Theorem rusgrnumwwlkb0 26866

Description: Induction base 0 for rusgrnumwwlk 26870. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Revised by AV, 7-May-2021.)

Hypotheses

Ref	Expression
rusgrnumwwlk.v	|- V = (Vtx ` G )
rusgrnumwwlk.l	|- L = ( v e. V , n e. NN0 |-> ( # ` { w e. ( n WWalksN G ) | ( w ` 0 ) = v } ) )

Assertion

Ref	Expression
rusgrnumwwlkb0	|- ( ( G e. USPGraph /\ P e. V ) -> ( P L 0 ) = 1 )

Distinct variable groups:   n, G, v, w   P, n, v, w   n, V, v, w

Allowed substitution hints:   L( w, v, n)

Proof of Theorem rusgrnumwwlkb0

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( G e. USPGraph /\ P e. V ) -> P e. V )
2		0nn0 11307	. . 3 |- 0 e. NN0
3		rusgrnumwwlk.v	. . . 4 |- V = (Vtx ` G )
4		rusgrnumwwlk.l	. . . 4 |- L = ( v e. V , n e. NN0 |-> ( # ` { w e. ( n WWalksN G ) | ( w ` 0 ) = v } ) )
5	3, 4	rusgrnumwwlklem 26865	. . 3 |- ( ( P e. V /\ 0 e. NN0 ) -> ( P L 0 ) = ( # ` { w e. ( 0 WWalksN G ) | ( w ` 0 ) = P } ) )
6	1, 2, 5	sylancl 694	. 2 |- ( ( G e. USPGraph /\ P e. V ) -> ( P L 0 ) = ( # ` { w e. ( 0 WWalksN G ) | ( w ` 0 ) = P } ) )
7		df-rab 2921	. . . . 5 |- { w e. ( 0 WWalksN G ) | ( w ` 0 ) = P } = { w | ( w e. ( 0 WWalksN G ) /\ ( w ` 0 ) = P ) }
8	7	a1i 11	. . . 4 |- ( ( G e. USPGraph /\ P e. V ) -> { w e. ( 0 WWalksN G ) | ( w ` 0 ) = P } = { w | ( w e. ( 0 WWalksN G ) /\ ( w ` 0 ) = P ) } )
9		wwlksn0s 26746	. . . . . . . . 9 |- ( 0 WWalksN G ) = { w e. Word (Vtx ` G ) | ( # ` w ) = 1 }
10	9	a1i 11	. . . . . . . 8 |- ( ( G e. USPGraph /\ P e. V ) -> ( 0 WWalksN G ) = { w e. Word (Vtx ` G ) | ( # ` w ) = 1 } )
11	10	eleq2d 2687	. . . . . . 7 |- ( ( G e. USPGraph /\ P e. V ) -> ( w e. ( 0 WWalksN G ) <-> w e. { w e. Word (Vtx ` G ) | ( # ` w ) = 1 } ) )
12		rabid 3116	. . . . . . 7 |- ( w e. { w e. Word (Vtx ` G ) | ( # ` w ) = 1 } <-> ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) )
13	11, 12	syl6bb 276	. . . . . 6 |- ( ( G e. USPGraph /\ P e. V ) -> ( w e. ( 0 WWalksN G ) <-> ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) ) )
14	13	anbi1d 741	. . . . 5 |- ( ( G e. USPGraph /\ P e. V ) -> ( ( w e. ( 0 WWalksN G ) /\ ( w ` 0 ) = P ) <-> ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) ) )
15	14	abbidv 2741	. . . 4 |- ( ( G e. USPGraph /\ P e. V ) -> { w | ( w e. ( 0 WWalksN G ) /\ ( w ` 0 ) = P ) } = { w | ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) } )
16		wrdl1s1 13394	. . . . . . . . 9 |- ( P e. (Vtx ` G ) -> ( v = <" P "> <-> ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 /\ ( v ` 0 ) = P ) ) )
17		df-3an 1039	. . . . . . . . 9 |- ( ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 /\ ( v ` 0 ) = P ) <-> ( ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 ) /\ ( v ` 0 ) = P ) )
