Metamath Proof Explorer

Theorem dipassr

Description: "Associative" law for second argument of inner product (compare dipass ). (Contributed by NM, 22-Nov-2007) (New usage is discouraged.)

Ref Expression
Hypotheses ipass.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
ipass.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
ipass.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion dipassr ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝑆 𝐶 ) ) = ( ( ∗ ‘ 𝐵 ) · ( 𝐴 𝑃 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ipass.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2 ipass.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3 ipass.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4 3anrot ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) )
5 1 2 3 dipass ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝐵 𝑆 𝐶 ) 𝑃 𝐴 ) = ( 𝐵 · ( 𝐶 𝑃 𝐴 ) ) )
6 4 5 sylan2b ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐵 𝑆 𝐶 ) 𝑃 𝐴 ) = ( 𝐵 · ( 𝐶 𝑃 𝐴 ) ) )
7 6 fveq2d ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑆 𝐶 ) 𝑃 𝐴 ) ) = ( ∗ ‘ ( 𝐵 · ( 𝐶 𝑃 𝐴 ) ) ) )
8 phnv ⊢ ( 𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec )
9 simpl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → 𝑈 ∈ NrmCVec )
10 1 2 nvscl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 𝑆 𝐶 ) ∈ 𝑋 )
11 10 3adant3r1 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐵 𝑆 𝐶 ) ∈ 𝑋 )
12 simpr1 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
13 1 3 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐵 𝑆 𝐶 ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( ( 𝐵 𝑆 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝑆 𝐶 ) ) )
14 9 11 12 13 syl3anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑆 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝑆 𝐶 ) ) )
15 8 14 sylan ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( ( 𝐵 𝑆 𝐶 ) 𝑃 𝐴 ) ) = ( 𝐴 𝑃 ( 𝐵 𝑆 𝐶 ) ) )
16 simpr2 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → 𝐵 ∈ ℂ )
17 1 3 dipcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
18 17 3com23 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
19 18 3adant3r2 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐶 𝑃 𝐴 ) ∈ ℂ )
20 16 19 cjmuld ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( 𝐵 · ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ 𝐵 ) · ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) )
21 1 3 dipcj ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
22 21 3com23 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
23 22 3adant3r2 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) = ( 𝐴 𝑃 𝐶 ) )
24 23 oveq2d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( ∗ ‘ 𝐵 ) · ( ∗ ‘ ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ 𝐵 ) · ( 𝐴 𝑃 𝐶 ) ) )
25 20 24 eqtrd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( 𝐵 · ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ 𝐵 ) · ( 𝐴 𝑃 𝐶 ) ) )
26 8 25 sylan ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ∗ ‘ ( 𝐵 · ( 𝐶 𝑃 𝐴 ) ) ) = ( ( ∗ ‘ 𝐵 ) · ( 𝐴 𝑃 𝐶 ) ) )
27 7 15 26 3eqtr3d ⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑃 ( 𝐵 𝑆 𝐶 ) ) = ( ( ∗ ‘ 𝐵 ) · ( 𝐴 𝑃 𝐶 ) ) )