Metamath Proof Explorer

Theorem nmmtri

Description: The triangle inequality for the norm of a subtraction. (Contributed by NM, 27-Dec-2007) (Revised by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses nmf.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
nmf.n ⊢ 𝑁 = ( norm ‘ 𝐺 )
nmmtri.m ⊢ − = ( -g ‘ 𝐺 )
Assertion nmmtri ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 nmf.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
2 nmf.n ⊢ 𝑁 = ( norm ‘ 𝐺 )
3 nmmtri.m ⊢ − = ( -g ‘ 𝐺 )
4 eqid ⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
5 2 1 3 4 ngpds ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( dist ‘ 𝐺 ) 𝐵 ) = ( 𝑁 ‘ ( 𝐴 − 𝐵 ) ) )
6 ngpms ⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp )
7 6 3ad2ant1 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐺 ∈ MetSp )
8 simp2 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
9 simp3 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 𝐵 ∈ 𝑋 )
10 ngpgrp ⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp )
11 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
12 1 11 grpidcl ⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ 𝑋 )
13 10 12 syl ⊢ ( 𝐺 ∈ NrmGrp → ( 0g ‘ 𝐺 ) ∈ 𝑋 )
14 13 3ad2ant1 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 0g ‘ 𝐺 ) ∈ 𝑋 )
15 1 4 mstri3 ⊢ ( ( 𝐺 ∈ MetSp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( 0g ‘ 𝐺 ) ∈ 𝑋 ) ) → ( 𝐴 ( dist ‘ 𝐺 ) 𝐵 ) ≤ ( ( 𝐴 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) + ( 𝐵 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) )
16 7 8 9 14 15 syl13anc ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( dist ‘ 𝐺 ) 𝐵 ) ≤ ( ( 𝐴 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) + ( 𝐵 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) )
17 2 1 11 4 nmval ⊢ ( 𝐴 ∈ 𝑋 → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) )
18 17 3ad2ant2 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) )
19 2 1 11 4 nmval ⊢ ( 𝐵 ∈ 𝑋 → ( 𝑁 ‘ 𝐵 ) = ( 𝐵 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) )
20 19 3ad2ant3 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐵 ) = ( 𝐵 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) )
21 18 20 oveq12d ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) = ( ( 𝐴 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) + ( 𝐵 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) )
22 16 21 breqtrrd ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( dist ‘ 𝐺 ) 𝐵 ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )
23 5 22 eqbrtrrd ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( 𝑁 ‘ 𝐴 ) + ( 𝑁 ‘ 𝐵 ) ) )