Metamath Proof Explorer

Theorem vcass

Description: Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006) (New usage is discouraged.)

Ref Expression
Hypotheses vciOLD.1 ⊢ 𝐺 = ( 1st ‘ 𝑊 )
vciOLD.2 ⊢ 𝑆 = ( 2nd ‘ 𝑊 )
vciOLD.3 ⊢ 𝑋 = ran 𝐺
Assertion vcass ( ( 𝑊 ∈ CVecOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 vciOLD.1 ⊢ 𝐺 = ( 1st ‘ 𝑊 )
2 vciOLD.2 ⊢ 𝑆 = ( 2nd ‘ 𝑊 )
3 vciOLD.3 ⊢ 𝑋 = ran 𝐺
4 1 2 3 vciOLD ⊢ ( 𝑊 ∈ CVecOLD → ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) ) ) )
5 simpr ⊢ ( ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) → ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
6 5 ralimi ⊢ ( ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) → ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
7 6 adantl ⊢ ( ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) → ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
8 7 ralimi ⊢ ( ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) → ∀ 𝑦 ∈ ℂ ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
9 8 adantl ⊢ ( ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) ) → ∀ 𝑦 ∈ ℂ ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
10 9 ralimi ⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
11 10 3ad2ant3 ⊢ ( ( 𝐺 ∈ AbelOp ∧ 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( ( 1 𝑆 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ 𝑋 ( 𝑦 𝑆 ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑦 𝑆 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 𝑆 𝑥 ) 𝐺 ( 𝑧 𝑆 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ) ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
12 4 11 syl ⊢ ( 𝑊 ∈ CVecOLD → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) )
13 oveq2 ⊢ ( 𝑥 = 𝐶 → ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( ( 𝑦 · 𝑧 ) 𝑆 𝐶 ) )
14 oveq2 ⊢ ( 𝑥 = 𝐶 → ( 𝑧 𝑆 𝑥 ) = ( 𝑧 𝑆 𝐶 ) )
15 14 oveq2d ⊢ ( 𝑥 = 𝐶 → ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝐶 ) ) )
16 13 15 eqeq12d ⊢ ( 𝑥 = 𝐶 → ( ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) ↔ ( ( 𝑦 · 𝑧 ) 𝑆 𝐶 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝐶 ) ) ) )
17 oveq1 ⊢ ( 𝑦 = 𝐴 → ( 𝑦 · 𝑧 ) = ( 𝐴 · 𝑧 ) )
18 17 oveq1d ⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 · 𝑧 ) 𝑆 𝐶 ) = ( ( 𝐴 · 𝑧 ) 𝑆 𝐶 ) )
19 oveq1 ⊢ ( 𝑦 = 𝐴 → ( 𝑦 𝑆 ( 𝑧 𝑆 𝐶 ) ) = ( 𝐴 𝑆 ( 𝑧 𝑆 𝐶 ) ) )
20 18 19 eqeq12d ⊢ ( 𝑦 = 𝐴 → ( ( ( 𝑦 · 𝑧 ) 𝑆 𝐶 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝐶 ) ) ↔ ( ( 𝐴 · 𝑧 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝑧 𝑆 𝐶 ) ) ) )
21 oveq2 ⊢ ( 𝑧 = 𝐵 → ( 𝐴 · 𝑧 ) = ( 𝐴 · 𝐵 ) )
22 21 oveq1d ⊢ ( 𝑧 = 𝐵 → ( ( 𝐴 · 𝑧 ) 𝑆 𝐶 ) = ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) )
23 oveq1 ⊢ ( 𝑧 = 𝐵 → ( 𝑧 𝑆 𝐶 ) = ( 𝐵 𝑆 𝐶 ) )
24 23 oveq2d ⊢ ( 𝑧 = 𝐵 → ( 𝐴 𝑆 ( 𝑧 𝑆 𝐶 ) ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) )
25 22 24 eqeq12d ⊢ ( 𝑧 = 𝐵 → ( ( ( 𝐴 · 𝑧 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝑧 𝑆 𝐶 ) ) ↔ ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) ) )
26 16 20 25 rspc3v ⊢ ( ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ ℂ ∀ 𝑧 ∈ ℂ ( ( 𝑦 · 𝑧 ) 𝑆 𝑥 ) = ( 𝑦 𝑆 ( 𝑧 𝑆 𝑥 ) ) → ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) ) )
27 12 26 syl5 ⊢ ( ( 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑊 ∈ CVecOLD → ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) ) )
28 27 3coml ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) → ( 𝑊 ∈ CVecOLD → ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) ) )
29 28 impcom ⊢ ( ( 𝑊 ∈ CVecOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 · 𝐵 ) 𝑆 𝐶 ) = ( 𝐴 𝑆 ( 𝐵 𝑆 𝐶 ) ) )