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Metamath Proof Explorer

Theorem ipval2lem3

Description: Lemma for ipval3 . (Contributed by NM, 1-Feb-2007) (New usage is discouraged.)

Ref Expression
Hypotheses dipfval.1 
|- X = ( BaseSet ` U )
dipfval.2 
|- G = ( +v ` U )
dipfval.4 
|- S = ( .sOLD ` U )
dipfval.6 
|- N = ( normCV ` U )
dipfval.7 
|- P = ( .iOLD ` U )
Assertion ipval2lem3 
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( N ` ( A G B ) ) ^ 2 ) e. RR )

Proof

Step Hyp Ref Expression
1 dipfval.1 
 |-  X = ( BaseSet ` U )
2 dipfval.2 
 |-  G = ( +v ` U )
3 dipfval.4 
 |-  S = ( .sOLD ` U )
4 dipfval.6 
 |-  N = ( normCV ` U )
5 dipfval.7 
 |-  P = ( .iOLD ` U )
6 1 3 nvsid 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( 1 S B ) = B )
7 6 oveq2d 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( A G ( 1 S B ) ) = ( A G B ) )
8 7 fveq2d 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( N ` ( A G ( 1 S B ) ) ) = ( N ` ( A G B ) ) )
9 8 oveq1d 
 |-  ( ( U e. NrmCVec /\ B e. X ) -> ( ( N ` ( A G ( 1 S B ) ) ) ^ 2 ) = ( ( N ` ( A G B ) ) ^ 2 ) )
10 9 3adant2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( N ` ( A G ( 1 S B ) ) ) ^ 2 ) = ( ( N ` ( A G B ) ) ^ 2 ) )
11 ax-1cn 
 |-  1 e. CC
12 1 2 3 4 5 ipval2lem2 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ 1 e. CC ) -> ( ( N ` ( A G ( 1 S B ) ) ) ^ 2 ) e. RR )
13 11 12 mpan2 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( N ` ( A G ( 1 S B ) ) ) ^ 2 ) e. RR )
14 10 13 eqeltrrd 
 |-  ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( N ` ( A G B ) ) ^ 2 ) e. RR )