Theorem ngptgp 22440

Description: A normed abelian group is a topological group (with the topology induced by the metric induced by the norm). (Contributed by Mario Carneiro, 4-Oct-2015.)

Assertion

Ref	Expression
ngptgp	⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp)

Proof of Theorem ngptgp

Dummy variables 𝑢 𝑟 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ngpgrp 22403	. . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
2	1	adantr 481	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ Grp)
3		ngpms 22404	. . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
4	3	adantr 481	. . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ MetSp)
5		mstps 22260	. . 3 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
6	4, 5	syl 17	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopSp)
7		eqid 2622	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
8		eqid 2622	. . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺)
9	7, 8	grpsubf 17494	. . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
10	2, 9	syl 17	. . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
11		rphalfcl 11858	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
12	11	adantl 482	. . . . . . 7 ⊢ ((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) → (𝑧 / 2) ∈ ℝ+)
13		simplll 798	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel))
14	13, 4	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ MetSp)
15		simpllr 799	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)))
16	15	simpld 475	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑥 ∈ (Base‘𝐺))
17		simprl 794	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
18		eqid 2622	. . . . . . . . . . . . 13 ⊢ (dist‘𝐺) = (dist‘𝐺)
19	7, 18	mscl 22266	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
20	14, 16, 17, 19	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
21	15	simprd 479	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
22		simprr 796	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑣 ∈ (Base‘𝐺))
23	7, 18	mscl 22266	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ MetSp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ)
24	14, 21, 22, 23	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ)
25		rpre 11839	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
26	25	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑧 ∈ ℝ)
27		lt2halves 11267	. . . . . . . . . . 11 ⊢ (((𝑥(dist‘𝐺)𝑢) ∈ ℝ ∧ (𝑦(dist‘𝐺)𝑣) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((𝑥(dist‘𝐺)𝑢) < (𝑧 / 2) ∧ (𝑦(dist‘𝐺)𝑣) < (𝑧 / 2)) → ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧))
28	20, 24, 26, 27	syl3anc 1326	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(dist‘𝐺)𝑢) < (𝑧 / 2) ∧ (𝑦(dist‘𝐺)𝑣) < (𝑧 / 2)) → ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧))
29	13, 2	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
30	7, 8	grpsubcl 17495	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
31	29, 16, 21, 30	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
32	7, 8	grpsubcl 17495	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺))
33	29, 17, 22, 32	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺))
34	7, 8	grpsubcl 17495	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
35	29, 17, 21, 34	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
36	7, 18	mstri 22274	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ MetSp ∧ ((𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺) ∧ (𝑢(-g‘𝐺)𝑦) ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ≤ (((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑦)) + ((𝑢(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣))))
37	14, 31, 33, 35, 36	syl13anc 1328	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ≤ (((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑦)) + ((𝑢(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣))))
38	13	simpld 475	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ NrmGrp)
39	7, 8, 18	ngpsubcan 22418	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑦)) = (𝑥(dist‘𝐺)𝑢))
40	38, 16, 17, 21, 39	syl13anc 1328	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑦)) = (𝑥(dist‘𝐺)𝑢))
41		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐺) = (+g‘𝐺)
42		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (invg‘𝐺) = (invg‘𝐺)
43	7, 41, 42, 8	grpsubval 17465	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦)))
44	17, 21, 43	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g‘𝐺)𝑦) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦)))
45	7, 41, 42, 8	grpsubval 17465	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣)))
46	45	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g‘𝐺)𝑣) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣)))
47	44, 46	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑢(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) = ((𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦))(dist‘𝐺)(𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣))))
48	7, 42	grpinvcl 17467	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺))
49	29, 21, 48	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺))
50	7, 42	grpinvcl 17467	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺))
