MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpsubfval Structured version   Unicode version

Theorem grpsubfval 14847
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 5728 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpsubval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2486 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5728 . . . . . . 7  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpsubval.p . . . . . . 7  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2486 . . . . . 6  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
8 eqidd 2437 . . . . . 6  |-  ( g  =  G  ->  x  =  x )
9 fveq2 5728 . . . . . . . 8  |-  ( g  =  G  ->  ( inv g `  g )  =  ( inv g `  G ) )
10 grpsubval.i . . . . . . . 8  |-  I  =  ( inv g `  G )
119, 10syl6eqr 2486 . . . . . . 7  |-  ( g  =  G  ->  ( inv g `  g )  =  I )
1211fveq1d 5730 . . . . . 6  |-  ( g  =  G  ->  (
( inv g `  g ) `  y
)  =  ( I `
 y ) )
137, 8, 12oveq123d 6102 . . . . 5  |-  ( g  =  G  ->  (
x ( +g  `  g
) ( ( inv g `  g ) `
 y ) )  =  ( x  .+  ( I `  y
) ) )
144, 4, 13mpt2eq123dv 6136 . . . 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 14814 . . . 4  |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
) ( ( inv g `  g ) `
 y ) ) ) )
16 fvex 5742 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2506 . . . . 5  |-  B  e. 
_V
1817, 17mpt2ex 6425 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x 
.+  ( I `  y ) ) )  e.  _V
1914, 15, 18fvmpt 5806 . . 3  |-  ( G  e.  _V  ->  ( -g `  G )  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
201, 19syl5eq 2480 . 2  |-  ( G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) ) )
21 fvprc 5722 . . . 4  |-  ( -.  G  e.  _V  ->  (
-g `  G )  =  (/) )
221, 21syl5eq 2480 . . 3  |-  ( -.  G  e.  _V  ->  .-  =  (/) )
23 fvprc 5722 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2480 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
25 mpt2eq12 6134 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
2624, 24, 25syl2anc 643 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) ) )
27 mpt20 6427 . . . 4  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .+  ( I `  y
) ) )  =  (/)
2826, 27syl6eq 2484 . . 3  |-  ( -.  G  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x  .+  ( I `
 y ) ) )  =  (/) )
2922, 28eqtr4d 2471 . 2  |-  ( -.  G  e.  _V  ->  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  ( I `  y
) ) ) )
3020, 29pm2.61i 158 1  |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  (
I `  y )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469   +g cplusg 13529   inv gcminusg 14686   -gcsg 14688
This theorem is referenced by:  grpsubval  14848  grpsubf  14868  grpsubpropd  14889  grpsubpropd2  14890  tgpsubcn  18120  tngtopn  18691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-sbg 14814
  Copyright terms: Public domain W3C validator