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Theorem ccatlid 12585
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 12547 . . . . 5  |-  (/)  e. Word  B
2 ccatcl 12575 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( (/) concat  S )  e. Word  B
)
31, 2mpan 670 . . . 4  |-  ( S  e. Word  B  ->  ( (/) concat  S )  e. Word  B
)
4 wrdf 12535 . . . 4  |-  ( (
(/) concat  S )  e. Word  B  ->  ( (/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B )
5 ffn 5721 . . . 4  |-  ( (
(/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B  ->  ( (/) concat  S )  Fn  ( 0..^ (
# `  ( (/) concat  S ) ) ) )
63, 4, 53syl 20 . . 3  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) ) )
7 ccatlen 12576 . . . . . . 7  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
81, 7mpan 670 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
9 hash0 12419 . . . . . . . 8  |-  ( # `  (/) )  =  0
109oveq1i 6291 . . . . . . 7  |-  ( (
# `  (/) )  +  ( # `  S
) )  =  ( 0  +  ( # `  S ) )
11 lencl 12544 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
1211nn0cnd 10861 . . . . . . . 8  |-  ( S  e. Word  B  ->  ( # `
 S )  e.  CC )
1312addid2d 9784 . . . . . . 7  |-  ( S  e. Word  B  ->  (
0  +  ( # `  S ) )  =  ( # `  S
) )
1410, 13syl5eq 2496 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )  +  ( # `  S
) )  =  (
# `  S )
)
158, 14eqtrd 2484 . . . . 5  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( # `  S ) )
1615oveq2d 6297 . . . 4  |-  ( S  e. Word  B  ->  (
0..^ ( # `  ( (/) concat  S ) ) )  =  ( 0..^ (
# `  S )
) )
1716fneq2d 5662 . . 3  |-  ( S  e. Word  B  ->  (
( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) )  <-> 
( (/) concat  S )  Fn  (
0..^ ( # `  S
) ) ) )
186, 17mpbid 210 . 2  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  S
) ) )
19 wrdfn 12542 . 2  |-  ( S  e. Word  B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
209a1i 11 . . . . . . 7  |-  ( S  e. Word  B  ->  ( # `
 (/) )  =  0 )
2120, 14oveq12d 6299 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  =  ( 0..^ ( # `  S ) ) )
2221eleq2d 2513 . . . . 5  |-  ( S  e. Word  B  ->  (
x  e.  ( (
# `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
2322biimpar 485 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )
24 ccatval2 12578 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B  /\  x  e.  ( ( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
251, 24mp3an1 1312 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
2623, 25syldan 470 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  (
x  -  ( # `  (/) ) ) ) )
279oveq2i 6292 . . . . 5  |-  ( x  -  ( # `  (/) ) )  =  ( x  - 
0 )
28 elfzoelz 11811 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
2928adantl 466 . . . . . . 7  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ZZ )
3029zcnd 10977 . . . . . 6  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  CC )
3130subid1d 9925 . . . . 5  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  0 )  =  x )
3227, 31syl5eq 2496 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  ( # `
 (/) ) )  =  x )
3332fveq2d 5860 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  -  ( # `  (/) ) ) )  =  ( S `  x ) )
3426, 33eqtrd 2484 . 2  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  x
) )
3518, 19, 34eqfnfvd 5969 1  |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   (/)c0 3770    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   0cc0 9495    + caddc 9498    - cmin 9810   ZZcz 10871  ..^cfzo 11806   #chash 12387  Word cword 12516   concat cconcat 12518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-concat 12526
This theorem is referenced by:  swrdccat  12700  swrdccat3a  12701  s0s1  12852  gsumccat  15988  frmdmnd  16006  frmd0  16007  efginvrel2  16724  efgcpbl2  16754  frgp0  16757  frgpnabllem1  16856  signstfvneq0  28507  elmrsubrn  28858
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