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Theorem ccatw2s1p1 12648
Description: Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.)
Assertion
Ref Expression
ccatw2s1p1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W ++ 
<" X "> ) ++  <" Y "> ) `  N )  =  X )

Proof of Theorem ccatw2s1p1
StepHypRef Expression
1 ccatws1cl 12632 . . . 4  |-  ( ( W  e. Word  V  /\  X  e.  V )  ->  ( W ++  <" X "> )  e. Word  V
)
21ad2ant2r 746 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( W ++  <" X "> )  e. Word  V
)
3 simpr 461 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  Y  e.  V )
43adantl 466 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  Y  e.  V )
5 lencl 12568 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
6 fzonn0p1 11894 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ( 0..^ ( ( # `  W
)  +  1 ) ) )
75, 6syl 16 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ( 0..^ ( (
# `  W )  +  1 ) ) )
87adantr 465 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( # `  W )  e.  ( 0..^ ( ( # `  W
)  +  1 ) ) )
98adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( # `  W )  e.  ( 0..^ ( ( # `  W
)  +  1 ) ) )
10 simpr 461 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( # `  W )  =  N )
1110eqcomd 2465 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  N  =  ( # `  W
) )
1211adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  =  ( # `  W
) )
13 ccatws1len 12634 . . . . . 6  |-  ( ( W  e. Word  V  /\  X  e.  V )  ->  ( # `  ( W ++  <" X "> ) )  =  ( ( # `  W
)  +  1 ) )
1413ad2ant2r 746 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( # `  ( W ++ 
<" X "> ) )  =  ( ( # `  W
)  +  1 ) )
1514oveq2d 6312 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( 0..^ ( # `  ( W ++  <" X "> ) ) )  =  ( 0..^ ( ( # `  W
)  +  1 ) ) )
169, 12, 153eltr4d 2560 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  e.  ( 0..^ ( # `  ( W ++  <" X "> ) ) ) )
17 ccats1val1 12638 . . 3  |-  ( ( ( W ++  <" X "> )  e. Word  V  /\  Y  e.  V  /\  N  e.  (
0..^ ( # `  ( W ++  <" X "> ) ) ) )  ->  ( ( ( W ++  <" X "> ) ++  <" Y "> ) `  N
)  =  ( ( W ++  <" X "> ) `  N ) )
182, 4, 16, 17syl3anc 1228 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W ++ 
<" X "> ) ++  <" Y "> ) `  N )  =  ( ( W ++ 
<" X "> ) `  N )
)
19 simpl 457 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  W  e. Word  V )
2019adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  W  e. Word  V )
21 simpl 457 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  X  e.  V )
2221adantl 466 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  X  e.  V )
23 eqcom 2466 . . . . . 6  |-  ( (
# `  W )  =  N  <->  N  =  ( # `
 W ) )
2423biimpi 194 . . . . 5  |-  ( (
# `  W )  =  N  ->  N  =  ( # `  W
) )
2524adantl 466 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  N  =  ( # `  W
) )
2625adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  =  ( # `  W
) )
27 ccats1val2 12639 . . 3  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  N  =  ( # `  W
) )  ->  (
( W ++  <" X "> ) `  N
)  =  X )
2820, 22, 26, 27syl3anc 1228 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( W ++  <" X "> ) `  N )  =  X )
2918, 28eqtrd 2498 1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W ++ 
<" X "> ) ++  <" Y "> ) `  N )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   NN0cn0 10816  ..^cfzo 11820   #chash 12407  Word cword 12537   ++ cconcat 12539   <"cs1 12540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-concat 12547  df-s1 12548
This theorem is referenced by:  numclwwlkovf2ex  25212  numclwlk1lem2foa  25217
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