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Theorem ccatval3 12389
Description: Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
ccatval3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
I  +  ( # `  S ) ) )  =  ( T `  I ) )

Proof of Theorem ccatval3
StepHypRef Expression
1 simp3 990 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ( 0..^ ( # `  T
) ) )
2 lencl 12360 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
323ad2ant1 1009 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  NN0 )
43nn0cnd 10742 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  CC )
5 lencl 12360 . . . . . . . . 9  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
653ad2ant2 1010 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  NN0 )
76nn0cnd 10742 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  CC )
84, 7pncan2d 9825 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) )  =  ( # `  T
) )
98oveq2d 6209 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) )  =  ( 0..^ ( # `  T
) ) )
101, 9eleqtrrd 2542 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ( 0..^ ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) ) ) )
113nn0zd 10849 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  ZZ )
126nn0zd 10849 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  ZZ )
1311, 12zaddcld 10855 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ZZ )
14 fzoaddel2 11708 . . . 4  |-  ( ( I  e.  ( 0..^ ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) )  /\  ( ( # `  S )  +  (
# `  T )
)  e.  ZZ  /\  ( # `  S )  e.  ZZ )  -> 
( I  +  (
# `  S )
)  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
1510, 13, 11, 14syl3anc 1219 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( I  +  ( # `  S ) )  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
16 ccatval2 12388 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  (
I  +  ( # `  S ) )  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  -> 
( ( S concat  T
) `  ( I  +  ( # `  S
) ) )  =  ( T `  (
( I  +  (
# `  S )
)  -  ( # `  S ) ) ) )
1715, 16syld3an3 1264 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
I  +  ( # `  S ) ) )  =  ( T `  ( ( I  +  ( # `  S ) )  -  ( # `  S ) ) ) )
18 elfzoelz 11663 . . . . . 6  |-  ( I  e.  ( 0..^ (
# `  T )
)  ->  I  e.  ZZ )
19183ad2ant3 1011 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ZZ )
2019zcnd 10852 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  CC )
2120, 4pncand 9824 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( I  +  ( # `  S
) )  -  ( # `
 S ) )  =  I )
2221fveq2d 5796 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( T `  ( ( I  +  ( # `  S ) )  -  ( # `  S ) ) )  =  ( T `  I ) )
2317, 22eqtrd 2492 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
I  +  ( # `  S ) ) )  =  ( T `  I ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   0cc0 9386    + caddc 9389    - cmin 9699   NN0cn0 10683   ZZcz 10750  ..^cfzo 11658   #chash 12213  Word cword 12332   concat cconcat 12334
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342
This theorem is referenced by:  swrdccat2  12463  cats1un  12481  splfv2a  12509  revccat  12517  cats1fvn  12596  gsumccat  15630  efgsval2  16343  efgsp1  16347  pgpfaclem1  16696  signstfvn  27107
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