Description: Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | nvpncan2.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| nvpncan2.2 | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | ||
| nvpncan2.3 | ⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) | ||
| Assertion | nvaddsub | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐶 ) = ( ( 𝐴 𝑀 𝐶 ) 𝐺 𝐵 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvpncan2.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
| 2 | nvpncan2.2 | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | |
| 3 | nvpncan2.3 | ⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) | |
| 4 | 2 | nvablo | ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp ) |
| 5 | 1 2 | bafval | ⊢ 𝑋 = ran 𝐺 |
| 6 | 2 3 | vsfval | ⊢ 𝑀 = ( /𝑔 ‘ 𝐺 ) |
| 7 | 5 6 | ablomuldiv | ⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐶 ) = ( ( 𝐴 𝑀 𝐶 ) 𝐺 𝐵 ) ) |
| 8 | 4 7 | sylan | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐶 ) = ( ( 𝐴 𝑀 𝐶 ) 𝐺 𝐵 ) ) |