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Theorem grpsubfval 16720
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 5870 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpsubval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2505 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5870 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpsubval.p . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2505 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
8 eqidd 2454 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
9 fveq2 5870 . . . . . . . 8  |-  ( g  =  G  ->  ( invg `  g )  =  ( invg `  G ) )
10 grpsubval.i . . . . . . . 8  |-  I  =  ( invg `  G )
119, 10syl6eqr 2505 . . . . . . 7  |-  ( g  =  G  ->  ( invg `  g )  =  I )
1211fveq1d 5872 . . . . . 6  |-  ( g  =  G  ->  (
( invg `  g ) `  y
)  =  ( I `
 y ) )
137, 8, 12oveq123d 6316 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) ( ( invg `  g ) `
 y ) )  =  ( x  .+  ( I `  y
) ) )
144, 4, 13mpt2eq123dv 6358 . . . 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 16687 . . . 4  |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( invg `  g ) `
 y ) ) ) )
16 fvex 5880 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2527 . . . . 5  |-  B  e. 
_V
1817, 17mpt2ex 6875 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  ( I `  y ) ) )  e.  _V
1914, 15, 18fvmpt 5953 . . 3  |-  ( G  e.  _V  ->  ( -g `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
201, 19syl5eq 2499 . 2  |-  ( G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
21 fvprc 5864 . . . 4  |-  ( -.  G  e.  _V  ->  (
-g `  G )  =  (/) )
221, 21syl5eq 2499 . . 3  |-  ( -.  G  e.  _V  ->  .-  =  (/) )
23 fvprc 5864 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2499 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
25 mpt2eq12 6356 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
2624, 24, 25syl2anc 667 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
27 mpt20 6366 . . . 4  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) )  =  (/)
2826, 27syl6eq 2503 . . 3  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  (/) )
2922, 28eqtr4d 2490 . 2  |-  ( -.  G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  ( I `  y
) ) ) )
3020, 29pm2.61i 168 1  |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   Basecbs 15133   +g cplusg 15202   invgcminusg 16682   -gcsg 16683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-sbg 16687
This theorem is referenced by:  grpsubval  16721  grpsubf  16745  grpsubpropd  16768  grpsubpropd2  16769  tgpsubcn  21117  tngtopn  21670
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