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Theorem clwlkf1clwwlk 24512
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.)
Hypotheses
Ref Expression
clwlkfclwwlk.1  |-  A  =  ( 1st `  c
)
clwlkfclwwlk.2  |-  B  =  ( 2nd `  c
)
clwlkfclwwlk.c  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
clwlkfclwwlk.f  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
Assertion
Ref Expression
clwlkf1clwwlk  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C -1-1-> ( ( V ClWWalksN  E ) `  N
) )
Distinct variable groups:    E, c    N, c    V, c    C, c    F, c
Allowed substitution hints:    A( c)    B( c)

Proof of Theorem clwlkf1clwwlk
Dummy variables  i  w  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.1 . . 3  |-  A  =  ( 1st `  c
)
2 clwlkfclwwlk.2 . . 3  |-  B  =  ( 2nd `  c
)
3 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
4 clwlkfclwwlk.f . . 3  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
51, 2, 3, 4clwlkfclwwlk 24506 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C
--> ( ( V ClWWalksN  E ) `
 N ) )
6 simprl 755 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  u  e.  C )
7 ovex 6300 . . . . . . 7  |-  ( ( 2nd `  u ) substr  <. 0 ,  ( # `  ( 1st `  u
) ) >. )  e.  _V
8 fveq2 5857 . . . . . . . . . 10  |-  ( c  =  u  ->  ( 2nd `  c )  =  ( 2nd `  u
) )
92, 8syl5eq 2513 . . . . . . . . 9  |-  ( c  =  u  ->  B  =  ( 2nd `  u
) )
10 fveq2 5857 . . . . . . . . . . . 12  |-  ( c  =  u  ->  ( 1st `  c )  =  ( 1st `  u
) )
111, 10syl5eq 2513 . . . . . . . . . . 11  |-  ( c  =  u  ->  A  =  ( 1st `  u
) )
1211fveq2d 5861 . . . . . . . . . 10  |-  ( c  =  u  ->  ( # `
 A )  =  ( # `  ( 1st `  u ) ) )
1312opeq2d 4213 . . . . . . . . 9  |-  ( c  =  u  ->  <. 0 ,  ( # `  A
) >.  =  <. 0 ,  ( # `  ( 1st `  u ) )
>. )
149, 13oveq12d 6293 . . . . . . . 8  |-  ( c  =  u  ->  ( B substr  <. 0 ,  (
# `  A ) >. )  =  ( ( 2nd `  u ) substr  <. 0 ,  ( # `  ( 1st `  u
) ) >. )
)
1514, 4fvmptg 5939 . . . . . . 7  |-  ( ( u  e.  C  /\  ( ( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
)  e.  _V )  ->  ( F `  u
)  =  ( ( 2nd `  u ) substr  <. 0 ,  ( # `  ( 1st `  u
) ) >. )
)
166, 7, 15sylancl 662 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  ( F `  u )  =  ( ( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
) )
17 simpr 461 . . . . . . . . 9  |-  ( ( u  e.  C  /\  w  e.  C )  ->  w  e.  C )
18 ovex 6300 . . . . . . . . 9  |-  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )  e.  _V
1917, 18jctir 538 . . . . . . . 8  |-  ( ( u  e.  C  /\  w  e.  C )  ->  ( w  e.  C  /\  ( ( 2nd `  w
) substr  <. 0 ,  (
# `  ( 1st `  w ) ) >.
)  e.  _V )
)
2019adantl 466 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  ( w  e.  C  /\  (
