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Theorem 0mhm 16188
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 2454 . . . . . 6  |-  ( Base `  N )  =  (
Base `  N )
3 0mhm.z . . . . . 6  |-  .0.  =  ( 0g `  N )
42, 3mndidcl 16137 . . . . 5  |-  ( N  e.  Mnd  ->  .0.  e.  ( Base `  N
) )
54adantl 464 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  e.  ( Base `  N ) )
6 fconst6g 5756 . . . 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 459 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  N  e.  Mnd )
9 eqid 2454 . . . . . . . . 9  |-  ( +g  `  N )  =  ( +g  `  N )
102, 9, 3mndlid 16140 . . . . . . . 8  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  (  .0.  ( +g  `  N
)  .0.  )  =  .0.  )
1110eqcomd 2462 . . . . . . 7  |-  ( ( N  e.  Mnd  /\  .0.  e.  ( Base `  N
) )  ->  .0.  =  (  .0.  ( +g  `  N )  .0.  ) )
128, 5, 11syl2anc 659 . . . . . 6  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  .0.  =  (  .0.  ( +g  `  N
)  .0.  ) )
1312adantr 463 . . . . 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 2454 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
1614, 15mndcl 16128 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  M ) y )  e.  B )
17163expb 1195 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
1817adantlr 712 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  N  e.  Mnd )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  M
) y )  e.  B )
19 fvex 5858 . . . . . . . 8  |-  ( 0g
`  N )  e. 
_V
203, 19eqeltri 2538 . . . . . . 7  |-  .0.  e.  _V
2120fvconst2 6103 . . . . . 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 6103 . . . . . . 7  |-  ( x  e.  B  ->  (
( B  X.  {  .0.  } ) `  x
)  =  .0.  )
2420fvconst2 6103 . . . . . . 7  |-  ( y  e.  B  ->  (
( B  X.  {  .0.  } ) `  y
)  =  .0.  )
2523, 24oveqan12d 6289 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( ( B  X.  {  .0.  }
) `  x )
( +g  `  N ) ( ( B  X.  {  .0.  } ) `  y ) )  =  (  .0.  ( +g  `  N )  .0.  )
)
2625adantl 464 . . . . 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 2505 . . . 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 2875 . . 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 2454 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  M
)
3014, 29mndidcl 16137 . . . . 5  |-  ( M  e.  Mnd  ->  ( 0g `  M )  e.  B )
3130adantr 463 . . . 4  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( 0g `  M
)  e.  B )
3220fvconst2 6103 . . . 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 1174 . 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 16167 . 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 662 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   {csn 4016    X. cxp 4986   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   0gc0g 14929   Mndcmnd 16118   MndHom cmhm 16163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165
This theorem is referenced by:  0ghm  16480
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