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Metamath Proof Explorer

Theorem ngpds2

Description: Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses ngpds2.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
ngpds2.z ⊢ 0 = ( 0g ‘ 𝐺 )
ngpds2.m ⊢ − = ( -g ‘ 𝐺 )
ngpds2.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
Assertion ngpds2 ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) )

Proof

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 ) )