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Theorem 0mhm 14451
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 19 . 2  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( M  e.  Mnd  /\  N  e.  Mnd )
)
2 eqid 2296 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
3 0mhm.z . . . . . 6  |-  .0.  =  ( 0g `  N )
42, 3mndidcl 14407 . . . . 5  |-  ( N  e.  Mnd  ->  .0.  e.  ( Base `  N
) )
54adantl 452 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  e.  ( Base `  N ) )
6 fconst6g 5446 . . . 4  |-  (  .0. 
e.  ( Base `  N
)  ->  ( B  X.  {  .0.  } ) : B --> ( Base `  N ) )
75, 6syl 15 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } ) : B --> ( Base `  N )
)
8 simpr 447 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  N  e.  Mnd )
9 eqid 2296 . . . . . . . . 9  |-  ( +g  `  N )  =  ( +g  `  N )
102, 9, 3mndlid 14409 . . . . . . . 8  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  (  .0.  ( +g  `  N
)  .0.  )  =  .0.  )
1110eqcomd 2301 . . . . . . 7  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  .0.  =  (  .0.  ( +g  `  N )  .0.  ) )
128, 5, 11syl2anc 642 . . . . . 6  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  =  (  .0.  ( +g  `  N
)  .0.  ) )
1312adantr 451 . . . . 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 2296 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
1614, 15mndcl 14388 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
17163expb 1152 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
1817adantlr 695 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
19 fvex 5555 . . . . . . . 8  |-  ( 0g
`  N )  e. 
_V
203, 19eqeltri 2366 . . . . . . 7  |-  .0.  e.  _V
2120fvconst2 5745 . . . . . 6  |-  ( ( x ( +g  `  M
) y )  e.  B  ->  ( ( B  X.  {  .0.  }
) `  ( x
( +g  `  M ) y ) )  =  .0.  )
2218, 21syl 15 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
( B  X.  {  .0.  } ) `  (
x ( +g  `  M
) y ) )  =  .0.  )
2320fvconst2 5745 . . . . . . 7  |-  ( x  e.  B  ->  (
( B  X.  {  .0.  } ) `  x
)  =  .0.  )
2420fvconst2 5745 . . . . . . 7  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2523, 24oveqan12d 5893 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( ( B  X.  {  .0.  }
) `  x )
( +g  `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  (  .0.  ( +g  `  N )  .0.  )
)
2625adantl 452 . . . . 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 2338 . . . 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 2648 . . 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 2296 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
3014, 29mndidcl 14407 . . . . 5  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
3130adantr 451 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( 0g `  M
)  e.  B )
3220fvconst2 5745 . . . 4  |-  ( ( 0g `  M )  e.  B  ->  (
( B  X.  {  .0.  } ) `  ( 0g `  M ) )  =  .0.  )
3331, 32syl 15 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( ( B  X.  {  .0.  } ) `  ( 0g `  M ) )  =  .0.  )
347, 28, 333jca 1132 . 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 14433 . 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 645 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   {csn 3653    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   MndHom cmhm 14429
This theorem is referenced by:  0ghm  14713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-mhm 14431
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