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Theorem mndplusf 14698
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 2435 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
31, 2mndcl 14687 . . . 4  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
433expb 1154 . . 3  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  G
) y )  e.  B )
54ralrimivva 2790 . 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 14694 . . 3  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x ( +g  `  G
) y ) )
87fmpt2 6410 . 2  |-  ( A. x  e.  B  A. y  e.  B  (
x ( +g  `  G
) y )  e.  B  <->  .+^  : ( B  X.  B ) --> B )
95, 8sylib 189 1  |-  ( G  e.  Mnd  ->  .+^  : ( B  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Mndcmnd 14676   + fcplusf 14679
This theorem is referenced by:  grpplusf  14814  submtmd  18126  mhmhmeotmd  24305
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-mnd 14682  df-plusf 14683
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