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Theorem clwlkcomp 25062
Description: A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
Hypotheses
Ref Expression
clwlkcomp.1  |-  F  =  ( 1st `  W
)
clwlkcomp.2  |-  P  =  ( 2nd `  W
)
Assertion
Ref Expression
clwlkcomp  |-  ( ( V  e.  X  /\  E  e.  Y  /\  W  e.  ( S  X.  T ) )  -> 
( W  e.  ( V ClWalks  E )  <->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
Distinct variable groups:    k, E    k, F    P, k
Allowed substitution hints:    S( k)    T( k)    V( k)    W( k)    X( k)    Y( k)

Proof of Theorem clwlkcomp
StepHypRef Expression
1 clwlkcomp.1 . . . . . . . 8  |-  F  =  ( 1st `  W
)
21eqcomi 2413 . . . . . . 7  |-  ( 1st `  W )  =  F
3 clwlkcomp.2 . . . . . . . 8  |-  P  =  ( 2nd `  W
)
43eqcomi 2413 . . . . . . 7  |-  ( 2nd `  W )  =  P
52, 4pm3.2i 453 . . . . . 6  |-  ( ( 1st `  W )  =  F  /\  ( 2nd `  W )  =  P )
6 eqop 6776 . . . . . 6  |-  ( W  e.  ( S  X.  T )  ->  ( W  =  <. F ,  P >. 
<->  ( ( 1st `  W
)  =  F  /\  ( 2nd `  W )  =  P ) ) )
75, 6mpbiri 233 . . . . 5  |-  ( W  e.  ( S  X.  T )  ->  W  =  <. F ,  P >. )
87eleq1d 2469 . . . 4  |-  ( W  e.  ( S  X.  T )  ->  ( W  e.  ( V ClWalks  E )  <->  <. F ,  P >.  e.  ( V ClWalks  E
) ) )
9 df-br 4393 . . . 4  |-  ( F ( V ClWalks  E ) P 
<-> 
<. F ,  P >.  e.  ( V ClWalks  E )
)
108, 9syl6bbr 263 . . 3  |-  ( W  e.  ( S  X.  T )  ->  ( W  e.  ( V ClWalks  E )  <->  F ( V ClWalks  E
) P ) )
11103ad2ant3 1018 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  W  e.  ( S  X.  T ) )  -> 
( W  e.  ( V ClWalks  E )  <->  F ( V ClWalks  E ) P ) )
12 isclwlk 25055 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V ClWalks  E ) P  <->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
13123adant3 1015 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  W  e.  ( S  X.  T ) )  -> 
( F ( V ClWalks  E ) P  <->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
1411, 13bitrd 253 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  W  e.  ( S  X.  T ) )  -> 
( W  e.  ( V ClWalks  E )  <->  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   {cpr 3971   <.cop 3975   class class class wbr 4392    X. cxp 4938   dom cdm 4940   -->wf 5519   ` cfv 5523  (class class class)co 6232   1stc1st 6734   2ndc2nd 6735   0cc0 9440   1c1 9441    + caddc 9443   ...cfz 11641  ..^cfzo 11765   #chash 12357  Word cword 12488   ClWalks cclwlk 25046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-hash 12358  df-word 12496  df-wlk 24807  df-clwlk 25049
This theorem is referenced by:  clwlkcompim  25063
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