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Theorem 0mhm 15468
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 22 . 2  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( M  e.  Mnd  /\  N  e.  Mnd )
)
2 eqid 2433 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
3 0mhm.z . . . . . 6  |-  .0.  =  ( 0g `  N )
42, 3mndidcl 15422 . . . . 5  |-  ( N  e.  Mnd  ->  .0.  e.  ( Base `  N
) )
54adantl 463 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  e.  ( Base `  N ) )
6 fconst6g 5587 . . . 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 458 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  N  e.  Mnd )
9 eqid 2433 . . . . . . . . 9  |-  ( +g  `  N )  =  ( +g  `  N )
102, 9, 3mndlid 15424 . . . . . . . 8  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  (  .0.  ( +g  `  N
)  .0.  )  =  .0.  )
1110eqcomd 2438 . . . . . . 7  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  .0.  =  (  .0.  ( +g  `  N )  .0.  ) )
128, 5, 11syl2anc 654 . . . . . 6  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  =  (  .0.  ( +g  `  N
)  .0.  ) )
1312adantr 462 . . . . 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 2433 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
1614, 15mndcl 15403 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
17163expb 1181 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
1817adantlr 707 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
19 fvex 5689 . . . . . . . 8  |-  ( 0g
`  N )  e. 
_V
203, 19eqeltri 2503 . . . . . . 7  |-  .0.  e.  _V
2120fvconst2 5920 . . . . . 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 5920 . . . . . . 7  |-  ( x  e.  B  ->  (
( B  X.  {  .0.  } ) `  x
)  =  .0.  )
2420fvconst2 5920 . . . . . . 7  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2523, 24oveqan12d 6099 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( ( B  X.  {  .0.  }
) `  x )
( +g  `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  (  .0.  ( +g  `  N )  .0.  )
)
2625adantl 463 . . . . 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 2475 . . . 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 2798 . . 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 2433 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
3014, 29mndidcl 15422 . . . . 5  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
3130adantr 462 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( 0g `  M
)  e.  B )
3220fvconst2 5920 . . . 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 1161 . 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 15449 . 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 657 1  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   _Vcvv 2962   {csn 3865    X. cxp 4825   -->wf 5402   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   0gc0g 14361   Mndcmnd 15392   MndHom cmhm 15445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-map 7204  df-0g 14363  df-mnd 15398  df-mhm 15447
This theorem is referenced by:  0ghm  15741
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