Theorem wwlksnextbi 26789

Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
wwlksnextbi	|- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( W e. ( ( N + 1 ) WWalksN G ) <-> T e. ( N WWalksN G ) ) )

Proof of Theorem wwlksnextbi

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnext.v	. . . . 5 |- V = (Vtx ` G )
2		wwlksnext.e	. . . . 5 |- E = (Edg ` G )
3	1, 2	wwlknp 26734	. . . 4 |- ( W e. ( ( N + 1 ) WWalksN G ) -> ( W e. Word V /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) )
4		wwlksnred 26787	. . . . . . . 8 |- ( N e. NN0 -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) ) )
5	4	ad2antrr 762	. . . . . . 7 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) ) )
6		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( W = ( T ++ <" S "> ) -> ( # ` W ) = ( # ` ( T ++ <" S "> ) ) )
7	6	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( W = ( T ++ <" S "> ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) <-> ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) ) )
8	7	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) <-> ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) ) )
9	8	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) <-> ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) ) )
10		s1cl 13382	. . . . . . . . . . . . . . . . . . . . . 22 |- ( S e. V -> <" S "> e. Word V )
11	10	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( N e. NN0 /\ S e. V ) -> <" S "> e. Word V )
12	11	anim2i 593	. . . . . . . . . . . . . . . . . . . 20 |- ( ( T e. Word V /\ ( N e. NN0 /\ S e. V ) ) -> ( T e. Word V /\ <" S "> e. Word V ) )
13	12	ancoms 469	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( T e. Word V /\ <" S "> e. Word V ) )
14		ccatlen 13360	. . . . . . . . . . . . . . . . . . 19 |- ( ( T e. Word V /\ <" S "> e. Word V ) -> ( # ` ( T ++ <" S "> ) ) = ( ( # ` T ) + ( # ` <" S "> ) ) )
15	13, 14	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( # ` ( T ++ <" S "> ) ) = ( ( # ` T ) + ( # ` <" S "> ) ) )
16	15	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) <-> ( ( # ` T ) + ( # ` <" S "> ) ) = ( ( N + 1 ) + 1 ) ) )
17		s1len 13385	. . . . . . . . . . . . . . . . . . . 20 |- ( # ` <" S "> ) = 1
18	17	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( # ` <" S "> ) = 1 )
19	18	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( # ` T ) + ( # ` <" S "> ) ) = ( ( # ` T ) + 1 ) )
20	19	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( ( # ` T ) + ( # ` <" S "> ) ) = ( ( N + 1 ) + 1 ) <-> ( ( # ` T ) + 1 ) = ( ( N + 1 ) + 1 ) ) )
21		lencl 13324	. . . . . . . . . . . . . . . . . . . 20 |- ( T e. Word V -> ( # ` T ) e. NN0 )
22	21	nn0cnd 11353	. . . . . . . . . . . . . . . . . . 19 |- ( T e. Word V -> ( # ` T ) e. CC )
23	22	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( # ` T ) e. CC )
24		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
25	24	nn0cnd 11353	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> ( N + 1 ) e. CC )
26	25	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( N + 1 ) e. CC )
27		1cnd 10056	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> 1 e. CC )
28	23, 26, 27	addcan2d 10240	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( ( # ` T ) + 1 ) = ( ( N + 1 ) + 1 ) <-> ( # ` T ) = ( N + 1 ) ) )
29	16, 20, 28	3bitrd 294	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) <-> ( # ` T ) = ( N + 1 ) ) )
30		opeq2 4403	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N + 1 ) = ( # ` T ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( # ` T ) >. )
31	30	eqcoms 2630	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` T ) = ( N + 1 ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( # ` T ) >. )
32	31	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` T ) = ( N + 1 ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = ( ( T ++ <" S "> ) substr <. 0 , ( # ` T ) >. ) )
33		swrdccat1 13457	. . . . . . . . . . . . . . . . . . 19 |- ( ( T e. Word V /\ <" S "> e. Word V ) -> ( ( T ++ <" S "> ) substr <. 0 , ( # ` T ) >. ) = T )
34	13, 33	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( T ++ <" S "> ) substr <. 0 , ( # ` T ) >. ) = T )
35	32, 34	sylan9eqr 2678	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) /\ ( # ` T ) = ( N + 1 ) ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T )
36	35	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( # ` T ) = ( N + 1 ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
37	29, 36	sylbid 230	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN0 /\ S e. V ) /\ T e. Word V ) -> ( ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
