Theorem hvaddcan2

Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	hvaddcan2	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐶 ) = ( 𝐵 +ℎ 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-hvcom	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐶 +ℎ 𝐴 ) = ( 𝐴 +ℎ 𝐶 ) )
2	1	3adant3	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐶 +ℎ 𝐴 ) = ( 𝐴 +ℎ 𝐶 ) )
3		ax-hvcom	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐶 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐶 ) )
4	3	3adant2	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐶 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐶 ) )
5	2 4	eqeq12d	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐶 +ℎ 𝐴 ) = ( 𝐶 +ℎ 𝐵 ) ↔ ( 𝐴 +ℎ 𝐶 ) = ( 𝐵 +ℎ 𝐶 ) ) )
6		hvaddcan	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐶 +ℎ 𝐴 ) = ( 𝐶 +ℎ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
7	5 6	bitr3d	⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐶 ) = ( 𝐵 +ℎ 𝐶 ) ↔ 𝐴 = 𝐵 ) )
8	7	3coml	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐶 ) = ( 𝐵 +ℎ 𝐶 ) ↔ 𝐴 = 𝐵 ) )