Step |
Hyp |
Ref |
Expression |
1 |
|
nvpncan2.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
nvpncan2.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
3 |
|
nvpncan2.3 |
⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) |
4 |
|
simprl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 ) |
5 |
|
simprr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → 𝐵 ∈ 𝑋 ) |
6 |
4 5 5
|
3jca |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) |
7 |
1 2 3
|
nvaddsub |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) ) |
8 |
6 7
|
syldan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) ) |
9 |
8
|
3impb |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) ) |
10 |
1 2 3
|
nvpncan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐺 𝐵 ) 𝑀 𝐵 ) = 𝐴 ) |
11 |
9 10
|
eqtr3d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝑀 𝐵 ) 𝐺 𝐵 ) = 𝐴 ) |