Theorem hvaddsub12

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	hvaddsub12	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 −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		hvadd12	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ ( - 1 ·ℎ 𝐶 ) ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
5	3 4	syl3an3	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
6		hvsubval	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 −ℎ 𝐶 ) = ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) )
7	6	oveq2d	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
8	7	3adant1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( 𝐴 +ℎ ( 𝐵 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
9		hvsubval	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 −ℎ 𝐶 ) = ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) )
10	9	oveq2d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 +ℎ ( 𝐴 −ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
11	10	3adant2	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 +ℎ ( 𝐴 −ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ ( - 1 ·ℎ 𝐶 ) ) ) )
12	5 8 11	3eqtr4d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 −ℎ 𝐶 ) ) )