18	16, 17	syl6rbb 277	. . . . . . . 8 |- ( P e. (Vtx ` G ) -> ( ( ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 ) /\ ( v ` 0 ) = P ) <-> v = <" P "> ) )
19		vex 3203	. . . . . . . . 9 |- v e. _V
20		eleq1 2689	. . . . . . . . . . 11 |- ( w = v -> ( w e. Word (Vtx ` G ) <-> v e. Word (Vtx ` G ) ) )
21		fveq2 6191	. . . . . . . . . . . 12 |- ( w = v -> ( # ` w ) = ( # ` v ) )
22	21	eqeq1d 2624	. . . . . . . . . . 11 |- ( w = v -> ( ( # ` w ) = 1 <-> ( # ` v ) = 1 ) )
23	20, 22	anbi12d 747	. . . . . . . . . 10 |- ( w = v -> ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) <-> ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 ) ) )
24		fveq1 6190	. . . . . . . . . . 11 |- ( w = v -> ( w ` 0 ) = ( v ` 0 ) )
25	24	eqeq1d 2624	. . . . . . . . . 10 |- ( w = v -> ( ( w ` 0 ) = P <-> ( v ` 0 ) = P ) )
26	23, 25	anbi12d 747	. . . . . . . . 9 |- ( w = v -> ( ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) <-> ( ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 ) /\ ( v ` 0 ) = P ) ) )
27	19, 26	elab 3350	. . . . . . . 8 |- ( v e. { w | ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) } <-> ( ( v e. Word (Vtx ` G ) /\ ( # ` v ) = 1 ) /\ ( v ` 0 ) = P ) )
28		velsn 4193	. . . . . . . 8 |- ( v e. { <" P "> } <-> v = <" P "> )
29	18, 27, 28	3bitr4g 303	. . . . . . 7 |- ( P e. (Vtx ` G ) -> ( v e. { w | ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) } <-> v e. { <" P "> } ) )
30	29, 3	eleq2s 2719	. . . . . 6 |- ( P e. V -> ( v e. { w | ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) } <-> v e. { <" P "> } ) )
31	30	eqrdv 2620	. . . . 5 |- ( P e. V -> { w | ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) } = { <" P "> } )
32	31	adantl 482	. . . 4 |- ( ( G e. USPGraph /\ P e. V ) -> { w | ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 1 ) /\ ( w ` 0 ) = P ) } = { <" P "> } )
33	8, 15, 32	3eqtrd 2660	. . 3 |- ( ( G e. USPGraph /\ P e. V ) -> { w e. ( 0 WWalksN G ) | ( w ` 0 ) = P } = { <" P "> } )
34	33	fveq2d 6195	. 2 |- ( ( G e. USPGraph /\ P e. V ) -> ( # ` { w e. ( 0 WWalksN G ) | ( w ` 0 ) = P } ) = ( # ` { <" P "> } ) )
35		s1cl 13382	. . . 4 |- ( P e. V -> <" P "> e. Word V )
36		hashsng 13159	. . . 4 |- ( <" P "> e. Word V -> ( # ` { <" P "> } ) = 1 )
37	35, 36	syl 17	. . 3 |- ( P e. V -> ( # ` { <" P "> } ) = 1 )
38	37	adantl 482	. 2 |- ( ( G e. USPGraph /\ P e. V ) -> ( # ` { <" P "> } ) = 1 )
39	6, 34, 38	3eqtrd 2660	1 |- ( ( G e. USPGraph /\ P e. V ) -> ( P L 0 ) = 1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   {csn 4177   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   NN0cn0 11292   #chash 13117  Word cword 13291   <"cs1 13294  Vtxcvtx 25874   USPGraph cuspgr 26043   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-s1 13302  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  rusgrnumwwlk  26870