( 2nd `  w
) substr  <. 0 ,  (
# `  ( 1st `  w ) ) >.
)  e.  _V )
)
21 fveq2 5857 . . . . . . . . . 10  |-  ( c  =  w  ->  ( 2nd `  c )  =  ( 2nd `  w
) )
222, 21syl5eq 2513 . . . . . . . . 9  |-  ( c  =  w  ->  B  =  ( 2nd `  w
) )
23 fveq2 5857 . . . . . . . . . . . 12  |-  ( c  =  w  ->  ( 1st `  c )  =  ( 1st `  w
) )
241, 23syl5eq 2513 . . . . . . . . . . 11  |-  ( c  =  w  ->  A  =  ( 1st `  w
) )
2524fveq2d 5861 . . . . . . . . . 10  |-  ( c  =  w  ->  ( # `
 A )  =  ( # `  ( 1st `  w ) ) )
2625opeq2d 4213 . . . . . . . . 9  |-  ( c  =  w  ->  <. 0 ,  ( # `  A
) >.  =  <. 0 ,  ( # `  ( 1st `  w ) )
>. )
2722, 26oveq12d 6293 . . . . . . . 8  |-  ( c  =  w  ->  ( B substr  <. 0 ,  (
# `  A ) >. )  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )
)
2827, 4fvmptg 5939 . . . . . . 7  |-  ( ( w  e.  C  /\  ( ( 2nd `  w
) substr  <. 0 ,  (
# `  ( 1st `  w ) ) >.
)  e.  _V )  ->  ( F `  w
)  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )
)
2920, 28syl 16 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  ( F `  w )  =  ( ( 2nd `  w
) substr  <. 0 ,  (
# `  ( 1st `  w ) ) >.
) )
3016, 29eqeq12d 2482 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  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 ) ) >.
) ) )
311, 2, 3, 4clwlkfclwwlk1hashn 24503 . . . . . . . . . 10  |-  ( w  e.  C  ->  ( # `
 ( 1st `  w
) )  =  N )
3231eqcomd 2468 . . . . . . . . 9  |-  ( w  e.  C  ->  N  =  ( # `  ( 1st `  w ) ) )
3332adantl 466 . . . . . . . 8  |-  ( ( u  e.  C  /\  w  e.  C )  ->  N  =  ( # `  ( 1st `  w
) ) )
3433ad2antlr 726 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  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 14068 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  N  e.  NN )
36353ad2ant3 1014 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  e.  NN )
3736adantr 465 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  N  e.  NN )
3817adantl 466 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  w  e.  C )
391, 2, 3, 4clwlkf1clwwlklem 24511 . . . . . . . . 9  |-  ( ( 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 ) ) )
4037, 6, 38, 39syl3anc 1223 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  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
) ) )
4140imp 429 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  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 ) )
42 simpll1 1030 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( u  e.  C  /\  w  e.  C
) )  /\  (
( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
)  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )
)  ->  V USGrph  E )
43 elrabi 3251 . . . . . . . . . . . 12  |-  ( u  e.  { c  e.  ( V ClWalks  E )  |  ( # `  A
)  =  N }  ->  u  e.  ( V ClWalks  E ) )
44 clwlkswlks 24420 . . . . . . . . . . . 12  |-  ( u  e.  ( V ClWalks  E
)  ->  u  e.  ( V Walks  E ) )
4543, 44syl 16 . . . . . . . . . . 11  |-  ( u  e.  { c  e.  ( V ClWalks  E )  |  ( # `  A
)  =  N }  ->  u  e.  ( V Walks 
E ) )
4645, 3eleq2s 2568 . . . . . . . . . 10  |-  ( u  e.  C  ->  u  e.  ( V Walks  E ) )
47 elrabi 3251 . . . . . . . . . . . 12  |-  ( w  e.  { c  e.  ( V ClWalks  E )  |  ( # `  A
)  =  N }  ->  w  e.  ( V ClWalks  E ) )
48 clwlkswlks 24420 . . . . . . . . . . . 12  |-  ( w  e.  ( V ClWalks  E
)  ->  w  e.  ( V Walks  E ) )
4947, 48syl 16 . . . . . . . . . . 11  |-  ( w  e.  { c  e.  ( V ClWalks  E )  |  ( # `  A
