Description: Cancellation law for vector subtraction. ( nnncan1 analog.) (Contributed by NM, 7-Mar-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | nvmf.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| nvmf.3 | ⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) | ||
| Assertion | nvnnncan1 | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑀 𝐵 ) 𝑀 ( 𝐴 𝑀 𝐶 ) ) = ( 𝐶 𝑀 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvmf.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| 2 | nvmf.3 | ⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) | |
| 3 | eqid | ⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 ) | |
| 4 | 3 | nvablo | ⊢ ( 𝑈 ∈ NrmCVec → ( +𝑣 ‘ 𝑈 ) ∈ AbelOp ) |
| 5 | 1 3 | bafval | ⊢ 𝑋 = ran ( +𝑣 ‘ 𝑈 ) |
| 6 | 3 2 | vsfval | ⊢ 𝑀 = ( /𝑔 ‘ ( +𝑣 ‘ 𝑈 ) ) |
| 7 | 5 6 | ablonnncan1 | ⊢ ( ( ( +𝑣 ‘ 𝑈 ) ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑀 𝐵 ) 𝑀 ( 𝐴 𝑀 𝐶 ) ) = ( 𝐶 𝑀 𝐵 ) ) |
| 8 | 4 7 | sylan | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑀 𝐵 ) 𝑀 ( 𝐴 𝑀 𝐶 ) ) = ( 𝐶 𝑀 𝐵 ) ) |