Description: The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | nvgcl.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | ||
Assertion | nvass | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvgcl.1 | ⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) | |
2 | nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) | |
3 | 2 | nvgrp | ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp ) |
4 | 1 2 | bafval | ⊢ 𝑋 = ran 𝐺 |
5 | 4 | grpoass | ⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) ) |
6 | 3 5 | sylan | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) ) |