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Theorem clwlkisclwwlk2 24452
Description: A closed walk corresponds to a closed walk as word in an undirected graph. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
Assertion
Ref Expression
clwlkisclwwlk2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( E. f 
f ( V ClWalks  E
) ( P concat  <" ( P `  0 ) "> )  <->  P  e.  ( V ClWWalks  E ) ) )
Distinct variable groups:    f, E    P, f    f, V

Proof of Theorem clwlkisclwwlk2
StepHypRef Expression
1 lswccats1fst 12589 . . . 4  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  ( lastS  `  ( P concat  <" ( P `  0 ) "> ) )  =  ( ( P concat  <" ( P `  0 ) "> ) `  0
) )
213adant1 1009 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( lastS  `  ( P concat  <" ( P ` 
0 ) "> ) )  =  ( ( P concat  <" ( P `  0 ) "> ) `  0
) )
32biantrurd 508 . 2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( ( ( P concat  <" ( P `
 0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )  e.  ( V ClWWalks  E )  <->  ( ( lastS  `  ( P concat  <" ( P ` 
0 ) "> ) )  =  ( ( P concat  <" ( P `  0 ) "> ) `  0
)  /\  ( ( P concat  <" ( P `
 0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )  e.  ( V ClWWalks  E )
) ) )
4 simpl 457 . . . . . . . . 9  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  P  e. Word  V )
5 wrdsymb1 12530 . . . . . . . . 9  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  ( P `  0 )  e.  V )
6 wrdlenccats1lenm1 12577 . . . . . . . . 9  |-  ( ( P  e. Word  V  /\  ( P `  0 )  e.  V )  -> 
( # `  P )  =  ( ( # `  ( P concat  <" ( P `  0 ) "> ) )  - 
1 ) )
74, 5, 6syl2anc 661 . . . . . . . 8  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  ( # `
 P )  =  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) )
87eqcomd 2468 . . . . . . 7  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  (
( # `  ( P concat  <" ( P ` 
0 ) "> ) )  -  1 )  =  ( # `  P ) )
98opeq2d 4213 . . . . . 6  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  <. 0 ,  ( ( # `  ( P concat  <" ( P `  0 ) "> ) )  - 
1 ) >.  =  <. 0 ,  ( # `  P
) >. )
109oveq2d 6291 . . . . 5  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  (
( P concat  <" ( P `  0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )  =  ( ( P concat  <" ( P ` 
0 ) "> ) substr  <. 0 ,  (
# `  P ) >. ) )
115s1cld 12565 . . . . . 6  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  <" ( P `  0 ) ">  e. Word  V )
12 eqidd 2461 . . . . . 6  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  ( # `
 P )  =  ( # `  P
) )
13 swrdccatid 12672 . . . . . 6  |-  ( ( P  e. Word  V  /\  <" ( P ` 
0 ) ">  e. Word  V  /\  ( # `  P )  =  (
# `  P )
)  ->  ( ( P concat  <" ( P `
 0 ) "> ) substr  <. 0 ,  ( # `  P
) >. )  =  P )
144, 11, 12, 13syl3anc 1223 . . . . 5  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  (
( P concat  <" ( P `  0 ) "> ) substr  <. 0 ,  ( # `  P
) >. )  =  P )
1510, 14eqtr2d 2502 . . . 4  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  P  =  ( ( P concat  <" ( P ` 
0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )
)
16153adant1 1009 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  P  =  ( ( P concat  <" ( P `  0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )
)
1716eleq1d 2529 . 2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( P  e.  ( V ClWWalks  E )  <->  ( ( P concat  <" ( P `  0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )  e.  ( V ClWWalks  E )
) )
18 simp1 991 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  V USGrph  E )
19 simp2 992 . . . 4  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  P  e. Word  V
)
20113adant1 1009 . . . 4  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  <" ( P `
 0 ) ">  e. Word  V )
21 ccatcl 12545 . . . 4  |-  ( ( P  e. Word  V  /\  <" ( P ` 
0 ) ">  e. Word  V )  ->  ( P concat  <" ( P `
 0 ) "> )  e. Word  V
)
2219, 20, 21syl2anc 661 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( P concat  <" ( P `  0 ) "> )  e. Word  V
)
23 lencl 12515 . . . . . . . 8  |-  ( P  e. Word  V  ->  ( # `
 P )  e. 
