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Theorem tngngp 24162

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 2732	. . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇)
4	1, 2, 3	tngngp2 24160	. . . 4 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋))))
5	4	simprbda 499	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
6		simplr 767	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ NrmGrp)
7		simpr 485	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8	2	fvexi 6902	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
9		reex 11197	. . . . . . . . . . 11 ⊢ ℝ ∈ V
10		fex2 7920	. . . . . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ V)
13	1, 2	tngbas 24142	. . . . . . . . 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 2835	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇))
16		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
17		eqid 2732	. . . . . . . 8 ⊢ (norm‘𝑇) = (norm‘𝑇)
18		eqid 2732	. . . . . . . 8 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
19	16, 17, 18	nmeq0 24118	. . . . . . 7 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑇)))
20	6, 15, 19	syl2anc 584	. . . . . 6 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑇)))
21	5	adantr 481	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp)
22		simpll 765	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁:𝑋⟶ℝ)
23	1, 2, 9	tngnm 24159	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
24	21, 22, 23	syl2anc 584	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 = (norm‘𝑇))
25	24	fveq1d 6890	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
26	25	eqeq1d 2734	. . . . . 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 24146	. . . . . . . 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 2743	. . . . . 6 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 = 0 ↔ 𝑥 = (0_g‘𝑇)))
31	20, 26, 30	3bitr4d 310	. . . . 5 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
32		simpllr 774	. . . . . . . 8 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑇 ∈ NrmGrp)
33	15	adantr 481	. . . . . . . 8 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇))
34	14	eleq2d 2819	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ (Base‘𝑇)))
35	34	biimpa 477	. . . . . . . 8 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ (Base‘𝑇))
36		eqid 2732	. . . . . . . . 9 ⊢ (-_g‘𝑇) = (-_g‘𝑇)
37	16, 17, 36	nmmtri 24122	. . . . . . . 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 2786	. . . . . . . . . . . 12 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (Base‘𝐺) = (Base‘𝑇))
41		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
42	1, 41	tngplusg 24144	. . . . . . . . . . . . 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 18924	. . . . . . . . . . 11 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (-_g‘𝐺) = (-_g‘𝑇))
45	39, 44	eqtrid 2784	. . . . . . . . . 10 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → − = (-_g‘𝑇))
46	45	oveqd 7422	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝑦) = (𝑥(-_g‘𝑇)𝑦))
47	24, 46	fveq12d 6895	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) = ((norm‘𝑇)‘(𝑥(-_g‘𝑇)𝑦)))
48	47	adantr 481	. . . . . . 7 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) = ((norm‘𝑇)‘(𝑥(-_g‘𝑇)𝑦)))
49	24	fveq1d 6890	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
50	25, 49	oveq12d 7423	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦)))
51	50	adantr 481	. . . . . . 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 5179	. . . . . 6 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
53	52	ralrimiva 3146	. . . . 5 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
54	31, 53	jca 512	. . . 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 3146	. . 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 512	. 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 769	. . 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 483	. . 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 483	. . . . . 6 ⊢ ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
60	59	ralimi 3083	. . . . 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 727	. . . 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 6888	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
63	62	eqeq1d 2734	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
64		eqeq1 2736	. . . . . 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 3611	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
67	61, 66	sylan 580	. . 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 485	. . . . . 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 3083	. . . . 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 727	. . . 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 7428	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 − 𝑦)) = (𝑁‘(𝑎 − 𝑦)))
72	62	oveq1d 7420	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
73	71, 72	breq12d 5160	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
74		oveq2 7413	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑎 − 𝑦) = (𝑎 − 𝑏))
75	74	fveq2d 6892	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑁‘(𝑎 − 𝑦)) = (𝑁‘(𝑎 − 𝑏)))
76		fveq2 6888	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑁‘𝑦) = (𝑁‘𝑏))
77	76	oveq2d 7421	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑏)))
78	75, 77	breq12d 5160	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
79	73, 78	rspc2va 3622	. . . . 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 459	. . . 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 580	. . 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 24161	. 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 799	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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 class class class wbr 5147 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 + caddc 11109 ≤ cle 11245 Basecbs 17140 +_gcplusg 17193 distcds 17202 0_gc0g 17381 Grpcgrp 18815 -_gcsg 18817 Metcmet 20922 normcnm 24076 NrmGrpcngp 24077 toNrmGrp ctng 24078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-tset 17212 df-ds 17215 df-rest 17364 df-topn 17365 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-xms 23817 df-ms 23818 df-nm 24082 df-ngp 24083 df-tng 24084

This theorem is referenced by: (None)

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