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Metamath Proof Explorer

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 ) ) )