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Theorem mndvrid 18663
Description: Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
mndvcl.b  |-  B  =  ( Base `  M
)
mndvcl.p  |-  .+  =  ( +g  `  M )
mndvlid.z  |-  .0.  =  ( 0g `  M )
Assertion
Ref Expression
mndvrid  |-  ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) )  -> 
( X  oF  .+  ( I  X.  {  .0.  } ) )  =  X )

Proof of Theorem mndvrid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elmapex 7436 . . . 4  |-  ( X  e.  ( B  ^m  I )  ->  ( B  e.  _V  /\  I  e.  _V ) )
21simprd 463 . . 3  |-  ( X  e.  ( B  ^m  I )  ->  I  e.  _V )
32adantl 466 . 2  |-  ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) )  ->  I  e.  _V )
4 elmapi 7437 . . 3  |-  ( X  e.  ( B  ^m  I )  ->  X : I --> B )
54adantl 466 . 2  |-  ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) )  ->  X : I --> B )
6 mndvcl.b . . . 4  |-  B  =  ( Base `  M
)
7 mndvlid.z . . . 4  |-  .0.  =  ( 0g `  M )
86, 7mndidcl 15752 . . 3  |-  ( M  e.  Mnd  ->  .0.  e.  B )
98adantr 465 . 2  |-  ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) )  ->  .0.  e.  B )
10 mndvcl.p . . . 4  |-  .+  =  ( +g  `  M )
116, 10, 7mndrid 15755 . . 3  |-  ( ( M  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
1211adantlr 714 . 2  |-  ( ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  B
)  ->  ( x  .+  .0.  )  =  x )
133, 5, 9, 12caofid0r 6551 1  |-  ( ( M  e.  Mnd  /\  X  e.  ( B  ^m  I ) )  -> 
( X  oF  .+  ( I  X.  {  .0.  } ) )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520    ^m cmap 7417   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Mndcmnd 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-1st 6781  df-2nd 6782  df-map 7419  df-0g 14693  df-mnd 15728
This theorem is referenced by: (None)
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