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Theorem 0ghm 15020
Description: The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
0ghm.z  |-  .0.  =  ( 0g `  N )
0ghm.b  |-  B  =  ( Base `  M
)
Assertion
Ref Expression
0ghm  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )

Proof of Theorem 0ghm
StepHypRef Expression
1 grpmnd 14817 . . 3  |-  ( M  e.  Grp  ->  M  e.  Mnd )
2 grpmnd 14817 . . 3  |-  ( N  e.  Grp  ->  N  e.  Mnd )
3 0ghm.z . . . 4  |-  .0.  =  ( 0g `  N )
4 0ghm.b . . . 4  |-  B  =  ( Base `  M
)
53, 40mhm 14758 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
61, 2, 5syl2an 464 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
7 ghmmhmb 15017 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( M  GrpHom  N )  =  ( M MndHom  N
) )
86, 7eleqtrrd 2513 1  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814    X. cxp 4876   ` cfv 5454  (class class class)co 6081   Basecbs 13469   0gc0g 13723   Mndcmnd 14684   Grpcgrp 14685   MndHom cmhm 14736    GrpHom cghm 15003
This theorem is referenced by:  0frgp  15411  0lmhm  16116  nmo0  18769  0nghm  18775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-map 7020  df-0g 13727  df-mnd 14690  df-mhm 14738  df-grp 14812  df-ghm 15004
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