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 β€˜ 𝐺 )
ngpds2.z ⊒ 0 = ( 0g β€˜ 𝐺 )
ngpds2.m ⊒ βˆ’ = ( -g β€˜ 𝐺 )
ngpds2.d ⊒ 𝐷 = ( dist β€˜ 𝐺 )
Assertion ngpds3 ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 𝐷 𝐡 ) = ( 0 𝐷 ( 𝐴 βˆ’ 𝐡 ) ) )

Proof

Step Hyp Ref Expression
1 ngpds2.x ⊒ 𝑋 = ( Base β€˜ 𝐺 )
2 ngpds2.z ⊒ 0 = ( 0g β€˜ 𝐺 )
3 ngpds2.m ⊒ βˆ’ = ( -g β€˜ 𝐺 )
4 ngpds2.d ⊒ 𝐷 = ( dist β€˜ 𝐺 )
5 1 2 3 4 ngpds2 ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 𝐷 𝐡 ) = ( ( 𝐴 βˆ’ 𝐡 ) 𝐷 0 ) )
6 ngpxms ⊒ ( 𝐺 ∈ NrmGrp β†’ 𝐺 ∈ ∞MetSp )
7 6 3ad2ant1 ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ 𝐺 ∈ ∞MetSp )
8 ngpgrp ⊒ ( 𝐺 ∈ NrmGrp β†’ 𝐺 ∈ Grp )
9 1 3 grpsubcl ⊒ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 βˆ’ 𝐡 ) ∈ 𝑋 )
10 8 9 syl3an1 ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 βˆ’ 𝐡 ) ∈ 𝑋 )
11 8 3ad2ant1 ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ 𝐺 ∈ Grp )
12 1 2 grpidcl ⊒ ( 𝐺 ∈ Grp β†’ 0 ∈ 𝑋 )
13 11 12 syl ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ 0 ∈ 𝑋 )
14 1 4 xmssym ⊒ ( ( 𝐺 ∈ ∞MetSp ∧ ( 𝐴 βˆ’ 𝐡 ) ∈ 𝑋 ∧ 0 ∈ 𝑋 ) β†’ ( ( 𝐴 βˆ’ 𝐡 ) 𝐷 0 ) = ( 0 𝐷 ( 𝐴 βˆ’ 𝐡 ) ) )
15 7 10 13 14 syl3anc ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( ( 𝐴 βˆ’ 𝐡 ) 𝐷 0 ) = ( 0 𝐷 ( 𝐴 βˆ’ 𝐡 ) ) )
16 5 15 eqtrd ⊒ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ) β†’ ( 𝐴 𝐷 𝐡 ) = ( 0 𝐷 ( 𝐴 βˆ’ 𝐡 ) ) )