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Metamath Proof Explorer

Theorem ngpds2r

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 ngpds2r ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐵 − 𝐴 ) 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 ngpxms ⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp )
6 1 4 xmssym ⊢ ( ( 𝐺 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) )
7 5 6 syl3an1 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐵 𝐷 𝐴 ) )
8 1 2 3 4 ngpds2 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( ( 𝐵 − 𝐴 ) 𝐷 0 ) )
9 8 3com23 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( ( 𝐵 − 𝐴 ) 𝐷 0 ) )
10 7 9 eqtrd ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐵 − 𝐴 ) 𝐷 0 ) )