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Theorem ccats1val2 12590
Description: Value of the symbol concatenated with a word. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Proof shortened by Alexander van der Vekens, 14-Oct-2018.)
Assertion
Ref Expression
ccats1val2  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( W concat  <" S "> ) `  I
)  =  S )

Proof of Theorem ccats1val2
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  W  e. Word  V )
2 s1cl 12573 . . . 4  |-  ( S  e.  V  ->  <" S ">  e. Word  V )
323ad2ant2 1018 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  <" S ">  e. Word  V )
4 lencl 12524 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
54nn0zd 10960 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ZZ )
6 elfzomin 11851 . . . . . . . 8  |-  ( (
# `  W )  e.  ZZ  ->  ( # `  W
)  e.  ( (
# `  W )..^ ( ( # `  W
)  +  1 ) ) )
75, 6syl 16 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  1 ) ) )
8 s1len 12576 . . . . . . . . 9  |-  ( # `  <" S "> )  =  1
98oveq2i 6293 . . . . . . . 8  |-  ( (
# `  W )  +  ( # `  <" S "> )
)  =  ( (
# `  W )  +  1 )
109oveq2i 6293 . . . . . . 7  |-  ( (
# `  W )..^ ( ( # `  W
)  +  ( # `  <" S "> ) ) )  =  ( ( # `  W
)..^ ( ( # `  W )  +  1 ) )
117, 10syl6eleqr 2566 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
1211adantr 465 . . . . 5  |-  ( ( W  e. Word  V  /\  I  =  ( # `  W
) )  ->  ( # `
 W )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
13 eleq1 2539 . . . . . 6  |-  ( I  =  ( # `  W
)  ->  ( I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) )  <-> 
( # `  W )  e.  ( ( # `  W )..^ ( (
# `  W )  +  ( # `  <" S "> )
) ) ) )
1413adantl 466 . . . . 5  |-  ( ( W  e. Word  V  /\  I  =  ( # `  W
) )  ->  (
I  e.  ( (
# `  W )..^ ( ( # `  W
)  +  ( # `  <" S "> ) ) )  <->  ( # `  W
)  e.  ( (
# `  W )..^ ( ( # `  W
)  +  ( # `  <" S "> ) ) ) ) )
1512, 14mpbird 232 . . . 4  |-  ( ( W  e. Word  V  /\  I  =  ( # `  W
) )  ->  I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
16153adant2 1015 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
17 ccatval2 12557 . . 3  |-  ( ( W  e. Word  V  /\  <" S ">  e. Word  V  /\  I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )  ->  ( ( W concat  <" S "> ) `  I )  =  ( <" S "> `  ( I  -  ( # `  W
) ) ) )
181, 3, 16, 17syl3anc 1228 . 2  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( W concat  <" S "> ) `  I
)  =  ( <" S "> `  ( I  -  ( # `
 W ) ) ) )
19 oveq1 6289 . . . . 5  |-  ( I  =  ( # `  W
)  ->  ( I  -  ( # `  W
) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
20193ad2ant3 1019 . . . 4  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
I  -  ( # `  W ) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
214nn0cnd 10850 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
2221subidd 9914 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  -  ( # `  W
) )  =  0 )
23223ad2ant1 1017 . . . 4  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( # `  W )  -  ( # `  W
) )  =  0 )
2420, 23eqtrd 2508 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
I  -  ( # `  W ) )  =  0 )
2524fveq2d 5868 . 2  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  ( <" S "> `  ( I  -  ( # `
 W ) ) )  =  ( <" S "> `  0 ) )
26 s1fv 12578 . . 3  |-  ( S  e.  V  ->  ( <" S "> `  0 )  =  S )
27263ad2ant2 1018 . 2  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  ( <" S "> `  0 )  =  S )
2818, 25, 273eqtrd 2512 1  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( W concat  <" S "> ) `  I
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801   ZZcz 10860  ..^cfzo 11788   #chash 12369  Word cword 12496   concat cconcat 12498   <"cs1 12499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507
This theorem is referenced by:  ccatws1ls  12596  gsmsymgrfixlem1  16247  gsmsymgreqlem2  16251  wwlknext  24400
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