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Theorem tngngp 24696

Description: Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

**Hypotheses**
Ref	Expression
tngngp.t	⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
tngngp.x	⊢ 𝑋 = (Base‘𝐺)
tngngp.m	⊢ − = (-_g‘𝐺)
tngngp.z	⊢ 0 = (0_g‘𝐺)

**Assertion**
Ref	Expression
tngngp	⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Distinct variable groups: 𝑥,𝑦, − 𝑥,𝑁,𝑦 𝑥,𝑇,𝑦 𝑥,𝑋,𝑦 𝑥, 0 ,𝑦 Allowed substitution hints: 𝐺(𝑥,𝑦)

Proof of Theorem tngngp Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tngngp.t	. . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
2		tngngp.x	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
3		eqid 2740	. . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇)
4	1, 2, 3	tngngp2 24694	. . . 4 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
5	4	simprbda 498	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6		simplr 768	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ NrmGrp)
7		simpr 484	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8	2	fvexi 6934	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
9		reex 11275	. . . . . . . . . . 11 ⊢ ℝ ∈ V
10		fex2 7974	. . . . . . . . . . 11 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
11	8, 9, 10	mp3an23 1453	. . . . . . . . . 10 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
12	11	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ V)
13	1, 2	tngbas 24676	. . . . . . . . 9 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑋 = (Base‘𝑇))
15	7, 14	eleqtrd 2846	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇))
16		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
17		eqid 2740	. . . . . . . 8 ⊢ (norm‘𝑇) = (norm‘𝑇)
18		eqid 2740	. . . . . . . 8 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
19	16, 17, 18	nmeq0 24652	. . . . . . 7 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑇)))
20	6, 15, 19	syl2anc 583	. . . . . 6 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑇)))
21	5	adantr 480	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
22		simpll 766	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁:𝑋⟶ℝ)
23	1, 2, 9	tngnm 24693	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
24	21, 22, 23	syl2anc 583	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 = (norm‘𝑇))
25	24	fveq1d 6922	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
26	25	eqeq1d 2742	. . . . . 6 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0))
27		tngngp.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝐺)
28	1, 27	tng0 24680	. . . . . . . 8 ⊢ (𝑁 ∈ V → 0 = (0_g‘𝑇))
29	12, 28	syl 17	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 0 = (0_g‘𝑇))
30	29	eqeq2d 2751	. . . . . 6 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 = 0 ↔ 𝑥 = (0_g‘𝑇)))
31	20, 26, 30	3bitr4d 311	. . . . 5 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
32		simpllr 775	. . . . . . . 8 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑇 ∈ NrmGrp)
33	15	adantr 480	. . . . . . . 8 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇))
34	14	eleq2d 2830	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ (Base‘𝑇)))
35	34	biimpa 476	. . . . . . . 8 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ (Base‘𝑇))
36		eqid 2740	. . . . . . . . 9 ⊢ (-_g‘𝑇) = (-_g‘𝑇)
37	16, 17, 36	nmmtri 24656	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-_g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
38	32, 33, 35, 37	syl3anc 1371	. . . . . . 7 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((norm‘𝑇)‘(𝑥(-_g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
39		tngngp.m	. . . . . . . . . . 11 ⊢ − = (-_g‘𝐺)
40	2, 14	eqtr3id 2794	. . . . . . . . . . . 12 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (Base‘𝐺) = (Base‘𝑇))
41		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
42	1, 41	tngplusg 24678	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ V → (+_g‘𝐺) = (+_g‘𝑇))
43	12, 42	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (+_g‘𝐺) = (+_g‘𝑇))
44	40, 43	grpsubpropd 19085	. . . . . . . . . . 11 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (-_g‘𝐺) = (-_g‘𝑇))
45	39, 44	eqtrid 2792	. . . . . . . . . 10 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → − = (-_g‘𝑇))
46	45	oveqd 7465	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝑦) = (𝑥(-_g‘𝑇)𝑦))
47	24, 46	fveq12d 6927	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) = ((norm‘𝑇)‘(𝑥(-_g‘𝑇)𝑦)))
48	47	adantr 480	. . . . . . 7 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) = ((norm‘𝑇)‘(𝑥(-_g‘𝑇)𝑦)))
49	24	fveq1d 6922	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
50	25, 49	oveq12d 7466	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
51	50	adantr 480	. . . . . . 7 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
52	38, 48, 51	3brtr4d 5198	. . . . . 6 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
53	52	ralrimiva 3152	. . . . 5 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
54	31, 53	jca 511	. . . 4 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
55	54	ralrimiva 3152	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
56	5, 55	jca 511	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
57		simprl 770	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝐺 ∈ Grp)
58		simpl 482	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝑁:𝑋⟶ℝ)
59		simpl 482	. . . . . 6 ⊢ ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
60	59	ralimi 3089	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
61	60	ad2antll 728	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → ∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
62		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
63	62	eqeq1d 2742	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
64		eqeq1 2744	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 = 0 ↔ 𝑎 = 0 ))
65	63, 64	bibi12d 345	. . . . 5 ⊢ (𝑥 = 𝑎 → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )))
66	65	rspccva 3634	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
67	61, 66	sylan 579	. . 3 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
68		simpr 484	. . . . . 6 ⊢ ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
69	68	ralimi 3089	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
70	69	ad2antll 728	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
71		fvoveq1 7471	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 − 𝑦)) = (𝑁‘(𝑎 − 𝑦)))
72	62	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
73	71, 72	breq12d 5179	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
74		oveq2 7456	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑎 − 𝑦) = (𝑎 − 𝑏))
75	74	fveq2d 6924	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑁‘(𝑎 − 𝑦)) = (𝑁‘(𝑎 − 𝑏)))
76		fveq2 6920	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑁‘𝑦) = (𝑁‘𝑏))
77	76	oveq2d 7464	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑏)))
78	75, 77	breq12d 5179	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
79	73, 78	rspc2va 3647	. . . . 5 ⊢ (((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))
80	79	ancoms 458	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))
81	70, 80	sylan 579	. . 3 ⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))
82	1, 2, 39, 27, 57, 58, 67, 81	tngngpd 24695	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝑇 ∈ NrmGrp)
83	56, 82	impbida 800	1 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 class class class wbr 5166 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 + caddc 11187 ≤ cle 11325 Basecbs 17258 +_gcplusg 17311 distcds 17320 0_gc0g 17499 Grpcgrp 18973 -_gcsg 18975 Metcmet 21373 normcnm 24610 NrmGrpcngp 24611 toNrmGrp ctng 24612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-tset 17330 df-ds 17333 df-rest 17482 df-topn 17483 df-0g 17501 df-topgen 17503 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-xms 24351 df-ms 24352 df-nm 24616 df-ngp 24617 df-tng 24618

This theorem is referenced by: (None)

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