MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0mhm Unicode version

Theorem 0mhm 14687
Description: The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
0mhm.z  |-  .0.  =  ( 0g `  N )
0mhm.b  |-  B  =  ( Base `  M
)
Assertion
Ref Expression
0mhm  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )

Proof of Theorem 0mhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 20 . 2  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( M  e.  Mnd  /\  N  e.  Mnd )
)
2 eqid 2389 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
3 0mhm.z . . . . . 6  |-  .0.  =  ( 0g `  N )
42, 3mndidcl 14643 . . . . 5  |-  ( N  e.  Mnd  ->  .0.  e.  ( Base `  N
) )
54adantl 453 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  e.  ( Base `  N ) )
6 fconst6g 5574 . . . 4  |-  (  .0. 
e.  ( Base `  N
)  ->  ( B  X.  {  .0.  } ) : B --> ( Base `  N ) )
75, 6syl 16 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } ) : B --> ( Base `  N )
)
8 simpr 448 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  N  e.  Mnd )
9 eqid 2389 . . . . . . . . 9  |-  ( +g  `  N )  =  ( +g  `  N )
102, 9, 3mndlid 14645 . . . . . . . 8  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  (  .0.  ( +g  `  N
)  .0.  )  =  .0.  )
1110eqcomd 2394 . . . . . . 7  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  .0.  =  (  .0.  ( +g  `  N )  .0.  ) )
128, 5, 11syl2anc 643 . . . . . 6  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  =  (  .0.  ( +g  `  N
)  .0.  ) )
1312adantr 452 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  .0.  =  (  .0.  ( +g  `  N )  .0.  ) )
14 0mhm.b . . . . . . . . 9  |-  B  =  ( Base `  M
)
15 eqid 2389 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
1614, 15mndcl 14624 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
17163expb 1154 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
1817adantlr 696 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
19 fvex 5684 . . . . . . . 8  |-  ( 0g
`  N )  e. 
_V
203, 19eqeltri 2459 . . . . . . 7  |-  .0.  e.  _V
2120fvconst2 5888 . . . . . 6  |-  ( ( x ( +g  `  M
) y )  e.  B  ->  ( ( B  X.  {  .0.  }
) `  ( x
( +g  `  M ) y ) )  =  .0.  )
2218, 21syl 16 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( B  X.  {  .0.  } ) `  (
x ( +g  `  M
) y ) )  =  .0.  )
2320fvconst2 5888 . . . . . . 7  |-  ( x  e.  B  ->  (
( B  X.  {  .0.  } ) `  x
)  =  .0.  )
2420fvconst2 5888 . . . . . . 7  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2523, 24oveqan12d 6041 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( ( B  X.  {  .0.  }
) `  x )
( +g  `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  (  .0.  ( +g  `  N )  .0.  )
)
2625adantl 453 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( ( B  X.  {  .0.  } ) `  x ) ( +g  `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  (  .0.  ( +g  `  N
)  .0.  ) )
2713, 22, 263eqtr4d 2431 . . . 4  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( B  X.  {  .0.  } ) `  (
x ( +g  `  M
) y ) )  =  ( ( ( B  X.  {  .0.  } ) `  x ) ( +g  `  N
) ( ( B  X.  {  .0.  }
) `  y )
) )
2827ralrimivva 2743 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  A. x  e.  B  A. y  e.  B  ( ( B  X.  {  .0.  } ) `  ( x ( +g  `  M ) y ) )  =  ( ( ( B  X.  {  .0.  } ) `  x
) ( +g  `  N
) ( ( B  X.  {  .0.  }
) `  y )
) )
29 eqid 2389 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
3014, 29mndidcl 14643 . . . . 5  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
3130adantr 452 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( 0g `  M
)  e.  B )
3220fvconst2 5888 . . . 4  |-  ( ( 0g `  M )  e.  B  ->  (
( B  X.  {  .0.  } ) `  ( 0g `  M ) )  =  .0.  )
3331, 32syl 16 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( ( B  X.  {  .0.  } ) `  ( 0g `  M ) )  =  .0.  )
347, 28, 333jca 1134 . 2  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( ( B  X.  {  .0.  } ) : B --> ( Base `  N
)  /\  A. x  e.  B  A. y  e.  B  ( ( B  X.  {  .0.  }
) `  ( x
( +g  `  M ) y ) )  =  ( ( ( B  X.  {  .0.  }
) `  x )
( +g  `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  /\  ( ( B  X.  {  .0.  } ) `  ( 0g `  M ) )  =  .0.  )
)
3514, 2, 15, 9, 29, 3ismhm 14669 . 2  |-  ( ( B  X.  {  .0.  } )  e.  ( M MndHom  N )  <->  ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  (
( B  X.  {  .0.  } ) : B --> ( Base `  N )  /\  A. x  e.  B  A. y  e.  B  ( ( B  X.  {  .0.  } ) `  ( x ( +g  `  M ) y ) )  =  ( ( ( B  X.  {  .0.  } ) `  x
) ( +g  `  N
) ( ( B  X.  {  .0.  }
) `  y )
)  /\  ( ( B  X.  {  .0.  }
) `  ( 0g `  M ) )  =  .0.  ) ) )
361, 34, 35sylanbrc 646 1  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   _Vcvv 2901   {csn 3759    X. cxp 4818   -->wf 5392   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   0gc0g 13652   Mndcmnd 14613   MndHom cmhm 14665
This theorem is referenced by:  0ghm  14949
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-map 6958  df-0g 13656  df-mnd 14619  df-mhm 14667
  Copyright terms: Public domain W3C validator