NN0 )
24 1e2m1 10640 . . . . . . . . . . 11  |-  1  =  ( 2  -  1 )
2524a1i 11 . . . . . . . . . 10  |-  ( (
# `  P )  e.  NN0  ->  1  =  ( 2  -  1 ) )
2625breq1d 4450 . . . . . . . . 9  |-  ( (
# `  P )  e.  NN0  ->  ( 1  <_  ( # `  P
)  <->  ( 2  -  1 )  <_  ( # `
 P ) ) )
27 2re 10594 . . . . . . . . . . 11  |-  2  e.  RR
2827a1i 11 . . . . . . . . . 10  |-  ( (
# `  P )  e.  NN0  ->  2  e.  RR )
29 1re 9584 . . . . . . . . . . 11  |-  1  e.  RR
3029a1i 11 . . . . . . . . . 10  |-  ( (
# `  P )  e.  NN0  ->  1  e.  RR )
31 nn0re 10793 . . . . . . . . . 10  |-  ( (
# `  P )  e.  NN0  ->  ( # `  P
)  e.  RR )
3228, 30, 31lesubaddd 10138 . . . . . . . . 9  |-  ( (
# `  P )  e.  NN0  ->  ( (
2  -  1 )  <_  ( # `  P
)  <->  2  <_  (
( # `  P )  +  1 ) ) )
3326, 32bitrd 253 . . . . . . . 8  |-  ( (
# `  P )  e.  NN0  ->  ( 1  <_  ( # `  P
)  <->  2  <_  (
( # `  P )  +  1 ) ) )
3423, 33syl 16 . . . . . . 7  |-  ( P  e. Word  V  ->  (
1  <_  ( # `  P
)  <->  2  <_  (
( # `  P )  +  1 ) ) )
3534biimpa 484 . . . . . 6  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  2  <_  ( ( # `  P
)  +  1 ) )
36 s1len 12567 . . . . . . 7  |-  ( # `  <" ( P `
 0 ) "> )  =  1
3736oveq2i 6286 . . . . . 6  |-  ( (
# `  P )  +  ( # `  <" ( P `  0
) "> )
)  =  ( (
# `  P )  +  1 )
3835, 37syl6breqr 4480 . . . . 5  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  2  <_  ( ( # `  P
)  +  ( # `  <" ( P `
 0 ) "> ) ) )
39383adant1 1009 . . . 4  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  2  <_  (
( # `  P )  +  ( # `  <" ( P `  0
) "> )
) )
404, 11jca 532 . . . . . 6  |-  ( ( P  e. Word  V  /\  1  <_  ( # `  P
) )  ->  ( P  e. Word  V  /\  <" ( P `  0
) ">  e. Word  V ) )
41403adant1 1009 . . . . 5  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( P  e. Word  V  /\  <" ( P `
 0 ) ">  e. Word  V )
)
42 ccatlen 12546 . . . . 5  |-  ( ( P  e. Word  V  /\  <" ( P ` 
0 ) ">  e. Word  V )  ->  ( # `
 ( P concat  <" ( P `  0 ) "> ) )  =  ( ( # `  P
)  +  ( # `  <" ( P `
 0 ) "> ) ) )
4341, 42syl 16 . . . 4  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( # `  ( P concat  <" ( P `
 0 ) "> ) )  =  ( ( # `  P
)  +  ( # `  <" ( P `
 0 ) "> ) ) )
4439, 43breqtrrd 4466 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  2  <_  ( # `
 ( P concat  <" ( P `  0 ) "> ) ) )
45 clwlkisclwwlk 24451 . . 3  |-  ( ( V USGrph  E  /\  ( P concat  <" ( P `
 0 ) "> )  e. Word  V  /\  2  <_  ( # `  ( P concat  <" ( P `  0 ) "> ) ) )  ->  ( E. f 
f ( V ClWalks  E
) ( P concat  <" ( P `  0 ) "> )  <->  ( ( lastS  `  ( P concat  <" ( P `  0 ) "> ) )  =  ( ( P concat  <" ( P `  0 ) "> ) `  0
)  /\  ( ( P concat  <" ( P `
 0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )  e.  ( V ClWWalks  E )
) ) )
4618, 22, 44, 45syl3anc 1223 . 2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( E. f 
f ( V ClWalks  E
) ( P concat  <" ( P `  0 ) "> )  <->  ( ( lastS  `  ( P concat  <" ( P `  0 ) "> ) )  =  ( ( P concat  <" ( P `  0 ) "> ) `  0
)  /\  ( ( P concat  <" ( P `
 0 ) "> ) substr  <. 0 ,  ( ( # `  ( P concat  <" ( P `
 0 ) "> ) )  - 
1 ) >. )  e.  ( V ClWWalks  E )
) ) )
473, 17, 463bitr4rd 286 1  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `
 P ) )  ->  ( E. f 
f ( V ClWalks  E
) ( P concat  <" ( P `  0 ) "> )  <->  P  e.  ( V ClWWalks  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9794   2c2 10574   NN0cn0 10784   #chash 12360  Word cword 12487   lastS clsw 12488   concat cconcat 12489   <"cs1 12490   substr csubstr 12491   USGrph cusg 23993   ClWalks cclwlk 24409   ClWWalks cclwwlk 24410
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-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-concat 12497  df-s1 12498  df-substr 12499  df-usgra 23996  df-wlk 24170  df-clwlk 24412  df-clwwlk 24413
This theorem is referenced by:  clwlkfoclwwlk  24507
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