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Metamath Proof Explorer

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 ) )