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Theorem ccatlid 11635
Description: Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
ccatlid  |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )

Proof of Theorem ccatlid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 wrd0 11619 . . . . 5  |-  (/)  e. Word  B
2 ccatcl 11630 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( (/) concat  S )  e. Word  B
)
31, 2mpan 651 . . . 4  |-  ( S  e. Word  B  ->  ( (/) concat  S )  e. Word  B
)
4 wrdf 11620 . . . 4  |-  ( (
(/) concat  S )  e. Word  B  ->  ( (/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B )
5 ffn 5495 . . . 4  |-  ( (
(/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B  ->  ( (/) concat  S )  Fn  ( 0..^ (
# `  ( (/) concat  S ) ) ) )
63, 4, 53syl 18 . . 3  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) ) )
7 ccatlen 11631 . . . . . . 7  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
81, 7mpan 651 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
9 hash0 11533 . . . . . . . 8  |-  ( # `  (/) )  =  0
109oveq1i 5991 . . . . . . 7  |-  ( (
# `  (/) )  +  ( # `  S
) )  =  ( 0  +  ( # `  S ) )
11 lencl 11622 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
1211nn0cnd 10169 . . . . . . . 8  |-  ( S  e. Word  B  ->  ( # `
 S )  e.  CC )
1312addid2d 9160 . . . . . . 7  |-  ( S  e. Word  B  ->  (
0  +  ( # `  S ) )  =  ( # `  S
) )
1410, 13syl5eq 2410 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )  +  ( # `  S
) )  =  (
# `  S )
)
158, 14eqtrd 2398 . . . . 5  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( # `  S ) )
1615oveq2d 5997 . . . 4  |-  ( S  e. Word  B  ->  (
0..^ ( # `  ( (/) concat  S ) ) )  =  ( 0..^ (
# `  S )
) )
1716fneq2d 5441 . . 3  |-  ( S  e. Word  B  ->  (
( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) )  <-> 
( (/) concat  S )  Fn  (
0..^ ( # `  S
) ) ) )
186, 17mpbid 201 . 2  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  S
) ) )
19 wrdf 11620 . . 3  |-  ( S  e. Word  B  ->  S : ( 0..^ (
# `  S )
) --> B )
20 ffn 5495 . . 3  |-  ( S : ( 0..^ (
# `  S )
) --> B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2119, 20syl 15 . 2  |-  ( S  e. Word  B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
229a1i 10 . . . . . . 7  |-  ( S  e. Word  B  ->  ( # `
 (/) )  =  0 )
2322, 14oveq12d 5999 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  =  ( 0..^ ( # `  S ) ) )
2423eleq2d 2433 . . . . 5  |-  ( S  e. Word  B  ->  (
x  e.  ( (
# `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
2524biimpar 471 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )
26 ccatval2 11633 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B  /\  x  e.  ( ( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
271, 26mp3an1 1265 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
2825, 27syldan 456 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  (
x  -  ( # `  (/) ) ) ) )
299oveq2i 5992 . . . . 5  |-  ( x  -  ( # `  (/) ) )  =  ( x  - 
0 )
30 elfzoelz 11030 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
3130adantl 452 . . . . . . 7  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ZZ )
3231zcnd 10269 . . . . . 6  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  CC )
3332subid1d 9293 . . . . 5  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  0 )  =  x )
3429, 33syl5eq 2410 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  ( # `
 (/) ) )  =  x )
3534fveq2d 5636 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  -  ( # `  (/) ) ) )  =  ( S `  x ) )
3628, 35eqtrd 2398 . 2  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  x
) )
3718, 21, 36eqfnfvd 5732 1  |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   (/)c0 3543    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981   0cc0 8884    + caddc 8887    - cmin 9184   ZZcz 10175  ..^cfzo 11025   #chash 11505  Word cword 11604   concat cconcat 11605
This theorem is referenced by:  s0s1  11756  gsumccat  14674  frmdmnd  14691  frmd0  14692  efginvrel2  15246  efgcpbl2  15276  frgp0  15279  frgpnabllem1  15371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-fzo 11026  df-hash 11506  df-word 11610  df-concat 11611
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