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Metamath Proof Explorer

Theorem ngpdsr

Description: Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses ngpds.n ⊢ 𝑁 = ( norm ‘ 𝐺 )
ngpds.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
ngpds.m ⊢ − = ( -g ‘ 𝐺 )
ngpds.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
Assertion ngpdsr ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 ngpds.n ⊢ 𝑁 = ( norm ‘ 𝐺 )
2 ngpds.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
3 ngpds.m ⊢ − = ( -g ‘ 𝐺 )
4 ngpds.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
5 ngpxms ⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp )
6 2 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 ngpds ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )
9 8 3com23 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 𝐷 𝐴 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )
10 7 9 eqtrd ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐵 − 𝐴 ) ) )