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Metamath Proof Explorer

Theorem ngpds3

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 ngpds3 ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )

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 1 2 3 4 ngpds2 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( ( 𝐴 − 𝐵 ) 𝐷 0 ) )
6 ngpxms ⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp )
7 6 3ad2ant1 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐺 ∈ ∞MetSp )
8 ngpgrp ⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp )
9 1 3 grpsubcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 )
10 8 9 syl3an1 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 − 𝐵 ) ∈ 𝑋 )
11 8 3ad2ant1 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐺 ∈ Grp )
12 1 2 grpidcl ⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝑋 )
13 11 12 syl ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 0 ∈ 𝑋 )
14 1 4 xmssym ⊢ ( ( 𝐺 ∈ ∞MetSp ∧ ( 𝐴 − 𝐵 ) ∈ 𝑋 ∧ 0 ∈ 𝑋 ) → ( ( 𝐴 − 𝐵 ) 𝐷 0 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )
15 7 10 13 14 syl3anc ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 − 𝐵 ) 𝐷 0 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )
16 5 15 eqtrd ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 0 𝐷 ( 𝐴 − 𝐵 ) ) )