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Theorem grpsubfval 16219
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 5872 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpsubval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2516 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5872 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpsubval.p . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2516 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
8 eqidd 2458 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
9 fveq2 5872 . . . . . . . 8  |-  ( g  =  G  ->  ( invg `  g )  =  ( invg `  G ) )
10 grpsubval.i . . . . . . . 8  |-  I  =  ( invg `  G )
119, 10syl6eqr 2516 . . . . . . 7  |-  ( g  =  G  ->  ( invg `  g )  =  I )
1211fveq1d 5874 . . . . . 6  |-  ( g  =  G  ->  (
( invg `  g ) `  y
)  =  ( I `
 y ) )
137, 8, 12oveq123d 6317 . . . . 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 16186 . . . 4  |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( invg `  g ) `
 y ) ) ) )
16 fvex 5882 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2541 . . . . 5  |-  B  e. 
_V
1817, 17mpt2ex 6876 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  ( I `  y ) ) )  e.  _V
1914, 15, 18fvmpt 5956 . . 3  |-  ( G  e.  _V  ->  ( -g `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
201, 19syl5eq 2510 . 2  |-  ( G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
21 fvprc 5866 . . . 4  |-  ( -.  G  e.  _V  ->  (
-g `  G )  =  (/) )
221, 21syl5eq 2510 . . 3  |-  ( -.  G  e.  _V  ->  .-  =  (/) )
23 fvprc 5866 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2510 . . . . 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 661 . . . 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 2514 . . 3  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  (/) )
2922, 28eqtr4d 2501 . 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 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   +g cplusg 14712   invgcminusg 16181   -gcsg 16182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-sbg 16186
This theorem is referenced by:  grpsubval  16220  grpsubf  16244  grpsubpropd  16267  grpsubpropd2  16268  tgpsubcn  20715  tngtopn  21290
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