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Theorem ccatlid 12283
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 12251 . . . . 5  |-  (/)  e. Word  B
2 ccatcl 12273 . . . . 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 12239 . . . 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 20 . . 3  |-  ( S  e. Word  B  ->  ( (/) concat  S )  Fn  (
0..^ ( # `  ( (/) concat  S ) ) ) )
7 ccatlen 12274 . . . . . . 7  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B )  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
81, 7mpan 670 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( (
# `  (/) )  +  ( # `  S
) ) )
9 hash0 12134 . . . . . . . 8  |-  ( # `  (/) )  =  0
109oveq1i 6100 . . . . . . 7  |-  ( (
# `  (/) )  +  ( # `  S
) )  =  ( 0  +  ( # `  S ) )
11 lencl 12248 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
1211nn0cnd 10637 . . . . . . . 8  |-  ( S  e. Word  B  ->  ( # `
 S )  e.  CC )
1312addid2d 9569 . . . . . . 7  |-  ( S  e. Word  B  ->  (
0  +  ( # `  S ) )  =  ( # `  S
) )
1410, 13syl5eq 2486 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )  +  ( # `  S
) )  =  (
# `  S )
)
158, 14eqtrd 2474 . . . . 5  |-  ( S  e. Word  B  ->  ( # `
 ( (/) concat  S ) )  =  ( # `  S ) )
1615oveq2d 6106 . . . 4  |-  ( S  e. Word  B  ->  (
0..^ ( # `  ( (/) concat  S ) ) )  =  ( 0..^ (
# `  S )
) )
1716fneq2d 5501 . . 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 12239 . . 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 6108 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) )  =  ( 0..^ ( # `  S ) ) )
2423eleq2d 2509 . . . . 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 12276 . . . . 5  |-  ( (
(/)  e. Word  B  /\  S  e. Word  B  /\  x  e.  ( ( # `  (/) )..^ ( ( # `  (/) )  +  ( # `  S
) ) ) )  ->  ( ( (/) concat  S ) `  x )  =  ( S `  ( x  -  ( # `
 (/) ) ) ) )
271, 26mp3an1 1301 . . . 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 6101 . . . . 5  |-  ( x  -  ( # `  (/) ) )  =  ( x  - 
0 )
30 elfzoelz 11552 . . . . . . . 8  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
3130adantl 466 . . . . . . 7  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ZZ )
3231zcnd 10747 . . . . . 6  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  CC )
3332subid1d 9707 . . . . 5  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  0 )  =  x )
3429, 33syl5eq 2486 . . . 4  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  -  ( # `
 (/) ) )  =  x )
3534fveq2d 5694 . . 3  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  -  ( # `  (/) ) ) )  =  ( S `  x ) )
3628, 35eqtrd 2474 . 2  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( (/) concat  S ) `
 x )  =  ( S `  x
) )
3718, 21, 36eqfnfvd 5799 1  |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   (/)c0 3636    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090   0cc0 9281    + caddc 9284    - cmin 9594   ZZcz 10645  ..^cfzo 11547   #chash 12102  Word cword 12220   concat cconcat 12222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-hash 12103  df-word 12228  df-concat 12230
This theorem is referenced by:  swrdccat  12383  swrdccat3a  12384  s0s1  12531  gsumccat  15518  frmdmnd  15536  frmd0  15537  efginvrel2  16223  efgcpbl2  16253  frgp0  16256  frgpnabllem1  16350  signstfvneq0  26972
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