Theorem wlklnwwlkln2lem 26768

Description: Lemma for wlklnwwlkln2 26769 and wlklnwwlklnupgr2 26771. Formerly part of proof for wlklnwwlkln2 26769. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)

Hypothesis

Ref	Expression
wlklnwwlkln2lem.1	|- ( ph -> ( P e. (WWalks ` G ) -> E. f f (Walks ` G ) P ) )

Assertion

Ref	Expression
wlklnwwlkln2lem	|- ( ph -> ( P e. ( N WWalksN G ) -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )

Distinct variable groups:   f, G   f, N   P, f   ph, f

Proof of Theorem wlklnwwlkln2lem

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2	1	wwlknbp 26733	. . 3 |- ( P e. ( N WWalksN G ) -> ( G e. _V /\ N e. NN0 /\ P e. Word (Vtx ` G ) ) )
3		iswwlksn 26730	. . . . . 6 |- ( N e. NN0 -> ( P e. ( N WWalksN G ) <-> ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) )
4	3	adantr 481	. . . . 5 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( P e. ( N WWalksN G ) <-> ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) )
5		lencl 13324	. . . . . . . . . . . . . 14 |- ( P e. Word (Vtx ` G ) -> ( # ` P ) e. NN0 )
6	5	nn0cnd 11353	. . . . . . . . . . . . 13 |- ( P e. Word (Vtx ` G ) -> ( # ` P ) e. CC )
7	6	adantl 482	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( # ` P ) e. CC )
8		1cnd 10056	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> 1 e. CC )
9		nn0cn 11302	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N e. CC )
10	9	adantr 481	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> N e. CC )
11	7, 8, 10	subadd2d 10411	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( ( ( # ` P ) - 1 ) = N <-> ( N + 1 ) = ( # ` P ) ) )
12		eqcom 2629	. . . . . . . . . . 11 |- ( ( N + 1 ) = ( # ` P ) <-> ( # ` P ) = ( N + 1 ) )
13	11, 12	syl6rbb 277	. . . . . . . . . 10 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( ( # ` P ) = ( N + 1 ) <-> ( ( # ` P ) - 1 ) = N ) )
14	13	biimpcd 239	. . . . . . . . 9 |- ( ( # ` P ) = ( N + 1 ) -> ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( ( # ` P ) - 1 ) = N ) )
15	14	adantl 482	. . . . . . . 8 |- ( ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) -> ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( ( # ` P ) - 1 ) = N ) )
16	15	impcom 446	. . . . . . 7 |- ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) -> ( ( # ` P ) - 1 ) = N )
17		wlklnwwlkln2lem.1	. . . . . . . . . . . . . 14 |- ( ph -> ( P e. (WWalks ` G ) -> E. f f (Walks ` G ) P ) )
18	17	com12 32	. . . . . . . . . . . . 13 |- ( P e. (WWalks ` G ) -> ( ph -> E. f f (Walks ` G ) P ) )
19	18	adantr 481	. . . . . . . . . . . 12 |- ( ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) -> ( ph -> E. f f (Walks ` G ) P ) )
20	19	adantl 482	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) -> ( ph -> E. f f (Walks ` G ) P ) )
21	20	imp 445	. . . . . . . . . 10 |- ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) -> E. f f (Walks ` G ) P )
22		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) /\ f (Walks ` G ) P ) -> f (Walks ` G ) P )
23		wlklenvm1 26517	. . . . . . . . . . . . 13 |- ( f (Walks ` G ) P -> ( # ` f ) = ( ( # ` P ) - 1 ) )
24	22, 23	jccir 562	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) /\ f (Walks ` G ) P ) -> ( f (Walks ` G ) P /\ ( # ` f ) = ( ( # ` P ) - 1 ) ) )
25	24	ex 450	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) -> ( f (Walks ` G ) P -> ( f (Walks ` G ) P /\ ( # ` f ) = ( ( # ` P ) - 1 ) ) ) )
26	25	eximdv 1846	. . . . . . . . . 10 |- ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) -> ( E. f f (Walks ` G ) P -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = ( ( # ` P ) - 1 ) ) ) )
27	21, 26	mpd 15	. . . . . . . . 9 |- ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = ( ( # ` P ) - 1 ) ) )
28		eqeq2 2633	. . . . . . . . . . 11 |- ( ( ( # ` P ) - 1 ) = N -> ( ( # ` f ) = ( ( # ` P ) - 1 ) <-> ( # ` f ) = N ) )
29	28	anbi2d 740	. . . . . . . . . 10 |- ( ( ( # ` P ) - 1 ) = N -> ( ( f (Walks ` G ) P /\ ( # ` f ) = ( ( # ` P ) - 1 ) ) <-> ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )
30	29	exbidv 1850	. . . . . . . . 9 |- ( ( ( # ` P ) - 1 ) = N -> ( E. f ( f (Walks ` G ) P /\ ( # ` f ) = ( ( # ` P ) - 1 ) ) <-> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )
31	27, 30	syl5ib 234	. . . . . . . 8 |- ( ( ( # ` P ) - 1 ) = N -> ( ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) /\ ph ) -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )
32	31	expd 452	. . . . . . 7 |- ( ( ( # ` P ) - 1 ) = N -> ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) -> ( ph -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) ) )
33	16, 32	mpcom 38	. . . . . 6 |- ( ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) /\ ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) ) -> ( ph -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )
34	33	ex 450	. . . . 5 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( ( P e. (WWalks ` G ) /\ ( # ` P ) = ( N + 1 ) ) -> ( ph -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) ) )
35	4, 34	sylbid 230	. . . 4 |- ( ( N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( P e. ( N WWalksN G ) -> ( ph -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) ) )
36	35	3adant1 1079	. . 3 |- ( ( G e. _V /\ N e. NN0 /\ P e. Word (Vtx ` G ) ) -> ( P e. ( N WWalksN G ) -> ( ph -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) ) )
37	2, 36	mpcom 38	. 2 |- ( P e. ( N WWalksN G ) -> ( ph -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )
38	37	com12 32	1 |- ( ph -> ( P e. ( N WWalksN G ) -> E. f ( f (Walks ` G ) P /\ ( # ` f ) = N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   #chash 13117  Word cword 13291  Vtxcvtx 25874  Walkscwlks 26492  WWalkscwwlks 26717   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-ifp 1013  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-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wlklnwwlkln2  26769  wlklnwwlklnupgr2  26771