Metamath Proof Explorer

Theorem ipval2lem2

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 ipval2lem2 
|- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> ( ( N ` ( A G ( C S 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 simpl1 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> U e. NrmCVec )
7 simpl2 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> A e. X )
8 1 3 nvscl 
 |-  ( ( U e. NrmCVec /\ C e. CC /\ B e. X ) -> ( C S B ) e. X )
9 8 3com23 
 |-  ( ( U e. NrmCVec /\ B e. X /\ C e. CC ) -> ( C S B ) e. X )
10 9 3expa 
 |-  ( ( ( U e. NrmCVec /\ B e. X ) /\ C e. CC ) -> ( C S B ) e. X )
11 10 3adantl2 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> ( C S B ) e. X )
12 1 2 nvgcl 
 |-  ( ( U e. NrmCVec /\ A e. X /\ ( C S B ) e. X ) -> ( A G ( C S B ) ) e. X )
13 6 7 11 12 syl3anc 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> ( A G ( C S B ) ) e. X )
14 1 4 nvcl 
 |-  ( ( U e. NrmCVec /\ ( A G ( C S B ) ) e. X ) -> ( N ` ( A G ( C S B ) ) ) e. RR )
15 6 13 14 syl2anc 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> ( N ` ( A G ( C S B ) ) ) e. RR )
16 15 resqcld 
 |-  ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ C e. CC ) -> ( ( N ` ( A G ( C S B ) ) ) ^ 2 ) e. RR )