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Theorem ccatval3 11523
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 957 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ( 0..^ ( # `  T
) ) )
2 lencl 11511 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
323ad2ant1 976 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  NN0 )
43nn0cnd 10109 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  CC )
5 lencl 11511 . . . . . . . . 9  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
653ad2ant2 977 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  NN0 )
76nn0cnd 10109 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  CC )
84, 7pncan2d 9246 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) )  =  ( # `  T
) )
98oveq2d 5958 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) )  =  ( 0..^ ( # `  T
) ) )
101, 9eleqtrrd 2435 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ( 0..^ ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) ) ) )
113nn0zd 10204 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  ZZ )
126nn0zd 10204 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  ZZ )
1311, 12zaddcld 10210 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ZZ )
14 fzoaddel2 10996 . . . 4  |-  ( ( I  e.  ( 0..^ ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) )  /\  ( ( # `  S )  +  (
# `  T )
)  e.  ZZ  /\  ( # `  S )  e.  ZZ )  -> 
( I  +  (
# `  S )
)  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
1510, 13, 11, 14syl3anc 1182 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( I  +  ( # `  S ) )  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
16 ccatval2 11522 . . 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 1227 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
I  +  ( # `  S ) ) )  =  ( T `  ( ( I  +  ( # `  S ) )  -  ( # `  S ) ) ) )
18 elfzoelz 10964 . . . . . 6  |-  ( I  e.  ( 0..^ (
# `  T )
)  ->  I  e.  ZZ )
19183ad2ant3 978 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ZZ )
2019zcnd 10207 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  CC )
2120, 4pncand 9245 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( I  +  ( # `  S
) )  -  ( # `
 S ) )  =  I )
2221fveq2d 5609 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( T `  ( ( I  +  ( # `  S ) )  -  ( # `  S ) ) )  =  ( T `  I ) )
2317, 22eqtrd 2390 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
I  +  ( # `  S ) ) )  =  ( T `  I ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5334  (class class class)co 5942   0cc0 8824    + caddc 8827    - cmin 9124   NN0cn0 10054   ZZcz 10113  ..^cfzo 10959   #chash 11427  Word cword 11493   concat cconcat 11494
This theorem is referenced by:  swrdccat2  11551  splfv2a  11561  cats1un  11566  revccat  11574  cats1fvn  11598  gsumccat  14557  efgsval2  15135  efgsp1  15139  pgpfaclem1  15409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-fzo 10960  df-hash 11428  df-word 11499  df-concat 11500
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