Theorem ngpi

Description: The properties of a normed group, which is a group accompanied by a norm. (Contributed by AV, 7-Oct-2021)

	Ref	Expression
Hypotheses	ngpi.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	ngpi.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
	ngpi.m	⊢ − = ( -g ‘ 𝑊 )
	ngpi.0	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	ngpi	⊢ ( 𝑊 ∈ NrmGrp → ( 𝑊 ∈ Grp ∧ 𝑁 : 𝑉 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑉 ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ngpi.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		ngpi.n	⊢ 𝑁 = ( norm ‘ 𝑊 )
3		ngpi.m	⊢ − = ( -g ‘ 𝑊 )
4		ngpi.0	⊢ 0 = ( 0g ‘ 𝑊 )
5		ngpgrp	⊢ ( 𝑊 ∈ NrmGrp → 𝑊 ∈ Grp )
6	1 2	nmf	⊢ ( 𝑊 ∈ NrmGrp → 𝑁 : 𝑉 ⟶ ℝ )
7	1 2 4	nmeq0	⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) )
8	1 2 3	nmmtri	⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) )
9	8	3expa	⊢ ( ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) ∧ 𝑦 ∈ 𝑉 ) → ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) )
10	9	ralrimiva	⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) → ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) )
11	7 10	jca	⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) → ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) )
12	11	ralrimiva	⊢ ( 𝑊 ∈ NrmGrp → ∀ 𝑥 ∈ 𝑉 ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) )
13	5 6 12	3jca	⊢ ( 𝑊 ∈ NrmGrp → ( 𝑊 ∈ Grp ∧ 𝑁 : 𝑉 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑉 ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) )