` F ) = (♯ ` W ) ) )
23	14, 22	sylbid 150	. . 3 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> ( (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) -> (♯ ` F ) = (♯ ` W ) ) )
24	23	expimpd 363	. 2 |- ( W e. Word (Vtx ` G ) -> ( ( (♯ ` F ) e. NN0 /\ (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) ) -> (♯ ` F ) = (♯ ` W ) ) )
25	5, 24	syl5 32	1 |- ( W e. Word (Vtx ` G ) -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) -> (♯ ` F ) = (♯ ` W ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   <.cop 3697   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   NN0cn0 9513   ZZcz 9594  ♯chash 11163  Word cword 11249   ++ cconcat 11303   <"cs1 11328  Vtxcvtx 16133  Walkscwlks 16438

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-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-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  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-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-s1 11329  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-wlks 16439

This theorem is referenced by: (None)