MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccats1val2 Structured version   Unicode version

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 ++  <" S "> ) `  I
)  =  S )

Proof of Theorem ccats1val2
StepHypRef Expression
1 simp1 995 . . 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 1017 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  <" S ">  e. Word  V )
4 lencl 12519 . . . . . . . . 9  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
54nn0zd 10924 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ZZ )
6 elfzomin 11834 . . . . . . . 8  |-  ( (
# `  W )  e.  ZZ  ->  ( # `  W
)  e.  ( (
# `  W )..^ ( ( # `  W
)  +  1 ) ) )
75, 6syl 17 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  1 ) ) )
8 s1len 12576 . . . . . . . . 9  |-  ( # `  <" S "> )  =  1
98oveq2i 6243 . . . . . . . 8  |-  ( (
# `  W )  +  ( # `  <" S "> )
)  =  ( (
# `  W )  +  1 )
109oveq2i 6243 . . . . . . 7  |-  ( (
# `  W )..^ ( ( # `  W
)  +  ( # `  <" S "> ) ) )  =  ( ( # `  W
)..^ ( ( # `  W )  +  1 ) )
117, 10syl6eleqr 2499 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
1211adantr 463 . . . . 5  |-  ( ( W  e. Word  V  /\  I  =  ( # `  W
) )  ->  ( # `
 W )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
13 eleq1 2472 . . . . . 6  |-  ( I  =  ( # `  W
)  ->  ( I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) )  <-> 
( # `  W )  e.  ( ( # `  W )..^ ( (
# `  W )  +  ( # `  <" S "> )
) ) ) )
1413adantl 464 . . . . 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 1014 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )
17 ccatval2 12555 . . 3  |-  ( ( W  e. Word  V  /\  <" S ">  e. Word  V  /\  I  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  <" S "> ) ) ) )  ->  ( ( W ++  <" S "> ) `  I )  =  ( <" S "> `  ( I  -  ( # `  W
) ) ) )
181, 3, 16, 17syl3anc 1228 . 2  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( W ++  <" S "> ) `  I
)  =  ( <" S "> `  ( I  -  ( # `
 W ) ) ) )
19 oveq1 6239 . . . . 5  |-  ( I  =  ( # `  W
)  ->  ( I  -  ( # `  W
) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
20193ad2ant3 1018 . . . 4  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
I  -  ( # `  W ) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
214nn0cnd 10813 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
2221subidd 9873 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  -  ( # `  W
) )  =  0 )
23223ad2ant1 1016 . . . 4  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( # `  W )  -  ( # `  W
) )  =  0 )
2420, 23eqtrd 2441 . . 3  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
I  -  ( # `  W ) )  =  0 )
2524fveq2d 5807 . 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 1017 . 2  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  ( <" S "> `  0 )  =  S )
2818, 25, 273eqtrd 2445 1  |-  ( ( W  e. Word  V  /\  S  e.  V  /\  I  =  ( # `  W
) )  ->  (
( W ++  <" S "> ) `  I
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441    + caddc 9443    - cmin 9759   ZZcz 10823  ..^cfzo 11765   #chash 12357  Word cword 12488   ++ cconcat 12490   <"cs1 12491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-fz 11642  df-fzo 11766  df-hash 12358  df-word 12496  df-concat 12498  df-s1 12499
This theorem is referenced by:  ccatws1ls  12596  ccatw2s1p1  12599  ccatw2s1p2  12600  gsmsymgrfixlem1  16666  gsmsymgreqlem2  16670  wwlknext  25023
  Copyright terms: Public domain W3C validator