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Theorem grpsubpropdg 13176
Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
Hypotheses
Ref Expression
grpsubpropd.b  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
grpsubpropd.p  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
grpsubpropdg.g  |-  ( ph  ->  G  e.  V )
grpsubpropdg.h  |-  ( ph  ->  H  e.  W )
Assertion
Ref Expression
grpsubpropdg  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )

Proof of Theorem grpsubpropdg
Dummy variables  a  b  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpsubpropd.b . . 3  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
2 grpsubpropd.p . . . 4  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
3 eqidd 2194 . . . 4  |-  ( ph  ->  a  =  a )
4 eqidd 2194 . . . . . 6  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
5 grpsubpropdg.g . . . . . 6  |-  ( ph  ->  G  e.  V )
6 grpsubpropdg.h . . . . . 6  |-  ( ph  ->  H  e.  W )
72oveqdr 5946 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  H ) y ) )
84, 1, 5, 6, 7grpinvpropdg 13147 . . . . 5  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
98fveq1d 5556 . . . 4  |-  ( ph  ->  ( ( invg `  G ) `  b
)  =  ( ( invg `  H
) `  b )
)
102, 3, 9oveq123d 5939 . . 3  |-  ( ph  ->  ( a ( +g  `  G ) ( ( invg `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
111, 1, 10mpoeq123dv 5980 . 2  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
12 eqid 2193 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
13 eqid 2193 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
14 eqid 2193 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
15 eqid 2193 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
1612, 13, 14, 15grpsubfvalg 13117 . . 3  |-  ( G  e.  V  ->  ( -g `  G )  =  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) ) )
175, 16syl 14 . 2  |-  ( ph  ->  ( -g `  G
)  =  ( a  e.  ( Base `  G
) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) ) ) )
18 eqid 2193 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
19 eqid 2193 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
20 eqid 2193 . . . 4  |-  ( invg `  H )  =  ( invg `  H )
21 eqid 2193 . . . 4  |-  ( -g `  H )  =  (
-g `  H )
2218, 19, 20, 21grpsubfvalg 13117 . . 3  |-  ( H  e.  W  ->  ( -g `  H )  =  ( a  e.  (
Base `  H ) ,  b  e.  ( Base `  H )  |->  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) ) )
236, 22syl 14 . 2  |-  ( ph  ->  ( -g `  H
)  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
2411, 17, 233eqtr4d 2236 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164   ` cfv 5254  (class class class)co 5918    e. cmpo 5920   Basecbs 12618   +g cplusg 12695   invgcminusg 13073   -gcsg 13074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-minusg 13076  df-sbg 13077
This theorem is referenced by:  rlmsubg  13954
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