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Theorem clwlkfclwwlk2sswd 24505
Description: The size of a subword of the second component of a closed walk with length of the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-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
clwlkfclwwlk2sswd  |-  ( c  e.  C  ->  ( # `
 A )  =  ( # `  ( B substr  <. 0 ,  (
# `  A ) >. ) ) )
Distinct variable groups:    E, c    N, c    V, c
Allowed substitution hints:    A( c)    B( c)    C( c)    F( c)

Proof of Theorem clwlkfclwwlk2sswd
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, 4clwlkfclwwlk2wrd 24502 . 2  |-  ( c  e.  C  ->  B  e. Word  V )
61, 2, 3, 4clwlkfclwwlk1hash 24504 . 2  |-  ( c  e.  C  ->  ( # `
 A )  e.  ( 0 ... ( # `
 B ) ) )
7 swrd0len 12599 . . 3  |-  ( ( B  e. Word  V  /\  ( # `  A )  e.  ( 0 ... ( # `  B
) ) )  -> 
( # `  ( B substr  <. 0 ,  ( # `  A ) >. )
)  =  ( # `  A ) )
87eqcomd 2468 . 2  |-  ( ( B  e. Word  V  /\  ( # `  A )  e.  ( 0 ... ( # `  B
) ) )  -> 
( # `  A )  =  ( # `  ( B substr  <. 0 ,  (
# `  A ) >. ) ) )
95, 6, 8syl2anc 661 1  |-  ( c  e.  C  ->  ( # `
 A )  =  ( # `  ( B substr  <. 0 ,  (
# `  A ) >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811   <.cop 4026    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   0cc0 9481   ...cfz 11661   #chash 12360  Word cword 12487   substr csubstr 12491   ClWalks cclwlk 24409
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-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-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-substr 12499  df-wlk 24170  df-clwlk 24412
This theorem is referenced by:  clwlkfclwwlk  24506
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