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Theorem grpsubfval 14540
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  =  ( inv g `  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 5541 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpsubval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2346 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5541 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpsubval.p . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2346 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
8 eqidd 2297 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
9 fveq2 5541 . . . . . . . 8  |-  ( g  =  G  ->  ( inv g `  g )  =  ( inv g `  G ) )
10 grpsubval.i . . . . . . . 8  |-  I  =  ( inv g `  G )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( g  =  G  ->  ( inv g `  g )  =  I )
1211fveq1d 5543 . . . . . 6  |-  ( g  =  G  ->  (
( inv g `  g ) `  y
)  =  ( I `
 y ) )
137, 8, 12oveq123d 5895 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) ( ( inv g `  g ) `
 y ) )  =  ( x  .+  ( I `  y
) ) )
144, 4, 13mpt2eq123dv 5926 . . . 4  |-  ( g  =  G  ->  (
x  e.  ( Base `  g ) ,  y  e.  ( Base `  g
)  |->  ( x ( +g  `  g ) ( ( inv g `  g ) `  y
) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
15 df-sbg 14507 . . . 4  |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( inv g `  g ) `
 y ) ) ) )
16 fvex 5555 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2366 . . . . 5  |-  B  e. 
_V
1817, 17mpt2ex 6214 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  ( I `  y ) ) )  e.  _V
1914, 15, 18fvmpt 5618 . . 3  |-  ( G  e.  _V  ->  ( -g `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
201, 19syl5eq 2340 . 2  |-  ( G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
21 fvprc 5535 . . . 4  |-  ( -.  G  e.  _V  ->  (
-g `  G )  =  (/) )
221, 21syl5eq 2340 . . 3  |-  ( -.  G  e.  _V  ->  .-  =  (/) )
23 fvprc 5535 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2340 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
25 mpt2eq12 5924 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
2624, 24, 25syl2anc 642 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
27 mpt20 6215 . . . 4  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) )  =  (/)
2826, 27syl6eq 2344 . . 3  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  (/) )
2922, 28eqtr4d 2331 . 2  |-  ( -.  G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  ( I `  y
) ) ) )
3020, 29pm2.61i 156 1  |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   +g cplusg 13224   inv gcminusg 14379   -gcsg 14381
This theorem is referenced by:  grpsubval  14541  grpsubf  14561  grpsubpropd  14582  grpsubpropd2  14583  tgpsubcn  17789  tngtopn  18182
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-rep 4147  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-reu 2563  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-sbg 14507
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