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Theorem clwlkf1clwwlklem1 25419
Description: Lemma 1 for clwlkf1clwwlklem 25422. (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.1 . . . . . 6  |-  A  =  ( 1st `  c
)
2 fveq2 5881 . . . . . 6  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
31, 2syl5eq 2482 . . . . 5  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
43fveq2d 5885 . . . 4  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
54eqeq1d 2431 . . 3  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
6 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
75, 6elrab2 3237 . 2  |-  ( W  e.  C  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `  ( 1st `  W ) )  =  N ) )
8 eqid 2429 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
9 eqid 2429 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
108, 9clwlkcompim 25337 . . . 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
) ) ) ) ) )
11 lencl 12674 . . . . . 6  |-  ( ( 1st `  W )  e. Word  dom  E  ->  (
# `  ( 1st `  W ) )  e. 
NN0 )
12 ffn 5746 . . . . . 6  |-  ( ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V  ->  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )
13 nn0re 10878 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  e.  RR )
1413lep1d 10538 . . . . . . . . 9  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  <_  ( ( # `  ( 1st `  W
) )  +  1 ) )
1514adantr 466 . . . . . . . 8  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( # `  ( 1st `  W ) )  <_  ( ( # `  ( 1st `  W
) )  +  1 ) )
16 fz0hash 12608 . . . . . . . 8  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( # `  ( 2nd `  W ) )  =  ( ( # `  ( 1st `  W
) )  +  1 ) )
1715, 16breqtrrd 4452 . . . . . . 7  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( # `  ( 1st `  W ) )  <_  ( # `  ( 2nd `  W ) ) )
18 breq1 4429 . . . . . . 7  |-  ( (
# `  ( 1st `  W ) )  =  N  ->  ( ( # `
 ( 1st `  W
) )  <_  ( # `
 ( 2nd `  W
) )  <->  N  <_  (
# `  ( 2nd `  W ) ) ) )
1917, 18syl5ibcom 223 . . . . . 6  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W )  Fn  ( 0 ... ( # `
 ( 1st `  W
) ) ) )  ->  ( ( # `  ( 1st `  W
) )  =  N  ->  N  <_  ( # `
 ( 2nd `  W
) ) ) )
2011, 12, 19syl2an 479 . . . . 5  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( ( # `  ( 1st `  W ) )  =  N  ->  N  <_  ( # `  ( 2nd `  W ) ) ) )
2120adantr 466 . . . 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
) ) ) )
2210, 21syl 17 . . 3  |-  ( W  e.  ( V ClWalks  E
)  ->  ( ( # `
 ( 1st `  W
) )  =  N  ->  N  <_  ( # `
 ( 2nd `  W
) ) ) )
2322imp 430 . 2  |-  ( ( W  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  N  <_  (
# `  ( 2nd `  W ) ) )
247, 23sylbi 198 1  |-  ( W  e.  C  ->  N  <_  ( # `  ( 2nd `  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   {cpr 4004   <.cop 4008   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   0cc0 9538   1c1 9539    + caddc 9541    <_ cle 9675   NN0cn0 10869   ...cfz 11782  ..^cfzo 11913   #chash 12512  Word cword 12643   substr csubstr 12647   ClWalks cclwlk 25320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-wlk 25081  df-clwlk 25323
This theorem is referenced by:  clwlkf1clwwlklem  25422
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