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Metamath Proof Explorer

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 ) )