Theorem hvaddcan

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

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

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) )
2		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 +ℎ 𝐶 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) )
3	1 2	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 +ℎ 𝐵 ) = ( 𝐴 +ℎ 𝐶 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ) )
4	3	bibi1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( 𝐴 +ℎ 𝐵 ) = ( 𝐴 +ℎ 𝐶 ) ↔ 𝐵 = 𝐶 ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ↔ 𝐵 = 𝐶 ) ) )
5		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
6	5	eqeq1d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ) )
7		eqeq1	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( 𝐵 = 𝐶 ↔ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = 𝐶 ) )
8	6 7	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ↔ 𝐵 = 𝐶 ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ↔ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = 𝐶 ) ) )
9		oveq2	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ) )
10	9	eqeq2d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ) ) )
11		eqeq2	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = 𝐶 ↔ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ) )
12	10 11	bibi12d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ 𝐶 ) ↔ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = 𝐶 ) ↔ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ) ↔ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ) ) )
13		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
14		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
15		ifhvhv0	⊢ if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ∈ ℋ
16	13 14 15	hvaddcani	⊢ ( ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) +ℎ if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ) ↔ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) )
17	4 8 12 16	dedth3h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) = ( 𝐴 +ℎ 𝐶 ) ↔ 𝐵 = 𝐶 ) )