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Theorem ccatlid 12557
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 12520 . . . . 5  |-  (/)  e. Word  B
2 ccatcl 12547 . . . . 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 12508 . . . 4  |-  ( (
(/) concat  S )  e. Word  B  ->  ( (/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B )
5 ffn 5724 . . . 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 12548 . . . . . . 7  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
81, 7mpan 670 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
9 hash0 12394 . . . . . . . 8  |-  ( # `  (/) )  =  0
109oveq1i 6287 . . . . . . 7  |-  ( (
# `  (/) )  +  ( # `  S
) )  =  ( 0  +  ( # `  S ) )
11 lencl 12517 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
1211nn0cnd 10845 . . . . . . . 8  |-  ( S  e. Word  B  ->  ( # `
 S )  e.  CC )
1312addid2d 9771 . . . . . . 7  |-  ( S  e. Word  B  ->  (
0  +  ( # `  S ) )  =  ( # `  S
) )
1410, 13syl5eq 2515 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )  +  ( # `  S
) )  =  (
# `  S )
)
158, 14eqtrd 2503 . . . . 5  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( # `  S ) )
1615oveq2d 6293 . . . 4  |-  ( S  e. Word  B  ->  (
0..^ ( # `  ( (/) concat  S ) ) )  =  ( 0..^ (
# `  S )
) )
1716fneq2d 5665 . . 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 wrdf 12508 . . 3  |-  ( S  e. Word  B  ->  S : ( 0..^ (
# `  S )
) --> B )
20 ffn 5724 . . 3  |-  ( S : ( 0..^ (
# `  S )
) --> B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2119, 20syl 16 . 2  |-  ( S  e. Word  B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
229a1i 11 . . . . . . 7  |-  ( S  e. Word  B  ->  ( # `
 (/) )  =  0 )
2322, 14oveq12d 6295 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  =  ( 0..^ ( # `  S ) ) )
2423eleq2d 2532 . . . . 5  |-  ( S  e. Word  B  ->  (
x  e.  ( (
# `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
2524biimpar 485 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )
26 ccatval2 12550 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B  /\  x  e.  ( ( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
271, 26mp3an1 1306 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
2825, 27syldan 470 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  (
x  -  ( # `  (/) ) ) ) )
299oveq2i 6288 . . . . 5  |-  ( x  -  ( # `  (/) ) )  =  ( x  - 
0 )
30 elfzoelz 11788 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
3130adantl 466 . . . . . . 7  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ZZ )
3231zcnd 10958 . . . . . 6  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  CC )
3332subid1d 9910 . . . . 5  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  0 )  =  x )
3429, 33syl5eq 2515 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  ( # `
 (/) ) )  =  x )
3534fveq2d 5863 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  -  ( # `  (/) ) ) )  =  ( S `  x ) )
3628, 35eqtrd 2503 . 2  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  x
) )
3718, 21, 36eqfnfvd 5971 1  |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   (/)c0 3780    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   0cc0 9483    + caddc 9486    - cmin 9796   ZZcz 10855  ..^cfzo 11783   #chash 12362  Word cword 12489   concat cconcat 12491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-concat 12499
This theorem is referenced by:  swrdccat  12670  swrdccat3a  12671  s0s1  12822  gsumccat  15827  frmdmnd  15845  frmd0  15846  efginvrel2  16536  efgcpbl2  16566  frgp0  16569  frgpnabllem1  16663  signstfvneq0  28157
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