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Theorem mndplusf 14399
Description: The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
mndplusf.1  |-  B  =  ( Base `  G
)
mndplusf.2  |-  .+^  =  ( + f `  G
)
Assertion
Ref Expression
mndplusf  |-  ( G  e.  Mnd  ->  .+^  : ( B  X.  B ) --> B )

Proof of Theorem mndplusf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndplusf.1 . . . . 5  |-  B  =  ( Base `  G
)
2 eqid 2296 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2mndcl 14388 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
433expb 1152 . . 3  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  G
) y )  e.  B )
54ralrimivva 2648 . 2  |-  ( G  e.  Mnd  ->  A. x  e.  B  A. y  e.  B  ( x
( +g  `  G ) y )  e.  B
)
6 mndplusf.2 . . . 4  |-  .+^  =  ( + f `  G
)
71, 2, 6plusffval 14395 . . 3  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x ( +g  `  G
) y ) )
87fmpt2 6207 . 2  |-  ( A. x  e.  B  A. y  e.  B  (
x ( +g  `  G
) y )  e.  B  <->  .+^  : ( B  X.  B ) --> B )
95, 8sylib 188 1  |-  ( G  e.  Mnd  ->  .+^  : ( B  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Mndcmnd 14377   + fcplusf 14380
This theorem is referenced by:  grpplusf  14515  submtmd  17803  mhmhmeotmd  23315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-mnd 14383  df-plusf 14384
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