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Theorem ccatw2s1p2 30441
Description: Extract the second of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
Assertion
Ref Expression
ccatw2s1p2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  ( N  +  1 ) )  =  Y )

Proof of Theorem ccatw2s1p2
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 12430 . . . 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 5804 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  ( N  +  1 ) )  =  ( ( W concat 
( <" X "> concat 
<" Y "> ) ) `  ( N  +  1 ) ) )
81adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  W  e. Word  V )
9 ccat2s1cl 30434 . . . 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 oveq1 6210 . . . . . . . 8  |-  ( N  =  ( # `  W
)  ->  ( N  +  1 )  =  ( ( # `  W
)  +  1 ) )
1211eqcoms 2466 . . . . . . 7  |-  ( (
# `  W )  =  N  ->  ( N  +  1 )  =  ( ( # `  W
)  +  1 ) )
1312adantl 466 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( N  +  1 )  =  ( (
# `  W )  +  1 ) )
14 lencl 12371 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
15 nn0z 10784 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ZZ )
16 1nn0 10710 . . . . . . . . . 10  |-  1  e.  NN0
17 zpnn0elfzo1 30393 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  ZZ  /\  1  e.  NN0 )  -> 
( ( # `  W
)  +  1 )  e.  ( ( # `  W )..^ ( (
# `  W )  +  ( 1  +  1 ) ) ) )
1815, 16, 17sylancl 662 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  1 )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  ( 1  +  1 ) ) ) )
19 1p1e2 10550 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
2019oveq2i 6214 . . . . . . . . . . 11  |-  ( (
# `  W )  +  ( 1  +  1 ) )  =  ( ( # `  W
)  +  2 )
2120a1i 11 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  ( 1  +  1 ) )  =  ( ( # `  W
)  +  2 ) )
2221oveq2d 6219 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )..^ ( ( # `  W
)  +  ( 1  +  1 ) ) )  =  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) ) )
2318, 22eleqtrd 2544 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  1 )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  2 ) ) )
2414, 23syl 16 . . . . . . 7  |-  ( W  e. Word  V  ->  (
( # `  W )  +  1 )  e.  ( ( # `  W
)..^ ( ( # `  W )  +  2 ) ) )
2524adantr 465 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( ( # `  W
)  +  1 )  e.  ( ( # `  W )..^ ( (
# `  W )  +  2 ) ) )
2613, 25eqeltrd 2542 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( N  +  1 )  e.  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) ) )
2726adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( N  +  1 )  e.  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) ) )
28 ccat2s1len 30435 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( # `  ( <" X "> concat  <" Y "> ) )  =  2 )
2928adantl 466 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( # `  ( <" X "> concat  <" Y "> ) )  =  2 )
3029oveq2d 6219 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( # `  W
)  +  ( # `  ( <" X "> concat  <" Y "> ) ) )  =  ( ( # `  W
)  +  2 ) )
3130oveq2d 6219 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( # `  W
)..^ ( ( # `  W )  +  (
# `  ( <" X "> concat  <" Y "> ) ) ) )  =  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) ) )
3227, 31eleqtrrd 2545 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( N  +  1 )  e.  ( (
# `  W )..^ ( ( # `  W
)  +  ( # `  ( <" X "> concat  <" Y "> ) ) ) ) )
33 ccatval2 12399 . . 3  |-  ( ( W  e. Word  V  /\  ( <" X "> concat 
<" Y "> )  e. Word  V  /\  ( N  +  1 )  e.  ( ( # `  W )..^ ( (
# `  W )  +  ( # `  ( <" X "> concat  <" Y "> ) ) ) ) )  ->  ( ( W concat  ( <" X "> concat  <" Y "> ) ) `  ( N  +  1 ) )  =  ( (
<" X "> concat  <" Y "> ) `  ( ( N  +  1 )  -  ( # `  W
) ) ) )
348, 10, 32, 33syl3anc 1219 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( W concat  ( <" X "> concat  <" Y "> ) ) `  ( N  +  1 ) )  =  ( (
<" X "> concat  <" Y "> ) `  ( ( N  +  1 )  -  ( # `  W
) ) ) )
3512oveq1d 6218 . . . . . 6  |-  ( (
# `  W )  =  N  ->  ( ( N  +  1 )  -  ( # `  W
) )  =  ( ( ( # `  W
)  +  1 )  -  ( # `  W
) ) )
36 nn0cn 10704 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  CC )
37 ax-1cn 9455 . . . . . . . 8  |-  1  e.  CC
3837a1i 11 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  1  e.  CC )
3936, 38pncan2d 9836 . . . . . 6  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  +  1 )  -  ( # `  W ) )  =  1 )
4035, 39sylan9eqr 2517 . . . . 5  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =  N )  -> 
( ( N  + 
1 )  -  ( # `
 W ) )  =  1 )
4114, 40sylan 471 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( ( N  + 
1 )  -  ( # `
 W ) )  =  1 )
4241fveq2d 5806 . . 3  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( ( <" X "> concat  <" Y "> ) `  ( ( N  +  1 )  -  ( # `  W
) ) )  =  ( ( <" X "> concat  <" Y "> ) `  1 ) )
43 ccat2s1p2 30437 . . 3  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( ( <" X "> concat  <" Y "> ) `  1 )  =  Y )
4442, 43sylan9eq 2515 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( <" X "> concat  <" Y "> ) `  ( ( N  +  1 )  -  ( # `  W
) ) )  =  Y )
457, 34, 443eqtrd 2499 1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  ( N  +  1 ) )  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9395   1c1 9398    + caddc 9400    - cmin 9710   2c2 10486   NN0cn0 10694   ZZcz 10761  ..^cfzo 11669   #chash 12224  Word cword 12343   concat cconcat 12345   <"cs1 12346
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-concat 12353  df-s1 12354
This theorem is referenced by:  numclwwlkovf2ex  30850  numclwlk1lem2foa  30855
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