Step |
Hyp |
Ref |
Expression |
1 |
|
ngpds2.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
ngpds2.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
ngpds2.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
ngpds2.d |
⊢ 𝐷 = ( dist ‘ 𝐺 ) |
5 |
|
eqid |
⊢ ( norm ‘ 𝐺 ) = ( norm ‘ 𝐺 ) |
6 |
5 1 3 4
|
ngpds |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( norm ‘ 𝐺 ) ‘ ( 𝐴 − 𝐵 ) ) ) |
7 |
|
ngpgrp |
⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp ) |
8 |
1 3
|
grpsubcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 ) |
9 |
7 8
|
syl3an1 |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 ) |
10 |
5 1 2 4
|
nmval |
⊢ ( ( 𝐴 − 𝐵 ) ∈ 𝑋 → ( ( norm ‘ 𝐺 ) ‘ ( 𝐴 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) ) |
11 |
9 10
|
syl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( norm ‘ 𝐺 ) ‘ ( 𝐴 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) ) |
12 |
6 11
|
eqtrd |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) ) |