Metamath Proof Explorer

Theorem lnopmul

Description: Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion lnopmul ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 ax-hv0cl ⊢ 0ℎ ∈ ℋ
2 lnopl ⊢ ( ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ) ∧ ( 𝐵 ∈ ℋ ∧ 0ℎ ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 0ℎ ) ) )
3 1 2 mpanr2 ⊢ ( ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ) ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 0ℎ ) ) )
4 3 3impa ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 0ℎ ) ) )
5 hvmulcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )
6 ax-hvaddid ⊢ ( ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ → ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ 𝐵 ) )
7 5 6 syl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ 𝐵 ) )
8 7 3adant1 ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ 𝐵 ) )
9 8 fveq2d ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 0ℎ ) ) = ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) )
10 lnop0 ⊢ ( 𝑇 ∈ LinOp → ( 𝑇 ‘ 0ℎ ) = 0ℎ )
11 10 oveq2d ⊢ ( 𝑇 ∈ LinOp → ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 0ℎ ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ 0ℎ ) )
12 11 3ad2ant1 ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 0ℎ ) ) = ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ 0ℎ ) )
13 lnopf ⊢ ( 𝑇 ∈ LinOp → 𝑇 : ℋ ⟶ ℋ )
14 13 ffvelcdmda ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ 𝐵 ) ∈ ℋ )
15 hvmulcl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ )
16 14 15 sylan2 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ ) ) → ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ )
17 16 3impb ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ )
18 17 3com12 ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ )
19 ax-hvaddid ⊢ ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) ∈ ℋ → ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )
20 18 19 syl ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ 0ℎ ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )
21 12 20 eqtrd ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) +ℎ ( 𝑇 ‘ 0ℎ ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )
22 4 9 21 3eqtr3d ⊢ ( ( 𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( 𝑇 ‘ 𝐵 ) ) )