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Theorem 0ghm 16407
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 16188 . . 3  |-  ( M  e.  Grp  ->  M  e.  Mnd )
2 grpmnd 16188 . . 3  |-  ( N  e.  Grp  ->  N  e.  Mnd )
3 0ghm.z . . . 4  |-  .0.  =  ( 0g `  N )
4 0ghm.b . . . 4  |-  B  =  ( Base `  M
)
53, 40mhm 16115 . . 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 16404 . 2  |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( M  GrpHom  N )  =  ( M MndHom  N
) )
86, 7eleqtrrd 2548 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 1395    e. wcel 1819   {csn 4032    X. cxp 5006   ` cfv 5594  (class class class)co 6296   Basecbs 14643   0gc0g 14856   Mndcmnd 16045   MndHom cmhm 16090   Grpcgrp 16179    GrpHom cghm 16390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-grp 16183  df-ghm 16391
This theorem is referenced by:  0frgp  16923  0lmhm  17812  nmo0  21367  0nghm  21373
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