This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dipdi

Description: Distributive law for inner product. (Contributed by NM, 20-Nov-2007) (New usage is discouraged.)

Ref Expression
Hypotheses dipdir.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
dipdir.2 ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
dipdir.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion dipdi ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑃 𝐵 ) + ( 𝐴 𝑃 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 dipdir.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2 dipdir.2 ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3 dipdir.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4 id ⊢ ( ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) )
5 4 3com13 ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) )
6 id ⊢ ( ( 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) )
7 6 3com12 ⊢ ( ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) )
8 1 2 3 dipdir ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝐵 𝐺 𝐶 ) 𝑃 𝐴 ) = ( ( 𝐵 𝑃 𝐴 ) + ( 𝐶 𝑃 𝐴 ) ) )
9 7 8 sylan2 ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝐵 𝐺 𝐶 ) 𝑃 𝐴 ) = ( ( 𝐵 𝑃 𝐴 ) + ( 𝐶 𝑃 𝐴 ) ) )
10 9 fveq2d ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝐺 𝐶 ) 𝑃 𝐴 ) ) = ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) + ( 𝐶 𝑃 𝐴 ) ) ) )
11 phnv ⊢ ( 𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec )
12 simpl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → 𝑈 ∈ NrmCVec )
13 1 2 nvgcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 𝐺 𝐶 ) ∈ 𝑋 )
14 13 3com23 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐺 𝐶 ) ∈ 𝑋 )
15 14 3adant3r3 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐵 𝐺 𝐶 ) ∈ 𝑋 )
16 simpr3 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
17 1 3 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐵 𝐺 𝐶 ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( ( 𝐵 𝐺 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝐺 𝐶 ) ) )
18 12 15 16 17 syl3anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝐺 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝐺 𝐶 ) ) )
19 11 18 sylan ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝐺 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝐺 𝐶 ) ) )
20 1 3 dipcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 𝑃 𝐴 ) ∈ ℂ )
21 20 3adant3r1 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐵 𝑃 𝐴 ) ∈ ℂ )
22 1 3 dipcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
23 22 3adant3r2 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
24 21 23 cjaddd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) + ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) + ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) )
25 1 3 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
26 25 3adant3r1 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
27 1 3 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
28 27 3adant3r2 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
29 26 28 oveq12d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) + ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) + ( 𝐴 𝑃 𝐶 ) ) )
30 24 29 eqtrd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) + ( 𝐶 𝑃 𝐴 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) + ( 𝐴 𝑃 𝐶 ) ) )
31 11 30 sylan ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) + ( 𝐶 𝑃 𝐴 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) + ( 𝐴 𝑃 𝐶 ) ) )
32 10 19 31 3eqtr3d ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑃 𝐵 ) + ( 𝐴 𝑃 𝐶 ) ) )
33 5 32 sylan2 ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑃 𝐵 ) + ( 𝐴 𝑃 𝐶 ) ) )