MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkf1clwwlklem3 Structured version   Visualization version   Unicode version

Theorem clwlkf1clwwlklem3 25576
Description: Lemma 3 for clwlkf1clwwlklem 25577. (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
clwlkf1clwwlklem3  |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
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 clwlkf1clwwlklem3
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.1 . . . . . 6  |-  A  =  ( 1st `  c
)
2 fveq2 5865 . . . . . 6  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
31, 2syl5eq 2497 . . . . 5  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
43fveq2d 5869 . . . 4  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
54eqeq1d 2453 . . 3  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
6 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
75, 6elrab2 3198 . 2  |-  ( W  e.  C  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `  ( 1st `  W ) )  =  N ) )
8 eqid 2451 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
9 eqid 2451 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
108, 9clwlkcompim 25492 . . . 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 12687 . . . . . 6  |-  ( ( 1st `  W )  e. Word  dom  E  ->  (
# `  ( 1st `  W ) )  e. 
NN0 )
12 nn0z 10960 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  e.  ZZ )
13 fzval3 11983 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e.  ZZ  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) )
1412, 13syl 17 . . . . . . . . 9  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) )
1514feq2d 5715 . . . . . . . 8  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V  <-> 
( 2nd `  W
) : ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) --> V ) )
1615biimpa 487 . . . . . . 7  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V )  ->  ( 2nd `  W ) : ( 0..^ ( ( # `  ( 1st `  W
) )  +  1 ) ) --> V )
17 iswrdi 12675 . . . . . . 7  |-  ( ( 2nd `  W ) : ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) --> V  ->  ( 2nd `  W )  e. Word  V
)
1816, 17syl 17 . . . . . 6  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V )  ->  ( 2nd `  W )  e. Word  V
)
1911, 18sylan 474 . . . . 5  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( 2nd `  W
)  e. Word  V )
2019adantr 467 . . . 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
) ) ) ) )  ->  ( 2nd `  W )  e. Word  V
)
2110, 20syl 17 . . 3  |-  ( W  e.  ( V ClWalks  E
)  ->  ( 2nd `  W )  e. Word  V
)
2221adantr 467 . 2  |-  ( ( W  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  ( 2nd `  W )  e. Word  V
)
237, 22sylbi 199 1  |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   {crab 2741   {cpr 3970   <.cop 3974    |-> cmpt 4461   dom cdm 4834   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   0cc0 9539   1c1 9540    + caddc 9542   NN0cn0 10869   ZZcz 10937   ...cfz 11784  ..^cfzo 11915   #chash 12515  Word cword 12656   substr csubstr 12660   ClWalks cclwlk 25475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-wlk 25236  df-clwlk 25478
This theorem is referenced by:  clwlkf1clwwlklem  25577
  Copyright terms: Public domain W3C validator