Theorem wlklenvclwlk 16170

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 4087	. . 3 |- ( F (Walks ` G ) ( W ++ <" ( W ` 0 ) "> ) <-> <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) )
2		wlkcl 16129	. . . 4 |- ( F (Walks ` G ) ( W ++ <" ( W ` 0 ) "> ) -> (♯ ` F ) e. NN0 )
3		wlklenvp1 16134	. . . 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 9480	. . . . . . . . 9 |- 0 e. ZZ
7		fvexg 5654	. . . . . . . . 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 11201	. . . . . . . 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 2238	. . . . . 6 |- ( W e. Word (Vtx ` G ) -> ( (♯ ` ( W ++ <" ( W ` 0 ) "> ) ) = ( (♯ ` F ) + 1 ) <-> ( (♯ ` W ) + 1 ) = ( (♯ ` F ) + 1 ) ) )
12		eqcom 2231	. . . . . 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 9402	. . . . . . 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 11107	. . . . . . . 8 |- ( W e. Word (Vtx ` G ) -> (♯ ` W ) e. NN0 )
18	17	nn0cnd 9447	. . . . . . 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 8185	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` F ) e. NN0 ) -> 1 e. CC )
21	16, 19, 20	addcan2d 8354	. . . . 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 1395   e. wcel 2200   _Vcvv 2800   <.cop 3670   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   0cc0 8022   1c1 8023   + caddc 8025   NN0cn0 9392   ZZcz 9469  ♯chash 11027  Word cword 11103   ++ cconcat 11157   <"cs1 11182  Vtxcvtx 15853  Walkscwlks 16114

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-map 6814  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-dec 9602  df-uz 9746  df-fz 10234  df-fzo 10368  df-ihash 11028  df-word 11104  df-concat 11158  df-s1 11183  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856  df-wlks 16115

This theorem is referenced by: (None)