Metamath Proof Explorer


Theorem pjmuli

Description: Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypothesis pjadjt.1 HC
Assertion pjmuli ABprojHAB=AprojHB

Proof

Step Hyp Ref Expression
1 pjadjt.1 HC
2 fvoveq1 A=ifAA0projHAB=projHifAA0B
3 oveq1 A=ifAA0AprojHB=ifAA0projHB
4 2 3 eqeq12d A=ifAA0projHAB=AprojHBprojHifAA0B=ifAA0projHB
5 oveq2 B=ifBB0ifAA0B=ifAA0ifBB0
6 5 fveq2d B=ifBB0projHifAA0B=projHifAA0ifBB0
7 fveq2 B=ifBB0projHB=projHifBB0
8 7 oveq2d B=ifBB0ifAA0projHB=ifAA0projHifBB0
9 6 8 eqeq12d B=ifBB0projHifAA0B=ifAA0projHBprojHifAA0ifBB0=ifAA0projHifBB0
10 ifhvhv0 ifBB0
11 0cn 0
12 11 elimel ifAA0
13 1 10 12 pjmulii projHifAA0ifBB0=ifAA0projHifBB0
14 4 9 13 dedth2h ABprojHAB=AprojHB