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Theorem 0ghm 15773
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 15562 . . 3  |-  ( M  e.  Grp  ->  M  e.  Mnd )
2 grpmnd 15562 . . 3  |-  ( N  e.  Grp  ->  N  e.  Mnd )
3 0ghm.z . . . 4  |-  .0.  =  ( 0g `  N )
4 0ghm.b . . . 4  |-  B  =  ( Base `  M
)
53, 40mhm 15498 . . 3  |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
61, 2, 5syl2an 477 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N ) )
7 ghmmhmb 15770 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( M  GrpHom  N )  =  ( M MndHom  N
) )
86, 7eleqtrrd 2520 1  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3889    X. cxp 4850   ` cfv 5430  (class class class)co 6103   Basecbs 14186   0gc0g 14390   Mndcmnd 15421   Grpcgrp 15422   MndHom cmhm 15474    GrpHom cghm 15756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-map 7228  df-0g 14392  df-mnd 15427  df-mhm 15476  df-grp 15557  df-ghm 15757
This theorem is referenced by:  0frgp  16288  0lmhm  17133  nmo0  20326  0nghm  20332
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