Metamath Proof Explorer

Theorem nmsub

Description: The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses nmf.x ⊢ X = Base G
nmf.n ⊢ N = norm ⁡ G
nmmtri.m ⊢ - ˙ = - G
Assertion nmsub ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → N ⁡ A - ˙ B = N ⁡ B - ˙ A

Proof

Step Hyp Ref Expression
1 nmf.x ⊢ X = Base G
2 nmf.n ⊢ N = norm ⁡ G
3 nmmtri.m ⊢ - ˙ = - G
4 simp1 ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → G ∈ NrmGrp
5 ngpgrp ⊢ G ∈ NrmGrp → G ∈ Grp
6 4 5 syl ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → G ∈ Grp
7 simp3 ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → B ∈ X
8 simp2 ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → A ∈ X
9 eqid ⊢ inv g ⁡ G = inv g ⁡ G
10 1 3 9 grpinvsub ⊢ G ∈ Grp ∧ B ∈ X ∧ A ∈ X → inv g ⁡ G ⁡ B - ˙ A = A - ˙ B
11 6 7 8 10 syl3anc ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → inv g ⁡ G ⁡ B - ˙ A = A - ˙ B
12 11 fveq2d ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → N ⁡ inv g ⁡ G ⁡ B - ˙ A = N ⁡ A - ˙ B
13 1 3 grpsubcl ⊢ G ∈ Grp ∧ B ∈ X ∧ A ∈ X → B - ˙ A ∈ X
14 6 7 8 13 syl3anc ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → B - ˙ A ∈ X
15 1 2 9 nminv ⊢ G ∈ NrmGrp ∧ B - ˙ A ∈ X → N ⁡ inv g ⁡ G ⁡ B - ˙ A = N ⁡ B - ˙ A
16 4 14 15 syl2anc ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → N ⁡ inv g ⁡ G ⁡ B - ˙ A = N ⁡ B - ˙ A
17 12 16 eqtr3d ⊢ G ∈ NrmGrp ∧ A ∈ X ∧ B ∈ X → N ⁡ A - ˙ B = N ⁡ B - ˙ A