MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccatval3 Structured version   Unicode version

Theorem ccatval3 11749
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 960 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ( 0..^ ( # `  T
) ) )
2 lencl 11737 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
323ad2ant1 979 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  NN0 )
43nn0cnd 10278 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  CC )
5 lencl 11737 . . . . . . . . 9  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
653ad2ant2 980 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  NN0 )
76nn0cnd 10278 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  CC )
84, 7pncan2d 9415 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) )  =  ( # `  T
) )
98oveq2d 6099 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( 0..^ ( ( ( # `  S
)  +  ( # `  T ) )  -  ( # `  S ) ) )  =  ( 0..^ ( # `  T
) ) )
101, 9eleqtrrd 2515 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ( 0..^ ( ( (
# `  S )  +  ( # `  T
) )  -  ( # `
 S ) ) ) )
113nn0zd 10375 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  S
)  e.  ZZ )
126nn0zd 10375 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( # `  T
)  e.  ZZ )
1311, 12zaddcld 10381 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ZZ )
14 fzoaddel2 11178 . . . 4  |-  ( ( I  e.  ( 0..^ ( ( ( # `  S )  +  (
# `  T )
)  -  ( # `  S ) ) )  /\  ( ( # `  S )  +  (
# `  T )
)  e.  ZZ  /\  ( # `  S )  e.  ZZ )  -> 
( I  +  (
# `  S )
)  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
1510, 13, 11, 14syl3anc 1185 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( I  +  ( # `  S ) )  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
16 ccatval2 11748 . . 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 1230 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  (
I  +  ( # `  S ) ) )  =  ( T `  ( ( I  +  ( # `  S ) )  -  ( # `  S ) ) ) )
18 elfzoelz 11142 . . . . . 6  |-  ( I  e.  ( 0..^ (
# `  T )
)  ->  I  e.  ZZ )
19183ad2ant3 981 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  ZZ )
2019zcnd 10378 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  I  e.  CC )
2120, 4pncand 9414 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( ( I  +  ( # `  S
) )  -  ( # `
 S ) )  =  I )
2221fveq2d 5734 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  T ) ) )  ->  ( T `  ( ( I  +  ( # `  S ) )  -  ( # `  S ) ) )  =  ( T `  I ) )
2317, 22eqtrd 2470 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 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   0cc0 8992    + caddc 8995    - cmin 9293   NN0cn0 10223   ZZcz 10284  ..^cfzo 11137   #chash 11620  Word cword 11719   concat cconcat 11720
This theorem is referenced by:  swrdccat2  11777  splfv2a  11787  cats1un  11792  revccat  11800  cats1fvn  11824  gsumccat  14789  efgsval2  15367  efgsp1  15371  pgpfaclem1  15641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726
  Copyright terms: Public domain W3C validator