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Theorem grpodivval 21221
Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdiv.1  |-  X  =  ran  G
grpdiv.2  |-  N  =  ( inv `  G
)
grpdiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivval  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )

Proof of Theorem grpodivval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdiv.1 . . . . 5  |-  X  =  ran  G
2 grpdiv.2 . . . . 5  |-  N  =  ( inv `  G
)
3 grpdiv.3 . . . . 5  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivfval 21220 . . . 4  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) )
54oveqd 5998 . . 3  |-  ( G  e.  GrpOp  ->  ( A D B )  =  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) ) B ) )
6 oveq1 5988 . . . 4  |-  ( x  =  A  ->  (
x G ( N `
 y ) )  =  ( A G ( N `  y
) ) )
7 fveq2 5632 . . . . 5  |-  ( y  =  B  ->  ( N `  y )  =  ( N `  B ) )
87oveq2d 5997 . . . 4  |-  ( y  =  B  ->  ( A G ( N `  y ) )  =  ( A G ( N `  B ) ) )
9 eqid 2366 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) )  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
) ) )
10 ovex 6006 . . . 4  |-  ( A G ( N `  B ) )  e. 
_V
116, 8, 9, 10ovmpt2 6109 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y ) ) ) B )  =  ( A G ( N `
 B ) ) )
125, 11sylan9eq 2418 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  =  ( A G ( N `
 B ) ) )
13123impb 1148 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   ran crn 4793   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   GrpOpcgr 21164   invcgn 21166    /g cgs 21167
This theorem is referenced by:  grpodivinv  21222  grpoinvdiv  21223  grpodivdiv  21226  grpomuldivass  21227  grpodivid  21228  grponpcan  21230  grpopnpcan2  21231  grponnncan2  21232  ablodivdiv4  21269  nvmval  21513  rngosub  26085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-gdiv 21172
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