Theorem hvsub32

Description: Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	hvsub32	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) −ℎ 𝐶 ) = ( ( 𝐴 −ℎ 𝐶 ) −ℎ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-hvcom	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 +ℎ 𝐶 ) = ( 𝐶 +ℎ 𝐵 ) )
2	1	3adant1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 +ℎ 𝐶 ) = ( 𝐶 +ℎ 𝐵 ) )
3	2	oveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 −ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( 𝐴 −ℎ ( 𝐶 +ℎ 𝐵 ) ) )
4		hvsubass	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) −ℎ 𝐶 ) = ( 𝐴 −ℎ ( 𝐵 +ℎ 𝐶 ) ) )
5		hvsubass	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐶 ) −ℎ 𝐵 ) = ( 𝐴 −ℎ ( 𝐶 +ℎ 𝐵 ) ) )
6	5	3com23	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐶 ) −ℎ 𝐵 ) = ( 𝐴 −ℎ ( 𝐶 +ℎ 𝐵 ) ) )
7	3 4 6	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) −ℎ 𝐶 ) = ( ( 𝐴 −ℎ 𝐶 ) −ℎ 𝐵 ) )