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Theorem ccatw2s1p1 30418
Description: Extract the first of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
Assertion
Ref Expression
ccatw2s1p1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  N )  =  X )

Proof of Theorem ccatw2s1p1
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  W  e. Word  V )
21anim1i 568 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( W  e. Word  V  /\  ( X  e.  V  /\  Y  e.  V
) ) )
3 3anass 969 . . . . 5  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  <->  ( W  e. Word  V  /\  ( X  e.  V  /\  Y  e.  V
) ) )
42, 3sylibr 212 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V
) )
5 ccatw2s1ass 12427 . . . 4  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W concat  <" X "> ) concat  <" Y "> )  =  ( W concat  ( <" X "> concat  <" Y "> ) ) )
64, 5syl 16 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( W concat  <" X "> ) concat  <" Y "> )  =  ( W concat  ( <" X "> concat  <" Y "> ) ) )
76fveq1d 5802 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  N )  =  ( ( W concat 
( <" X "> concat 
<" Y "> ) ) `  N
) )
81adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  W  e. Word  V )
9 ccat2s1cl 30412 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( <" X "> concat  <" Y "> )  e. Word  V )
109adantl 466 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( <" X "> concat 
<" Y "> )  e. Word  V )
11 lencl 12368 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
12 eleq1 2526 . . . . . . . 8  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  e.  NN0  <->  N  e.  NN0 ) )
13 nn0z 10781 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
14 2z 10790 . . . . . . . . . . 11  |-  2  e.  ZZ
1514a1i 11 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  2  e.  ZZ )
1613, 15zaddcld 10863 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  2 )  e.  ZZ )
17 2pos 10525 . . . . . . . . . 10  |-  0  <  2
18 2re 10503 . . . . . . . . . . . 12  |-  2  e.  RR
1918a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  2  e.  RR )
20 nn0re 10700 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
2119, 20ltaddposd 10035 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 0  <  2  <->  N  <  ( N  +  2 ) ) )
2217, 21mpbii 211 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  < 
( N  +  2 ) )
23 fzolb 11676 . . . . . . . . 9  |-  ( N  e.  ( N..^ ( N  +  2 ) )  <->  ( N  e.  ZZ  /\  ( N  +  2 )  e.  ZZ  /\  N  < 
( N  +  2 ) ) )
2413, 16, 22, 23syl3anbrc 1172 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( N..^ ( N  +  2 ) ) )
2512, 24syl6bi 228 . . . . . . 7  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  e.  NN0  ->  N  e.  ( N..^ ( N  + 
2 ) ) ) )
2611, 25mpan9 469 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  N  e.  ( N..^ ( N  +  2
) ) )
27 id 22 . . . . . . . . 9  |-  ( (
# `  W )  =  N  ->  ( # `  W )  =  N )
28 oveq1 6208 . . . . . . . . 9  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  +  2 )  =  ( N  +  2 ) )
2927, 28oveq12d 6219 . . . . . . . 8  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) )  =  ( N..^ ( N  +  2 ) ) )
3029eleq2d 2524 . . . . . . 7  |-  ( (
# `  W )  =  N  ->  ( N  e.  ( ( # `  W )..^ ( (
# `  W )  +  2 ) )  <-> 
N  e.  ( N..^ ( N  +  2 ) ) ) )
3130adantl 466 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( N  e.  ( ( # `  W
)..^ ( ( # `  W )  +  2 ) )  <->  N  e.  ( N..^ ( N  + 
2 ) ) ) )
3226, 31mpbird 232 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  N  e.  ( ( # `
 W )..^ ( ( # `  W
)  +  2 ) ) )
3332adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  e.  ( ( # `
 W )..^ ( ( # `  W
)  +  2 ) ) )
34 ccat2s1len 30413 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( # `  ( <" X "> concat  <" Y "> ) )  =  2 )
3534adantl 466 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( # `  ( <" X "> concat  <" Y "> ) )  =  2 )
3635oveq2d 6217 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( # `  W
)  +  ( # `  ( <" X "> concat  <" Y "> ) ) )  =  ( ( # `  W
)  +  2 ) )
3736oveq2d 6217 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( # `  W
)..^ ( ( # `  W )  +  (
# `  ( <" X "> concat  <" Y "> ) ) ) )  =  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) ) )
3833, 37eleqtrrd 2545 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  e.  ( ( # `
 W )..^ ( ( # `  W
)  +  ( # `  ( <" X "> concat  <" Y "> ) ) ) ) )
39 ccatval2 12396 . . 3  |-  ( ( W  e. Word  V  /\  ( <" X "> concat 
<" Y "> )  e. Word  V  /\  N  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  ( <" X "> concat  <" Y "> ) ) ) ) )  ->  (
( W concat  ( <" X "> concat  <" Y "> ) ) `  N )  =  ( ( <" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) ) )
408, 10, 38, 39syl3anc 1219 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( W concat  ( <" X "> concat  <" Y "> ) ) `  N
)  =  ( (
<" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) ) )
41 oveq1 6208 . . . . . 6  |-  ( N  =  ( # `  W
)  ->  ( N  -  ( # `  W
) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
4241eqcoms 2466 . . . . 5  |-  ( (
# `  W )  =  N  ->  ( N  -  ( # `  W
) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
4311nn0cnd 10750 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
4443subidd 9819 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  -  ( # `  W
) )  =  0 )
4542, 44sylan9eqr 2517 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( N  -  ( # `
 W ) )  =  0 )
4645fveq2d 5804 . . 3  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( ( <" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) )  =  ( ( <" X "> concat  <" Y "> ) `  0 ) )
47 ccat2s1p1 30414 . . 3  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( ( <" X "> concat  <" Y "> ) `  0 )  =  X )
4846, 47sylan9eq 2515 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( <" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) )  =  X )
497, 40, 483eqtrd 2499 1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  N )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   RRcr 9393   0cc0 9394    + caddc 9397    < clt 9530    - cmin 9707   2c2 10483   NN0cn0 10691   ZZcz 10758  ..^cfzo 11666   #chash 12221  Word cword 12340   concat cconcat 12342   <"cs1 12343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-concat 12350  df-s1 12351
This theorem is referenced by:  numclwwlkovf2ex  30828  numclwlk1lem2foa  30833
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