Theorem lnfnsubi

Description: Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	lnfnl.1	⊢ 𝑇 ∈ LinFn
Assertion	lnfnsubi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) − ( 𝑇 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lnfnl.1	⊢ 𝑇 ∈ LinFn
2		neg1cn	⊢ - 1 ∈ ℂ
3	1	lnfnaddmuli	⊢ ( ( - 1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( - 1 · ( 𝑇 ‘ 𝐵 ) ) ) )
4	2 3	mp3an1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( - 1 · ( 𝑇 ‘ 𝐵 ) ) ) )
5		hvsubval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 −ℎ 𝐵 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) )
6	5	fveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝑇 ‘ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐵 ) ) ) )
7	1	lnfnfi	⊢ 𝑇 : ℋ ⟶ ℂ
8	7	ffvelrni	⊢ ( 𝐴 ∈ ℋ → ( 𝑇 ‘ 𝐴 ) ∈ ℂ )
9	7	ffvelrni	⊢ ( 𝐵 ∈ ℋ → ( 𝑇 ‘ 𝐵 ) ∈ ℂ )
10		mulm1	⊢ ( ( 𝑇 ‘ 𝐵 ) ∈ ℂ → ( - 1 · ( 𝑇 ‘ 𝐵 ) ) = - ( 𝑇 ‘ 𝐵 ) )
11	10	oveq2d	⊢ ( ( 𝑇 ‘ 𝐵 ) ∈ ℂ → ( ( 𝑇 ‘ 𝐴 ) + ( - 1 · ( 𝑇 ‘ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) + - ( 𝑇 ‘ 𝐵 ) ) )
12	11	adantl	⊢ ( ( ( 𝑇 ‘ 𝐴 ) ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℂ ) → ( ( 𝑇 ‘ 𝐴 ) + ( - 1 · ( 𝑇 ‘ 𝐵 ) ) ) = ( ( 𝑇 ‘ 𝐴 ) + - ( 𝑇 ‘ 𝐵 ) ) )
13		negsub	⊢ ( ( ( 𝑇 ‘ 𝐴 ) ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℂ ) → ( ( 𝑇 ‘ 𝐴 ) + - ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) − ( 𝑇 ‘ 𝐵 ) ) )
14	12 13	eqtr2d	⊢ ( ( ( 𝑇 ‘ 𝐴 ) ∈ ℂ ∧ ( 𝑇 ‘ 𝐵 ) ∈ ℂ ) → ( ( 𝑇 ‘ 𝐴 ) − ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( - 1 · ( 𝑇 ‘ 𝐵 ) ) ) )
15	8 9 14	syl2an	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝑇 ‘ 𝐴 ) − ( 𝑇 ‘ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) + ( - 1 · ( 𝑇 ‘ 𝐵 ) ) ) )
16	4 6 15	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝑇 ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( ( 𝑇 ‘ 𝐴 ) − ( 𝑇 ‘ 𝐵 ) ) )