tngngp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tngngp		Structured version Visualization version GIF version

Theorem tngngp 24633

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 2737	. . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇)
4	1, 2, 3	tngngp2 24631	. . . 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 769	. . . . . . 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 6850	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
9		reex 11124	. . . . . . . . . . 11 ⊢ ℝ ∈ V
10		fex2 7882	. . . . . . . . . . 11 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
11	8, 9, 10	mp3an23 1456	. . . . . . . . . 10 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
12	11	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ V)
13	1, 2	tngbas 24620	. . . . . . . . 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 2839	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇))
16		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
17		eqid 2737	. . . . . . . 8 ⊢ (norm‘𝑇) = (norm‘𝑇)
18		eqid 2737	. . . . . . . 8 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
19	16, 17, 18	nmeq0 24597	. . . . . . 7 ⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑇)))
20	6, 15, 19	syl2anc 585	. . . . . 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 767	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁:𝑋⟶ℝ)
23	1, 2, 9	tngnm 24630	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
24	21, 22, 23	syl2anc 585	. . . . . . . 8 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 = (norm‘𝑇))
25	24	fveq1d 6838	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
26	25	eqeq1d 2739	. . . . . 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 24622	. . . . . . . 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 2748	. . . . . 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 776	. . . . . . . 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 2823	. . . . . . . . 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 2737	. . . . . . . . 9 ⊢ (-_g‘𝑇) = (-_g‘𝑇)
37	16, 17, 36	nmmtri 24601	. . . . . . . 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 1374	. . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
42	1, 41	tngplusg 24621	. . . . . . . . . . . . 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 19016	. . . . . . . . . . 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 7379	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝑦) = (𝑥(-_g‘𝑇)𝑦))
47	24, 46	fveq12d 6843	. . . . . . . 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 6838	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
50	25, 49	oveq12d 7380	. . . . . . . 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 5118	. . . . . 6 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
53	52	ralrimiva 3130	. . . . 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 3130	. . 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 771	. . 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 3075	. . . . 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 730	. . . 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 6836	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
63	62	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
64		eqeq1 2741	. . . . . 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 3564	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))
67	61, 66	sylan 581	. . 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 3075	. . . . 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 730	. . . 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 7385	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 − 𝑦)) = (𝑁‘(𝑎 − 𝑦)))
72	62	oveq1d 7377	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
73	71, 72	breq12d 5099	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
74		oveq2 7370	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑎 − 𝑦) = (𝑎 − 𝑏))
75	74	fveq2d 6840	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑁‘(𝑎 − 𝑦)) = (𝑁‘(𝑎 − 𝑏)))
76		fveq2 6836	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑁‘𝑦) = (𝑁‘𝑏))
77	76	oveq2d 7378	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑏)))
78	75, 77	breq12d 5099	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
79	73, 78	rspc2va 3577	. . . . 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 581	. . 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 24632	. 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 801	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 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 class class class wbr 5086 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 ℝcr 11032 0cc0 11033 + caddc 11036 ≤ cle 11175 Basecbs 17174 +_gcplusg 17215 distcds 17224 0_gc0g 17397 Grpcgrp 18904 -_gcsg 18906 Metcmet 21334 normcnm 24555 NrmGrpcngp 24556 toNrmGrp ctng 24557

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-sup 9350 df-inf 9351 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-plusg 17228 df-tset 17234 df-ds 17237 df-rest 17380 df-topn 17381 df-0g 17399 df-topgen 17401 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-sbg 18909 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-xms 24299 df-ms 24300 df-nm 24561 df-ngp 24562 df-tng 24563

This theorem is referenced by: (None)

W3C validator