Theorem clwlksf1clwwlk 26969

Description: There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)

Hypotheses

Ref	Expression
clwlksfclwwlk.1	|- A = ( 1st ` c )
clwlksfclwwlk.2	|- B = ( 2nd ` c )
clwlksfclwwlk.c	|- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
clwlksfclwwlk.f	|- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )

Assertion

Ref	Expression
clwlksf1clwwlk	|- ( ( G e. FinUSGraph /\ N e. Prime ) -> F : C -1-1-> ( N ClWWalksN G ) )

Distinct variable groups:   G, c   N, c   C, c   F, c

Allowed substitution hints:   A( c)   B( c)

Proof of Theorem clwlksf1clwwlk

Dummy variables i w u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . 3 |- A = ( 1st ` c )
2		clwlksfclwwlk.2	. . 3 |- B = ( 2nd ` c )
3		clwlksfclwwlk.c	. . 3 |- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
4		clwlksfclwwlk.f	. . 3 |- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )
5	1, 2, 3, 4	clwlksfclwwlk 26962	. 2 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> F : C --> ( N ClWWalksN G ) )
6		simprl 794	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> u e. C )
7		ovex 6678	. . . . . 6 |- ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) e. _V
8		fveq2 6191	. . . . . . . . 9 |- ( c = u -> ( 2nd ` c ) = ( 2nd ` u ) )
9	2, 8	syl5eq 2668	. . . . . . . 8 |- ( c = u -> B = ( 2nd ` u ) )
10		fveq2 6191	. . . . . . . . . . 11 |- ( c = u -> ( 1st ` c ) = ( 1st ` u ) )
11	1, 10	syl5eq 2668	. . . . . . . . . 10 |- ( c = u -> A = ( 1st ` u ) )
12	11	fveq2d 6195	. . . . . . . . 9 |- ( c = u -> ( # ` A ) = ( # ` ( 1st ` u ) ) )
13	12	opeq2d 4409	. . . . . . . 8 |- ( c = u -> <. 0 , ( # ` A ) >. = <. 0 , ( # ` ( 1st ` u ) ) >. )
14	9, 13	oveq12d 6668	. . . . . . 7 |- ( c = u -> ( B substr <. 0 , ( # ` A ) >. ) = ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) )
15	14, 4	fvmptg 6280	. . . . . 6 |- ( ( u e. C /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) e. _V ) -> ( F ` u ) = ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) )
16	6, 7, 15	sylancl 694	. . . . 5 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( F ` u ) = ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) )
17		simpr 477	. . . . . . . 8 |- ( ( u e. C /\ w e. C ) -> w e. C )
18		ovex 6678	. . . . . . . 8 |- ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) e. _V
19	17, 18	jctir 561	. . . . . . 7 |- ( ( u e. C /\ w e. C ) -> ( w e. C /\ ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) e. _V ) )
20	19	adantl 482	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( w e. C /\ ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) e. _V ) )
21		fveq2 6191	. . . . . . . . 9 |- ( c = w -> ( 2nd ` c ) = ( 2nd ` w ) )
22	2, 21	syl5eq 2668	. . . . . . . 8 |- ( c = w -> B = ( 2nd ` w ) )
23		fveq2 6191	. . . . . . . . . . 11 |- ( c = w -> ( 1st ` c ) = ( 1st ` w ) )
24	1, 23	syl5eq 2668	. . . . . . . . . 10 |- ( c = w -> A = ( 1st ` w ) )
25	24	fveq2d 6195	. . . . . . . . 9 |- ( c = w -> ( # ` A ) = ( # ` ( 1st ` w ) ) )
26	25	opeq2d 4409	. . . . . . . 8 |- ( c = w -> <. 0 , ( # ` A ) >. = <. 0 , ( # ` ( 1st ` w ) ) >. )
27	22, 26	oveq12d 6668	. . . . . . 7 |- ( c = w -> ( B substr <. 0 , ( # ` A ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) )
28	27, 4	fvmptg 6280	. . . . . 6 |- ( ( w e. C /\ ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) e. _V ) -> ( F ` w ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) )
29	20, 28	syl 17	. . . . 5 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( F ` w ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) )
30	16, 29	eqeq12d 2637	. . . 4 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( ( F ` u ) = ( F ` w ) <-> ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) )
31	1, 2, 3, 4	clwlksfclwwlk1hashn 26959	. . . . . . . . 9 |- ( w e. C -> ( # ` ( 1st ` w ) ) = N )
32	31	eqcomd 2628	. . . . . . . 8 |- ( w e. C -> N = ( # ` ( 1st ` w ) ) )
33	32	adantl 482	. . . . . . 7 |- ( ( u e. C /\ w e. C ) -> N = ( # ` ( 1st ` w ) ) )
34	33	ad2antlr 763	. . . . . 6 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> N = ( # ` ( 1st ` w ) ) )
35		prmnn 15388	. . . . . . . . 9 |- ( N e. Prime -> N e. NN )
36	35	ad2antlr 763	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> N e. NN )
37	17	adantl 482	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> w e. C )
