Theorem hvaddsubass

Description: Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	hvaddsubass	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐶 ) = ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		neg1cn	⊢ - 1 ∈ ℂ
2		hvmulcl	⊢ ( ( - 1 ∈ ℂ ∧ 𝐶 ∈ ℋ ) → ( - 1 ·ℎ 𝐶 ) ∈ ℋ )
3	1 2	mpan	⊢ ( 𝐶 ∈ ℋ → ( - 1 ·ℎ 𝐶 ) ∈ ℋ )
4		ax-hvass	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ ( - 1 ·ℎ 𝐶 ) ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( - 1 ·ℎ 𝐶 ) ) = ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
5	3 4	syl3an3	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( - 1 ·ℎ 𝐶 ) ) = ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
6		hvaddcl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) ∈ ℋ )
7		hvsubval	⊢ ( ( ( 𝐴 +ℎ 𝐵 ) ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐶 ) = ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( - 1 ·ℎ 𝐶 ) ) )
8	6 7	stoic3	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐶 ) = ( ( 𝐴 +ℎ 𝐵 ) +ℎ ( - 1 ·ℎ 𝐶 ) ) )
9		hvsubval	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 −ℎ 𝐶 ) = ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) )
10	9	3adant1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 −ℎ 𝐶 ) = ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) )
11	10	oveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
12	5 8 11	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐶 ) = ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) )