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Theorem ccatlid 11711
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 11695 . . . . 5  |-  (/)  e. Word  B
2 ccatcl 11706 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( (/) concat  S )  e. Word  B
)
31, 2mpan 652 . . . 4  |-  ( S  e. Word  B  ->  ( (/) concat  S )  e. Word  B
)
4 wrdf 11696 . . . 4  |-  ( (
(/) concat  S )  e. Word  B  ->  ( (/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B )
5 ffn 5558 . . . 4  |-  ( (
(/) concat  S ) : ( 0..^ ( # `  ( (/) concat  S ) ) ) --> B  ->  ( (/) concat  S )  Fn  ( 0..^ (
# `  ( (/) concat  S ) ) ) )
63, 4, 53syl 19 . . 3  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) ) )
7 ccatlen 11707 . . . . . . 7  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
81, 7mpan 652 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
9 hash0 11609 . . . . . . . 8  |-  ( # `  (/) )  =  0
109oveq1i 6058 . . . . . . 7  |-  ( (
# `  (/) )  +  ( # `  S
) )  =  ( 0  +  ( # `  S ) )
11 lencl 11698 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
1211nn0cnd 10240 . . . . . . . 8  |-  ( S  e. Word  B  ->  ( # `
 S )  e.  CC )
1312addid2d 9231 . . . . . . 7  |-  ( S  e. Word  B  ->  (
0  +  ( # `  S ) )  =  ( # `  S
) )
1410, 13syl5eq 2456 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )  +  ( # `  S
) )  =  (
# `  S )
)
158, 14eqtrd 2444 . . . . 5  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( # `  S ) )
1615oveq2d 6064 . . . 4  |-  ( S  e. Word  B  ->  (
0..^ ( # `  ( (/) concat  S ) ) )  =  ( 0..^ (
# `  S )
) )
1716fneq2d 5504 . . 3  |-  ( S  e. Word  B  ->  (
( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) )  <-> 
( (/) concat  S )  Fn  (
0..^ ( # `  S
) ) ) )
186, 17mpbid 202 . 2  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  S
) ) )
19 wrdf 11696 . . 3  |-  ( S  e. Word  B  ->  S : ( 0..^ (
# `  S )
) --> B )
20 ffn 5558 . . 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 6066 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  =  ( 0..^ ( # `  S ) ) )
2423eleq2d 2479 . . . . 5  |-  ( S  e. Word  B  ->  (
x  e.  ( (
# `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
2524biimpar 472 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )
26 ccatval2 11709 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B  /\  x  e.  ( ( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
271, 26mp3an1 1266 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( ( # `
 (/) )..^ ( (
# `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
2825, 27syldan 457 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  (
x  -  ( # `  (/) ) ) ) )
299oveq2i 6059 . . . . 5  |-  ( x  -  ( # `  (/) ) )  =  ( x  - 
0 )
30 elfzoelz 11103 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
3130adantl 453 . . . . . . 7  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ZZ )
3231zcnd 10340 . . . . . 6  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  CC )
3332subid1d 9364 . . . . 5  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  0 )  =  x )
3429, 33syl5eq 2456 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  ( # `
 (/) ) )  =  x )
3534fveq2d 5699 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  -  ( # `  (/) ) ) )  =  ( S `  x ) )
3628, 35eqtrd 2444 . 2  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  x
) )
3718, 21, 36eqfnfvd 5797 1  |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   (/)c0 3596    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   0cc0 8954    + caddc 8957    - cmin 9255   ZZcz 10246  ..^cfzo 11098   #chash 11581  Word cword 11680   concat cconcat 11681
This theorem is referenced by:  s0s1  11832  gsumccat  14750  frmdmnd  14767  frmd0  14768  efginvrel2  15322  efgcpbl2  15352  frgp0  15355  frgpnabllem1  15447  swrdccat3a0  28023  swrdccatin12b  28035  swrdccat3a  28038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-fzo 11099  df-hash 11582  df-word 11686  df-concat 11687
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