38	1, 2, 3, 4	clwlksf1clwwlklem 26968	. . . . . . . 8 |- ( ( N e. NN /\ u e. C /\ w e. C ) -> ( ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) -> A. i e. ( 0 ... N ) ( ( 2nd ` u ) ` i ) = ( ( 2nd ` w ) ` i ) ) )
39	36, 6, 37, 38	syl3anc 1326	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) -> A. i e. ( 0 ... N ) ( ( 2nd ` u ) ` i ) = ( ( 2nd ` w ) ` i ) ) )
40	39	imp 445	. . . . . 6 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> A. i e. ( 0 ... N ) ( ( 2nd ` u ) ` i ) = ( ( 2nd ` w ) ` i ) )
41		fusgrusgr 26214	. . . . . . . . 9 |- ( G e. FinUSGraph -> G e. USGraph )
42		usgruspgr 26073	. . . . . . . . 9 |- ( G e. USGraph -> G e. USPGraph )
43	41, 42	syl 17	. . . . . . . 8 |- ( G e. FinUSGraph -> G e. USPGraph )
44	43	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> G e. USPGraph )
45		elrabi 3359	. . . . . . . . . . 11 |- ( u e. { c e. (ClWalks ` G ) | ( # ` A ) = N } -> u e. (ClWalks ` G ) )
46		clwlkwlk 26671	. . . . . . . . . . 11 |- ( u e. (ClWalks ` G ) -> u e. (Walks ` G ) )
47	45, 46	syl 17	. . . . . . . . . 10 |- ( u e. { c e. (ClWalks ` G ) | ( # ` A ) = N } -> u e. (Walks ` G ) )
48	47, 3	eleq2s 2719	. . . . . . . . 9 |- ( u e. C -> u e. (Walks ` G ) )
49		elrabi 3359	. . . . . . . . . . 11 |- ( w e. { c e. (ClWalks ` G ) | ( # ` A ) = N } -> w e. (ClWalks ` G ) )
50		clwlkwlk 26671	. . . . . . . . . . 11 |- ( w e. (ClWalks ` G ) -> w e. (Walks ` G ) )
51	49, 50	syl 17	. . . . . . . . . 10 |- ( w e. { c e. (ClWalks ` G ) | ( # ` A ) = N } -> w e. (Walks ` G ) )
52	51, 3	eleq2s 2719	. . . . . . . . 9 |- ( w e. C -> w e. (Walks ` G ) )
53	48, 52	anim12i 590	. . . . . . . 8 |- ( ( u e. C /\ w e. C ) -> ( u e. (Walks ` G ) /\ w e. (Walks ` G ) ) )
54	53	ad2antlr 763	. . . . . . 7 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> ( u e. (Walks ` G ) /\ w e. (Walks ` G ) ) )
55	1, 2, 3, 4	clwlksfclwwlk1hashn 26959	. . . . . . . . . 10 |- ( u e. C -> ( # ` ( 1st ` u ) ) = N )
56	55	eqcomd 2628	. . . . . . . . 9 |- ( u e. C -> N = ( # ` ( 1st ` u ) ) )
57	56	adantr 481	. . . . . . . 8 |- ( ( u e. C /\ w e. C ) -> N = ( # ` ( 1st ` u ) ) )
58	57	ad2antlr 763	. . . . . . 7 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> N = ( # ` ( 1st ` u ) ) )
59		uspgr2wlkeq 26542	. . . . . . 7 |- ( ( G e. USPGraph /\ ( u e. (Walks ` G ) /\ w e. (Walks ` G ) ) /\ N = ( # ` ( 1st ` u ) ) ) -> ( u = w <-> ( N = ( # ` ( 1st ` w ) ) /\ A. i e. ( 0 ... N ) ( ( 2nd ` u ) ` i ) = ( ( 2nd ` w ) ` i ) ) ) )
60	44, 54, 58, 59	syl3anc 1326	. . . . . 6 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> ( u = w <-> ( N = ( # ` ( 1st ` w ) ) /\ A. i e. ( 0 ... N ) ( ( 2nd ` u ) ` i ) = ( ( 2nd ` w ) ` i ) ) ) )
61	34, 40, 60	mpbir2and 957	. . . . 5 |- ( ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) /\ ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) ) -> u = w )
62	61	ex 450	. . . 4 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( ( ( 2nd ` u ) substr <. 0 , ( # ` ( 1st ` u ) ) >. ) = ( ( 2nd ` w ) substr <. 0 , ( # ` ( 1st ` w ) ) >. ) -> u = w ) )
63	30, 62	sylbid 230	. . 3 |- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ ( u e. C /\ w e. C ) ) -> ( ( F ` u ) = ( F ` w ) -> u = w ) )
64	63	ralrimivva 2971	. 2 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> A. u e. C A. w e. C ( ( F ` u ) = ( F ` w ) -> u = w ) )
65		dff13 6512	. 2 |- ( F : C -1-1-> ( N ClWWalksN G ) <-> ( F : C --> ( N ClWWalksN G ) /\ A. u e. C A. w e. C ( ( F ` u ) = ( F ` w ) -> u = w ) ) )
66	5, 64, 65	sylanbrc 698	1 |- ( ( G e. FinUSGraph /\ N e. Prime ) -> F : C -1-1-> ( N ClWWalksN G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   NNcn 11020   ...cfz 12326   #chash 13117   substr csubstr 13295   Primecprime 15385   USPGraph cuspgr 26043   USGraph cusgr 26044   FinUSGraph cfusgr 26208  Walkscwlks 26492  ClWalkscclwlks 26666   ClWWalksN cclwwlksn 26876

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  ax-pre-sup 10014

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-rmo 2920  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-2o 7561  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-sup 8348  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-substr 13303  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-wlks 26495  df-clwlks 26667  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  clwlksf1oclwwlk  26970