MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccatrid Structured version   Unicode version

Theorem ccatrid 11749
Description: Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
ccatrid  |-  ( S  e. Word  B  ->  ( S concat 
(/) )  =  S )

Proof of Theorem ccatrid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 wrd0 11732 . . . . 5  |-  (/)  e. Word  B
2 ccatcl 11743 . . . . . 6  |-  ( ( S  e. Word  B  /\  (/) 
e. Word  B )  ->  ( S concat 
(/) )  e. Word  B
)
3 wrdf 11733 . . . . . 6  |-  ( ( S concat  (/) )  e. Word  B  ->  ( S concat  (/) ) : ( 0..^ ( # `  ( S concat  (/) ) ) ) --> B )
42, 3syl 16 . . . . 5  |-  ( ( S  e. Word  B  /\  (/) 
e. Word  B )  ->  ( S concat 
(/) ) : ( 0..^ ( # `  ( S concat 
(/) ) ) ) --> B )
51, 4mpan2 653 . . . 4  |-  ( S  e. Word  B  ->  ( S concat 
(/) ) : ( 0..^ ( # `  ( S concat 
(/) ) ) ) --> B )
6 ffn 5591 . . . 4  |-  ( ( S concat  (/) ) : ( 0..^ ( # `  ( S concat 
(/) ) ) ) --> B  ->  ( S concat  (/) )  Fn  ( 0..^ ( # `  ( S concat 
(/) ) ) ) )
75, 6syl 16 . . 3  |-  ( S  e. Word  B  ->  ( S concat 
(/) )  Fn  (
0..^ ( # `  ( S concat 
(/) ) ) ) )
8 ccatlen 11744 . . . . . . 7  |-  ( ( S  e. Word  B  /\  (/) 
e. Word  B )  ->  ( # `
 ( S concat  (/) ) )  =  ( ( # `  S )  +  (
# `  (/) ) ) )
91, 8mpan2 653 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 ( S concat  (/) ) )  =  ( ( # `  S )  +  (
# `  (/) ) ) )
10 hash0 11646 . . . . . . . 8  |-  ( # `  (/) )  =  0
1110oveq2i 6092 . . . . . . 7  |-  ( (
# `  S )  +  ( # `  (/) ) )  =  ( ( # `  S )  +  0 )
12 lencl 11735 . . . . . . . . 9  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
1312nn0cnd 10276 . . . . . . . 8  |-  ( S  e. Word  B  ->  ( # `
 S )  e.  CC )
1413addid1d 9266 . . . . . . 7  |-  ( S  e. Word  B  ->  (
( # `  S )  +  0 )  =  ( # `  S
) )
1511, 14syl5eq 2480 . . . . . 6  |-  ( S  e. Word  B  ->  (
( # `  S )  +  ( # `  (/) ) )  =  ( # `  S
) )
169, 15eqtrd 2468 . . . . 5  |-  ( S  e. Word  B  ->  ( # `
 ( S concat  (/) ) )  =  ( # `  S
) )
1716oveq2d 6097 . . . 4  |-  ( S  e. Word  B  ->  (
0..^ ( # `  ( S concat 
(/) ) ) )  =  ( 0..^ (
# `  S )
) )
1817fneq2d 5537 . . 3  |-  ( S  e. Word  B  ->  (
( S concat  (/) )  Fn  ( 0..^ ( # `  ( S concat  (/) ) ) )  <->  ( S concat  (/) )  Fn  ( 0..^ ( # `  S ) ) ) )
197, 18mpbid 202 . 2  |-  ( S  e. Word  B  ->  ( S concat 
(/) )  Fn  (
0..^ ( # `  S
) ) )
20 wrdf 11733 . . 3  |-  ( S  e. Word  B  ->  S : ( 0..^ (
# `  S )
) --> B )
21 ffn 5591 . . 3  |-  ( S : ( 0..^ (
# `  S )
) --> B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2220, 21syl 16 . 2  |-  ( S  e. Word  B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
23 ccatval1 11745 . . 3  |-  ( ( S  e. Word  B  /\  (/) 
e. Word  B  /\  x  e.  ( 0..^ ( # `  S ) ) )  ->  ( ( S concat  (/) ) `  x )  =  ( S `  x ) )
241, 23mp3an2 1267 . 2  |-  ( ( S  e. Word  B  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S concat  (/) ) `  x )  =  ( S `  x ) )
2519, 22, 24eqfnfvd 5830 1  |-  ( S  e. Word  B  ->  ( S concat 
(/) )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   (/)c0 3628    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990    + caddc 8993  ..^cfzo 11135   #chash 11618  Word cword 11717   concat cconcat 11718
This theorem is referenced by:  gsumccat  14787  frmdmnd  14804  frmd0  14805  efginvrel2  15359  efgredleme  15375  efgcpbllemb  15387  efgcpbl2  15389  frgpnabllem1  15484  swrdccat  28216  swrdccat3blem  28218  cshw0  28238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724
  Copyright terms: Public domain W3C validator