38	37	3ad2antr1 1226	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( ( # ` ( T ++ <" S "> ) ) = ( ( N + 1 ) + 1 ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
39	9, 38	sylbid 230	. . . . . . . . . . . . 13 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
40	39	imp 445	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T )
41		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( W = ( T ++ <" S "> ) -> ( W substr <. 0 , ( N + 1 ) >. ) = ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) )
42	41	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( W = ( T ++ <" S "> ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) = T <-> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
43	42	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) = T <-> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
44	43	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) = T <-> ( ( T ++ <" S "> ) substr <. 0 , ( N + 1 ) >. ) = T ) )
45	40, 44	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( W substr <. 0 , ( N + 1 ) >. ) = T )
46	45	eleq1d 2686	. . . . . . . . . 10 |- ( ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) <-> T e. ( N WWalksN G ) ) )
47	46	biimpd 219	. . . . . . . . 9 |- ( ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) /\ ( # ` W ) = ( ( N + 1 ) + 1 ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) -> T e. ( N WWalksN G ) ) )
48	47	ex 450	. . . . . . . 8 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) -> T e. ( N WWalksN G ) ) ) )
49	48	com23 86	. . . . . . 7 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( ( W substr <. 0 , ( N + 1 ) >. ) e. ( N WWalksN G ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> T e. ( N WWalksN G ) ) ) )
50	5, 49	syld 47	. . . . . 6 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> T e. ( N WWalksN G ) ) ) )
51	50	com13 88	. . . . 5 |- ( ( # ` W ) = ( ( N + 1 ) + 1 ) -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> T e. ( N WWalksN G ) ) ) )
52	51	3ad2ant2 1083	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = ( ( N + 1 ) + 1 ) /\ A. i e. ( 0..^ ( N + 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( W e. ( ( N + 1 ) WWalksN G ) -> ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> T e. ( N WWalksN G ) ) ) )
53	3, 52	mpcom 38	. . 3 |- ( W e. ( ( N + 1 ) WWalksN G ) -> ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> T e. ( N WWalksN G ) ) )
54	53	com12 32	. 2 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( W e. ( ( N + 1 ) WWalksN G ) -> T e. ( N WWalksN G ) ) )
55	1, 2	wwlksnext 26788	. . . . . . . . . . 11 |- ( ( T e. ( N WWalksN G ) /\ S e. V /\ { ( lastS ` T ) , S } e. E ) -> ( T ++ <" S "> ) e. ( ( N + 1 ) WWalksN G ) )
56		eleq1 2689	. . . . . . . . . . 11 |- ( W = ( T ++ <" S "> ) -> ( W e. ( ( N + 1 ) WWalksN G ) <-> ( T ++ <" S "> ) e. ( ( N + 1 ) WWalksN G ) ) )
57	55, 56	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( T e. ( N WWalksN G ) /\ S e. V /\ { ( lastS ` T ) , S } e. E ) -> ( W = ( T ++ <" S "> ) -> W e. ( ( N + 1 ) WWalksN G ) ) )
58	57	3exp 1264	. . . . . . . . 9 |- ( T e. ( N WWalksN G ) -> ( S e. V -> ( { ( lastS ` T ) , S } e. E -> ( W = ( T ++ <" S "> ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) ) )
59	58	com23 86	. . . . . . . 8 |- ( T e. ( N WWalksN G ) -> ( { ( lastS ` T ) , S } e. E -> ( S e. V -> ( W = ( T ++ <" S "> ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) ) )
60	59	com14 96	. . . . . . 7 |- ( W = ( T ++ <" S "> ) -> ( { ( lastS ` T ) , S } e. E -> ( S e. V -> ( T e. ( N WWalksN G ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) ) )
61	60	imp 445	. . . . . 6 |- ( ( W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) -> ( S e. V -> ( T e. ( N WWalksN G ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) )
62	61	3adant1 1079	. . . . 5 |- ( ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) -> ( S e. V -> ( T e. ( N WWalksN G ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) )
63	62	com12 32	. . . 4 |- ( S e. V -> ( ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) -> ( T e. ( N WWalksN G ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) )
64	63	adantl 482	. . 3 |- ( ( N e. NN0 /\ S e. V ) -> ( ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) -> ( T e. ( N WWalksN G ) -> W e. ( ( N + 1 ) WWalksN G ) ) ) )
65	64	imp 445	. 2 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( T e. ( N WWalksN G ) -> W e. ( ( N + 1 ) WWalksN G ) ) )
66	54, 65	impbid 202	1 |- ( ( ( N e. NN0 /\ S e. V ) /\ ( T e. Word V /\ W = ( T ++ <" S "> ) /\ { ( lastS ` T ) , S } e. E ) ) -> ( W e. ( ( N + 1 ) WWalksN G ) <-> T e. ( N WWalksN G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {cpr 4179   <.cop 4183   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnextwrd  26792