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Theorem clwlkf1clwwlklem1 24508
Description: Lemma 1 for clwlkf1clwwlklem 24511. (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
clwlkf1clwwlklem1  |-  ( W  e.  C  ->  N  <_  ( # `  ( 2nd `  W ) ) )
Distinct variable groups:    E, c    N, c    V, c    W, c    C, c    F, c
Allowed substitution hints:    A( c)    B( c)

Proof of Theorem clwlkf1clwwlklem1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.c . . . 4  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
21eleq2i 2538 . . 3  |-  ( W  e.  C  <->  W  e.  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N } )
3 clwlkfclwwlk.1 . . . . . . 7  |-  A  =  ( 1st `  c
)
4 fveq2 5857 . . . . . . 7  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
53, 4syl5eq 2513 . . . . . 6  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
65fveq2d 5861 . . . . 5  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
76eqeq1d 2462 . . . 4  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
87elrab 3254 . . 3  |-  ( W  e.  { c  e.  ( V ClWalks  E )  |  ( # `  A
)  =  N }  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  W
) )  =  N ) )
92, 8bitri 249 . 2  |-  ( W  e.  C  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `  ( 1st `  W ) )  =  N ) )
10 eqid 2460 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
11 eqid 2460 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
1210, 11clwlkcompim 24426 . . . 4  |-  ( W  e.  ( V ClWalks  E
)  ->  ( (
( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  i )
)  =  { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  /\  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  ( # `
 ( 1st `  W
) ) ) ) ) )
13 lencl 12515 . . . . . 6  |-  ( ( 1st `  W )  e. Word  dom  E  ->  (
# `  ( 1st `  W ) )  e. 
NN0 )
14 ffn 5722 . . . . . 6  |-  ( ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  ->  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )
15 nn0re 10793 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  e.  RR )
1615lep1d 10466 . . . . . . . . 9  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  <_  ( ( # `  ( 1st `  W
) )  +  1 ) )
1716adantr 465 . . . . . . . 8  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( # `  ( 1st `  W ) )  <_  ( ( # `  ( 1st `  W
) )  +  1 ) )
18 fz0hash 12452 . . . . . . . 8  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( # `  ( 2nd `  W ) )  =  ( ( # `  ( 1st `  W
) )  +  1 ) )
1917, 18breqtrrd 4466 . . . . . . 7  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( # `  ( 1st `  W ) )  <_  ( # `  ( 2nd `  W ) ) )
20 breq1 4443 . . . . . . 7  |-  ( (
# `  ( 1st `  W ) )  =  N  ->  ( ( # `
 ( 1st `  W
) )  <_  ( # `
 ( 2nd `  W
) )  <->  N  <_  (
# `  ( 2nd `  W ) ) ) )
2119, 20syl5ibcom 220 . . . . . 6  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( ( # `  ( 1st `  W
) )  =  N  ->  N  <_  ( # `
 ( 2nd `  W
) ) ) )
2213, 14, 21syl2an 477 . . . . 5  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( ( # `  ( 1st `  W ) )  =  N  ->  N  <_  ( # `  ( 2nd `  W ) ) ) )
2322adantr 465 . . . 4  |-  ( ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  i )
)  =  { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  /\  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  ( # `
 ( 1st `  W
) ) ) ) )  ->  ( ( # `
 ( 1st `  W
) )  =  N  ->  N  <_  ( # `
 ( 2nd `  W
) ) ) )
2412, 23syl 16 . . 3  |-  ( W  e.  ( V ClWalks  E
)  ->  ( ( # `
 ( 1st `  W
) )  =  N  ->  N  <_  ( # `
 ( 2nd `  W
) ) ) )
2524imp 429 . 2  |-  ( ( W  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  N  <_  (
# `  ( 2nd `  W ) ) )
269, 25sylbi 195 1  |-  ( W  e.  C  ->  N  <_  ( # `  ( 2nd `  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   {cpr 4022   <.cop 4026   class class class wbr 4440    |-> cmpt 4498   dom cdm 4992    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618   NN0cn0 10784   ...cfz 11661  ..^cfzo 11781   #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-wlk 24170  df-clwlk 24412
This theorem is referenced by:  clwlkf1clwwlklem  24511
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