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Theorem grpsubfval 15592
Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
grpsubval.b  |-  B  =  ( Base `  G
)
grpsubval.p  |-  .+  =  ( +g  `  G )
grpsubval.i  |-  I  =  ( invg `  G )
grpsubval.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpsubfval  |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) )
Distinct variable groups:    x, y, B    x, G, y    x, I, y    x,  .+ , y
Allowed substitution hints:    .- ( x, y)

Proof of Theorem grpsubfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpsubval.m . . 3  |-  .-  =  ( -g `  G )
2 fveq2 5703 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpsubval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2493 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5703 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpsubval.p . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2493 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
8 eqidd 2444 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
9 fveq2 5703 . . . . . . . 8  |-  ( g  =  G  ->  ( invg `  g )  =  ( invg `  G ) )
10 grpsubval.i . . . . . . . 8  |-  I  =  ( invg `  G )
119, 10syl6eqr 2493 . . . . . . 7  |-  ( g  =  G  ->  ( invg `  g )  =  I )
1211fveq1d 5705 . . . . . 6  |-  ( g  =  G  ->  (
( invg `  g ) `  y
)  =  ( I `
 y ) )
137, 8, 12oveq123d 6124 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) ( ( invg `  g ) `
 y ) )  =  ( x  .+  ( I `  y
) ) )
144, 4, 13mpt2eq123dv 6160 . . . 4  |-  ( g  =  G  ->  (
x  e.  ( Base `  g ) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) ( ( invg `  g ) `  y
) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
15 df-sbg 15559 . . . 4  |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( invg `  g ) `
 y ) ) ) )
16 fvex 5713 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2513 . . . . 5  |-  B  e. 
_V
1817, 17mpt2ex 6662 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  ( I `  y ) ) )  e.  _V
1914, 15, 18fvmpt 5786 . . 3  |-  ( G  e.  _V  ->  ( -g `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
201, 19syl5eq 2487 . 2  |-  ( G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
21 fvprc 5697 . . . 4  |-  ( -.  G  e.  _V  ->  (
-g `  G )  =  (/) )
221, 21syl5eq 2487 . . 3  |-  ( -.  G  e.  _V  ->  .-  =  (/) )
23 fvprc 5697 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2487 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
25 mpt2eq12 6158 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
2624, 24, 25syl2anc 661 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
27 mpt20 6168 . . . 4  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) )  =  (/)
2826, 27syl6eq 2491 . . 3  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  (/) )
2922, 28eqtr4d 2478 . 2  |-  ( -.  G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  ( I `  y
) ) ) )
3020, 29pm2.61i 164 1  |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   Basecbs 14186   +g cplusg 14250   invgcminusg 15423   -gcsg 15425
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-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-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-sbg 15559
This theorem is referenced by:  grpsubval  15593  grpsubf  15617  grpsubpropd  15638  grpsubpropd2  15639  tgpsubcn  19673  tngtopn  20248
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