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Theorem 0ghm 14697
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 14494 . . 3  |-  ( M  e.  Grp  ->  M  e.  Mnd )
2 grpmnd 14494 . . 3  |-  ( N  e.  Grp  ->  N  e.  Mnd )
3 0ghm.z . . . 4  |-  .0.  =  ( 0g `  N )
4 0ghm.b . . . 4  |-  B  =  ( Base `  M
)
53, 40mhm 14435 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
61, 2, 5syl2an 463 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
7 ghmmhmb 14694 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( M  GrpHom  N )  =  ( M MndHom  N
) )
86, 7eleqtrrd 2360 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 358    = wceq 1623    e. wcel 1684   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362   MndHom cmhm 14413    GrpHom cghm 14680
This theorem is referenced by:  0frgp  15088  0lmhm  15797  nmo0  18244  0nghm  18250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681
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