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Metamath Proof Explorer

Theorem nvmfval

Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses nvmval.1 
|- X = ( BaseSet ` U )
nvmval.2 
|- G = ( +v ` U )
nvmval.4 
|- S = ( .sOLD ` U )
nvmval.3 
|- M = ( -v ` U )
Assertion nvmfval 
|- ( U e. NrmCVec -> M = ( x e. X , y e. X |-> ( x G ( -u 1 S y ) ) ) )

Proof

Step Hyp Ref Expression
1 nvmval.1 
 |-  X = ( BaseSet ` U )
2 nvmval.2 
 |-  G = ( +v ` U )
3 nvmval.4 
 |-  S = ( .sOLD ` U )
4 nvmval.3 
 |-  M = ( -v ` U )
5 2 nvgrp 
 |-  ( U e. NrmCVec -> G e. GrpOp )
6 1 2 bafval 
 |-  X = ran G
7 eqid 
 |-  ( inv ` G ) = ( inv ` G )
8 2 4 vsfval 
 |-  M = ( /g ` G )
9 6 7 8 grpodivfval 
 |-  ( G e. GrpOp -> M = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
10 5 9 syl 
 |-  ( U e. NrmCVec -> M = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
11 1 2 3 7 nvinv 
 |-  ( ( U e. NrmCVec /\ y e. X ) -> ( -u 1 S y ) = ( ( inv ` G ) ` y ) )
12 11 3adant2 
 |-  ( ( U e. NrmCVec /\ x e. X /\ y e. X ) -> ( -u 1 S y ) = ( ( inv ` G ) ` y ) )
13 12 oveq2d 
 |-  ( ( U e. NrmCVec /\ x e. X /\ y e. X ) -> ( x G ( -u 1 S y ) ) = ( x G ( ( inv ` G ) ` y ) ) )
14 13 mpoeq3dva 
 |-  ( U e. NrmCVec -> ( x e. X , y e. X |-> ( x G ( -u 1 S y ) ) ) = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
15 10 14 eqtr4d 
 |-  ( U e. NrmCVec -> M = ( x e. X , y e. X |-> ( x G ( -u 1 S y ) ) ) )