Theorem clwwlkextfrlem1 27208

Description: Lemma for numclwwlk2lem1 27235. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-May-2021.)

Assertion

Ref	Expression
clwwlkextfrlem1	|- ( ( ( N e. NN0 /\ X e. (Vtx ` G ) ) /\ ( W e. ( N WWalksN G ) /\ ( lastS ` W ) =/= ( W ` 0 ) ) ) -> ( ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) )

Proof of Theorem clwwlkextfrlem1

Step	Hyp	Ref	Expression
1		wwlknbp2 26752	. . . 4 |- ( W e. ( N WWalksN G ) -> ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) )
2		simpll 790	. . . . . . 7 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> W e. Word (Vtx ` G ) )
3		s1cl 13382	. . . . . . . . . 10 |- ( X e. (Vtx ` G ) -> <" X "> e. Word (Vtx ` G ) )
4	3	adantl 482	. . . . . . . . 9 |- ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> <" X "> e. Word (Vtx ` G ) )
5	4	adantl 482	. . . . . . . 8 |- ( ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> <" X "> e. Word (Vtx ` G ) )
6	5	adantl 482	. . . . . . 7 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> <" X "> e. Word (Vtx ` G ) )
7		nn0p1gt0 11322	. . . . . . . . . . 11 |- ( N e. NN0 -> 0 < ( N + 1 ) )
8	7	adantr 481	. . . . . . . . . 10 |- ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> 0 < ( N + 1 ) )
9	8	adantl 482	. . . . . . . . 9 |- ( ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> 0 < ( N + 1 ) )
10	9	adantl 482	. . . . . . . 8 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> 0 < ( N + 1 ) )
11		breq2 4657	. . . . . . . . . 10 |- ( ( # ` W ) = ( N + 1 ) -> ( 0 < ( # ` W ) <-> 0 < ( N + 1 ) ) )
12	11	adantl 482	. . . . . . . . 9 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( 0 < ( # ` W ) <-> 0 < ( N + 1 ) ) )
13	12	adantr 481	. . . . . . . 8 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> ( 0 < ( # ` W ) <-> 0 < ( N + 1 ) ) )
14	10, 13	mpbird 247	. . . . . . 7 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> 0 < ( # ` W ) )
15		ccatfv0 13367	. . . . . . 7 |- ( ( W e. Word (Vtx ` G ) /\ <" X "> e. Word (Vtx ` G ) /\ 0 < ( # ` W ) ) -> ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) )
16	2, 6, 14, 15	syl3anc 1326	. . . . . 6 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) )
17		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) = ( N + 1 ) -> ( ( # ` W ) - 1 ) = ( ( N + 1 ) - 1 ) )
18	17	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( ( # ` W ) - 1 ) = ( ( N + 1 ) - 1 ) )
19	18	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( ( # ` W ) - 1 ) = ( ( N + 1 ) - 1 ) )
20		nn0cn 11302	. . . . . . . . . . . . . . . . 17 |- ( N e. NN0 -> N e. CC )
21		pncan1 10454	. . . . . . . . . . . . . . . . 17 |- ( N e. CC -> ( ( N + 1 ) - 1 ) = N )
22	20, 21	syl 17	. . . . . . . . . . . . . . . 16 |- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N )
23	22	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( N + 1 ) - 1 ) = N )
24	23	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( ( N + 1 ) - 1 ) = N )
25	19, 24	eqtr2d 2657	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> N = ( ( # ` W ) - 1 ) )
26	25	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( ( W ++ <" X "> ) ` N ) = ( ( W ++ <" X "> ) ` ( ( # ` W ) - 1 ) ) )
27		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> W e. Word (Vtx ` G ) )
28	4	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> <" X "> e. Word (Vtx ` G ) )
29	8	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> 0 < ( N + 1 ) )
30	12	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( 0 < ( # ` W ) <-> 0 < ( N + 1 ) ) )
31	29, 30	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> 0 < ( # ` W ) )
32		hashneq0 13155	. . . . . . . . . . . . . . . . 17 |- ( W e. Word (Vtx ` G ) -> ( 0 < ( # ` W ) <-> W =/= (/) ) )
33	32	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( W e. Word (Vtx ` G ) -> ( W =/= (/) <-> 0 < ( # ` W ) ) )
34	33	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( W =/= (/) <-> 0 < ( # ` W ) ) )
35	34	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( W =/= (/) <-> 0 < ( # ` W ) ) )
36	31, 35	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> W =/= (/) )
37		ccatval1lsw 13368	. . . . . . . . . . . . 13 |- ( ( W e. Word (Vtx ` G ) /\ <" X "> e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( ( W ++ <" X "> ) ` ( ( # ` W ) - 1 ) ) = ( lastS ` W ) )
38	27, 28, 36, 37	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( ( W ++ <" X "> ) ` ( ( # ` W ) - 1 ) ) = ( lastS ` W ) )
39	26, 38	eqtr2d 2657	. . . . . . . . . . 11 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( lastS ` W ) = ( ( W ++ <" X "> ) ` N ) )
40	39	neeq1d 2853	. . . . . . . . . 10 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( ( lastS ` W ) =/= ( W ` 0 ) <-> ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) )
41	40	biimpd 219	. . . . . . . . 9 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) -> ( ( lastS ` W ) =/= ( W ` 0 ) -> ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) )
42	41	ex 450	. . . . . . . 8 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( lastS ` W ) =/= ( W ` 0 ) -> ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) ) )
43	42	com23 86	. . . . . . 7 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( ( lastS ` W ) =/= ( W ` 0 ) -> ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) ) )
44	43	imp32 449	. . . . . 6 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) )
45	16, 44	jca 554	. . . . 5 |- ( ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) /\ ( ( lastS ` W ) =/= ( W ` 0 ) /\ ( N e. NN0 /\ X e. (Vtx ` G ) ) ) ) -> ( ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) )
46	45	exp32 631	. . . 4 |- ( ( W e. Word (Vtx ` G ) /\ ( # ` W ) = ( N + 1 ) ) -> ( ( lastS ` W ) =/= ( W ` 0 ) -> ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) ) ) )
47	1, 46	syl 17	. . 3 |- ( W e. ( N WWalksN G ) -> ( ( lastS ` W ) =/= ( W ` 0 ) -> ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) ) ) )
48	47	imp 445	. 2 |- ( ( W e. ( N WWalksN G ) /\ ( lastS ` W ) =/= ( W ` 0 ) ) -> ( ( N e. NN0 /\ X e. (Vtx ` G ) ) -> ( ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) ) )
49	48	impcom 446	1 |- ( ( ( N e. NN0 /\ X e. (Vtx ` G ) ) /\ ( W e. ( N WWalksN G ) /\ ( lastS ` W ) =/= ( W ` 0 ) ) ) -> ( ( ( W ++ <" X "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" X "> ) ` N ) =/= ( W ` 0 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NN0cn0 11292   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294  Vtxcvtx 25874   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-xnn0 11364  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-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  numclwwlk2lem1  27235