Metamath Proof Explorer


Theorem dipsubdi

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

Ref Expression
Hypotheses ipsubdir.1 𝑋 = ( BaseSet ‘ 𝑈 )
ipsubdir.3 𝑀 = ( −𝑣𝑈 )
ipsubdir.7 𝑃 = ( ·𝑖OLD𝑈 )
Assertion dipsubdi ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝑀 𝐶 ) ) = ( ( 𝐴 𝑃 𝐵 ) − ( 𝐴 𝑃 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ipsubdir.1 𝑋 = ( BaseSet ‘ 𝑈 )
2 ipsubdir.3 𝑀 = ( −𝑣𝑈 )
3 ipsubdir.7 𝑃 = ( ·𝑖OLD𝑈 )
4 id ( ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) → ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) )
5 4 3com13 ( ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) → ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) )
6 id ( ( 𝐵𝑋𝐶𝑋𝐴𝑋 ) → ( 𝐵𝑋𝐶𝑋𝐴𝑋 ) )
7 6 3com12 ( ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) → ( 𝐵𝑋𝐶𝑋𝐴𝑋 ) )
8 1 2 3 dipsubdir ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐵𝑋𝐶𝑋𝐴𝑋 ) ) → ( ( 𝐵 𝑀 𝐶 ) 𝑃 𝐴 ) = ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) )
9 7 8 sylan2 ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ( 𝐵 𝑀 𝐶 ) 𝑃 𝐴 ) = ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) )
10 9 fveq2d ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑀 𝐶 ) 𝑃 𝐴 ) ) = ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) ) )
11 phnv ( 𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec )
12 simpl ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → 𝑈 ∈ NrmCVec )
13 1 2 nvmcl ( ( 𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐶𝑋 ) → ( 𝐵 𝑀 𝐶 ) ∈ 𝑋 )
14 13 3com23 ( ( 𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐵𝑋 ) → ( 𝐵 𝑀 𝐶 ) ∈ 𝑋 )
15 14 3adant3r3 ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( 𝐵 𝑀 𝐶 ) ∈ 𝑋 )
16 simpr3 ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → 𝐴𝑋 )
17 1 3 dipcj ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐵 𝑀 𝐶 ) ∈ 𝑋𝐴𝑋 ) → ( ∗ ‘ ( ( 𝐵 𝑀 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝑀 𝐶 ) ) )
18 12 15 16 17 syl3anc ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑀 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝑀 𝐶 ) ) )
19 11 18 sylan ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑀 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝑀 𝐶 ) ) )
20 1 3 dipcl ( ( 𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐴𝑋 ) → ( 𝐵 𝑃 𝐴 ) ∈ ℂ )
21 20 3adant3r1 ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( 𝐵 𝑃 𝐴 ) ∈ ℂ )
22 1 3 dipcl ( ( 𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐴𝑋 ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
23 22 3adant3r2 ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
24 cjsub ( ( ( 𝐵 𝑃 𝐴 ) ∈ ℂ ∧ ( 𝐶 𝑃 𝐴 ) ∈ ℂ ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) − ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) )
25 21 23 24 syl2anc ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) − ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) )
26 1 3 dipcj ( ( 𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐴𝑋 ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
27 26 3adant3r1 ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐵 ) )
28 1 3 dipcj ( ( 𝑈 ∈ NrmCVec ∧ 𝐶𝑋𝐴𝑋 ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
29 28 3adant3r2 ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
30 27 29 oveq12d ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ( ∗ ‘ ( 𝐵 𝑃 𝐴 ) ) − ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) − ( 𝐴 𝑃 𝐶 ) ) )
31 25 30 eqtrd ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) − ( 𝐴 𝑃 𝐶 ) ) )
32 11 31 sylan ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑃 𝐴 ) − ( 𝐶 𝑃 𝐴 ) ) ) = ( ( 𝐴 𝑃 𝐵 ) − ( 𝐴 𝑃 𝐶 ) ) )
33 10 19 32 3eqtr3d ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐶𝑋𝐵𝑋𝐴𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝑀 𝐶 ) ) = ( ( 𝐴 𝑃 𝐵 ) − ( 𝐴 𝑃 𝐶 ) ) )
34 5 33 sylan2 ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴𝑋𝐵𝑋𝐶𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝑀 𝐶 ) ) = ( ( 𝐴 𝑃 𝐵 ) − ( 𝐴 𝑃 𝐶 ) ) )