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Theorem tngngp 24591

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 2735	. . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇)
4	1, 2, 3	tngngp2 24589	. . . 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 6889	. . . . . . . . . . 11 ⊢ 𝑋 ∈ V
9		reex 11218	. . . . . . . . . . 11 ⊢ ℝ ∈ V
10		fex2 7930	. . . . . . . . . . 11 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
11	8, 9, 10	mp3an23 1455	. . . . . . . . . 10 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
12	11	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ V)
13	1, 2	tngbas 24578	. . . . . . . . 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 2836	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇))
16		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑇) = (Base‘𝑇)
17		eqid 2735	. . . . . . . 8 ⊢ (norm‘𝑇) = (norm‘𝑇)
18		eqid 2735	. . . . . . . 8 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
19	16, 17, 18	nmeq0 24555	. . . . . . 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 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 24588	. . . . . . . . 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 6877	. . . . . . 7 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥))
26	25	eqeq1d 2737	. . . . . 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 24580	. . . . . . . 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 2746	. . . . . 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 2820	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (-_g‘𝑇) = (-_g‘𝑇)
37	16, 17, 36	nmmtri 24559	. . . . . . . 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 1373	. . . . . . 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 2784	. . . . . . . . . . . 12 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (Base‘𝐺) = (Base‘𝑇))
41		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
42	1, 41	tngplusg 24579	. . . . . . . . . . . . 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 19026	. . . . . . . . . . 11 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (-_g‘𝐺) = (-_g‘𝑇))
45	39, 44	eqtrid 2782	. . . . . . . . . 10 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → − = (-_g‘𝑇))
46	45	oveqd 7420	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝑦) = (𝑥(-_g‘𝑇)𝑦))
47	24, 46	fveq12d 6882	. . . . . . . 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 6877	. . . . . . . . 9 ⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦))
50	25, 49	oveq12d 7421	. . . . . . . 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 5151	. . . . . 6 ⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
53	52	ralrimiva 3132	. . . . 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 3132	. . 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 3073	. . . . 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 729	. . . 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 6875	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎))
63	62	eqeq1d 2737	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0))
64		eqeq1 2739	. . . . . 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 3600	. . . 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 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 3073	. . . . 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 729	. . . 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 7426	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 − 𝑦)) = (𝑁‘(𝑎 − 𝑦)))
72	62	oveq1d 7418	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦)))
73	71, 72	breq12d 5132	. . . . . 6 ⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦))))
74		oveq2 7411	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑎 − 𝑦) = (𝑎 − 𝑏))
75	74	fveq2d 6879	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑁‘(𝑎 − 𝑦)) = (𝑁‘(𝑎 − 𝑏)))
76		fveq2 6875	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (𝑁‘𝑦) = (𝑁‘𝑏))
77	76	oveq2d 7419	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑏)))
78	75, 77	breq12d 5132	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))))
79	73, 78	rspc2va 3613	. . . . 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 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 24590	. 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 1540 ∈ wcel 2108 ∀wral 3051 Vcvv 3459 class class class wbr 5119 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ℝcr 11126 0cc0 11127 + caddc 11130 ≤ cle 11268 Basecbs 17226 +_gcplusg 17269 distcds 17278 0_gc0g 17451 Grpcgrp 18914 -_gcsg 18916 Metcmet 21299 normcnm 24513 NrmGrpcngp 24514 toNrmGrp ctng 24515

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9452 df-inf 9453 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-plusg 17282 df-tset 17288 df-ds 17291 df-rest 17434 df-topn 17435 df-0g 17453 df-topgen 17455 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-grp 18917 df-minusg 18918 df-sbg 18919 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-xms 24257 df-ms 24258 df-nm 24519 df-ngp 24520 df-tng 24521

This theorem is referenced by: (None)

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