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Theorem clwlkf1clwwlklem2 30520
Description: Lemma 2 for clwlkf1clwwlklem 30522. (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
clwlkf1clwwlklem2  |-  ( W  e.  C  ->  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) )
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 clwlkf1clwwlklem2
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.c . . . 4  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
21eleq2i 2507 . . 3  |-  ( W  e.  C  <->  W  e.  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N } )
3 clwlkfclwwlk.1 . . . . . . 7  |-  A  =  ( 1st `  c
)
4 fveq2 5691 . . . . . . 7  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
53, 4syl5eq 2487 . . . . . 6  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
65fveq2d 5695 . . . . 5  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
76eqeq1d 2451 . . . 4  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
87elrab 3117 . . 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 2443 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
11 eqid 2443 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
1210, 11clwlkcompim 30427 . . . 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 fveq2 5691 . . . . . . 7  |-  ( (
# `  ( 1st `  W ) )  =  N  ->  ( ( 2nd `  W ) `  ( # `  ( 1st `  W ) ) )  =  ( ( 2nd `  W ) `  N
) )
1413eqeq2d 2454 . . . . . 6  |-  ( (
# `  ( 1st `  W ) )  =  N  ->  ( (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  ( # `
 ( 1st `  W
) ) )  <->  ( ( 2nd `  W ) ` 
0 )  =  ( ( 2nd `  W
) `  N )
) )
1514biimpcd 224 . . . . 5  |-  ( ( ( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  ( # `
 ( 1st `  W
) ) )  -> 
( ( # `  ( 1st `  W ) )  =  N  ->  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) ) )
1615ad2antll 728 . . . 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  ->  ( ( 2nd `  W ) `  0
)  =  ( ( 2nd `  W ) `
 N ) ) )
1712, 16syl 16 . . 3  |-  ( W  e.  ( V ClWalks  E
)  ->  ( ( # `
 ( 1st `  W
) )  =  N  ->  ( ( 2nd `  W ) `  0
)  =  ( ( 2nd `  W ) `
 N ) ) )
1817imp 429 . 2  |-  ( ( W  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  ( ( 2nd `  W ) ` 
0 )  =  ( ( 2nd `  W
) `  N )
)
199, 18sylbi 195 1  |-  ( W  e.  C  ->  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   {cpr 3879   <.cop 3883    e. cmpt 4350   dom cdm 4840   -->wf 5414   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   0cc0 9282   1c1 9283    + caddc 9285   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   substr csubstr 12225   ClWalks cclwlk 30412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-wlk 23415  df-clwlk 30415
This theorem is referenced by:  clwlkf1clwwlklem  30522
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