)  =  N }  ->  w  e.  ( V Walks 
E ) )
5049, 3eleq2s 2568 . . . . . . . . . 10  |-  ( w  e.  C  ->  w  e.  ( V Walks  E ) )
5146, 50anim12i 566 . . . . . . . . 9  |-  ( ( u  e.  C  /\  w  e.  C )  ->  ( u  e.  ( V Walks  E )  /\  w  e.  ( V Walks  E ) ) )
5251ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( u  e.  C  /\  w  e.  C
) )  /\  (
( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
)  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )
)  ->  ( u  e.  ( V Walks  E )  /\  w  e.  ( V Walks  E ) ) )
531, 2, 3, 4clwlkfclwwlk1hashn 24503 . . . . . . . . . . 11  |-  ( u  e.  C  ->  ( # `
 ( 1st `  u
) )  =  N )
5453eqcomd 2468 . . . . . . . . . 10  |-  ( u  e.  C  ->  N  =  ( # `  ( 1st `  u ) ) )
5554adantr 465 . . . . . . . . 9  |-  ( ( u  e.  C  /\  w  e.  C )  ->  N  =  ( # `  ( 1st `  u
) ) )
5655ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( u  e.  C  /\  w  e.  C
) )  /\  (
( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
)  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )
)  ->  N  =  ( # `  ( 1st `  u ) ) )
57 usg2wlkeq 24370 . . . . . . . 8  |-  ( ( V USGrph  E  /\  (
u  e.  ( V Walks 
E )  /\  w  e.  ( V Walks  E ) )  /\  N  =  ( # `  ( 1st `  u ) ) )  ->  ( u  =  w  <->  ( N  =  ( # `  ( 1st `  w ) )  /\  A. i  e.  ( 0 ... N
) ( ( 2nd `  u ) `  i
)  =  ( ( 2nd `  w ) `
 i ) ) ) )
5842, 52, 56, 57syl3anc 1223 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  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 ) ) ) )
5934, 41, 58mpbir2and 915 . . . . . 6  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( u  e.  C  /\  w  e.  C
) )  /\  (
( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
)  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )
)  ->  u  =  w )
6059ex 434 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  ( (
( 2nd `  u
) substr  <. 0 ,  (
# `  ( 1st `  u ) ) >.
)  =  ( ( 2nd `  w ) substr  <. 0 ,  ( # `  ( 1st `  w
) ) >. )  ->  u  =  w ) )
6130, 60sylbid 215 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
u  e.  C  /\  w  e.  C )
)  ->  ( ( F `  u )  =  ( F `  w )  ->  u  =  w ) )
6261ex 434 . . 3  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  ( (
u  e.  C  /\  w  e.  C )  ->  ( ( F `  u )  =  ( F `  w )  ->  u  =  w ) ) )
6362ralrimivv 2877 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  A. u  e.  C  A. w  e.  C  ( ( F `  u )  =  ( F `  w )  ->  u  =  w ) )
64 dff13 6145 . 2  |-  ( F : C -1-1-> ( ( V ClWWalksN  E ) `  N
)  <->  ( F : C
--> ( ( V ClWWalksN  E ) `
 N )  /\  A. u  e.  C  A. w  e.  C  (
( F `  u
)  =  ( F `
 w )  ->  u  =  w )
) )
655, 63, 64sylanbrc 664 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C -1-1-> ( ( V ClWWalksN  E ) `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106   <.cop 4026   class class class wbr 4440    |-> cmpt 4498   -->wf 5575   -1-1->wf1 5576   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Fincfn 7506   0cc0 9481   NNcn 10525   ...cfz 11661   #chash 12360   substr csubstr 12491   Primecprime 14065   USGrph cusg 23993   Walks cwalk 24160   ClWalks cclwlk 24409   ClWWalksN cclwwlkn 24411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-lsw 12496  df-substr 12499  df-dvds 13837  df-prm 14066  df-usgra 23996  df-wlk 24170  df-clwlk 24412  df-clwwlk 24413  df-clwwlkn 24414
This theorem is referenced by:  clwlkf1oclwwlk  24513
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