51	29, 22, 50	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺))
52	7, 41, 18	ngplcan 22415	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺) ∧ ((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦))(dist‘𝐺)(𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣))) = (((invg‘𝐺)‘𝑦)(dist‘𝐺)((invg‘𝐺)‘𝑣)))
53	13, 49, 51, 17, 52	syl13anc 1328	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦))(dist‘𝐺)(𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣))) = (((invg‘𝐺)‘𝑦)(dist‘𝐺)((invg‘𝐺)‘𝑣)))
54	7, 42, 18	ngpinvds 22417	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((invg‘𝐺)‘𝑦)(dist‘𝐺)((invg‘𝐺)‘𝑣)) = (𝑦(dist‘𝐺)𝑣))
55	13, 21, 22, 54	syl12anc 1324	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((invg‘𝐺)‘𝑦)(dist‘𝐺)((invg‘𝐺)‘𝑣)) = (𝑦(dist‘𝐺)𝑣))
56	47, 53, 55	3eqtrd 2660	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑢(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) = (𝑦(dist‘𝐺)𝑣))
57	40, 56	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑦)) + ((𝑢(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣))) = ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)))
58	37, 57	breqtrd 4679	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ≤ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)))
59	7, 18	mscl 22266	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ MetSp ∧ (𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺)) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ∈ ℝ)
60	14, 31, 33, 59	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ∈ ℝ)
61	20, 24	readdcld 10069	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∈ ℝ)
62		lelttr 10128	. . . . . . . . . . . 12 ⊢ ((((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ∈ ℝ ∧ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ≤ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∧ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) < 𝑧))
63	60, 61, 26, 62	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) ≤ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∧ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) < 𝑧))
64	58, 63	mpand 711	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧 → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) < 𝑧))
65	28, 64	syld 47	. . . . . . . . 9 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(dist‘𝐺)𝑢) < (𝑧 / 2) ∧ (𝑦(dist‘𝐺)𝑣) < (𝑧 / 2)) → ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) < 𝑧))
66	16, 17	ovresd 6801	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) = (𝑥(dist‘𝐺)𝑢))
67	66	breq1d 4663	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ↔ (𝑥(dist‘𝐺)𝑢) < (𝑧 / 2)))
68	21, 22	ovresd 6801	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) = (𝑦(dist‘𝐺)𝑣))
69	68	breq1d 4663	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2) ↔ (𝑦(dist‘𝐺)𝑣) < (𝑧 / 2)))
70	67, 69	anbi12d 747	. . . . . . . . 9 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) ↔ ((𝑥(dist‘𝐺)𝑢) < (𝑧 / 2) ∧ (𝑦(dist‘𝐺)𝑣) < (𝑧 / 2))))
71	31, 33	ovresd 6801	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) = ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)))
72	71	breq1d 4663	. . . . . . . . 9 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧 ↔ ((𝑥(-g‘𝐺)𝑦)(dist‘𝐺)(𝑢(-g‘𝐺)𝑣)) < 𝑧))
73	65, 70, 72	3imtr4d 283	. . . . . . . 8 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))
74	73	ralrimivva 2971	. . . . . . 7 ⊢ ((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) → ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))
75		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑧 / 2) → ((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ↔ (𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2)))
76		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑧 / 2) → ((𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟 ↔ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)))
77	75, 76	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑟 = (𝑧 / 2) → (((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) ↔ ((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2))))
78	77	imbi1d 331	. . . . . . . . 9 ⊢ (𝑟 = (𝑧 / 2) → ((((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧) ↔ (((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧)))
79	78	2ralbidv 2989	. . . . . . . 8 ⊢ (𝑟 = (𝑧 / 2) → (∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧) ↔ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧)))
80	79	rspcev 3309	. . . . . . 7 ⊢ (((𝑧 / 2) ∈ ℝ+ ∧ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧)) → ∃𝑟 ∈ ℝ+ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))
81	12, 74, 80	syl2anc 693	. . . . . 6 ⊢ ((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))
82	81	ralrimiva 2966	. . . . 5 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ∀𝑧 ∈ ℝ+ ∃𝑟 ∈ ℝ+ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))
83	82	ralrimivva 2971	. . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ ℝ+ ∃𝑟 ∈ ℝ+ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))
84		msxms 22259	. . . . . 6 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
85		eqid 2622	. . . . . . 7 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
86	7, 85	xmsxmet 22261	. . . . . 6 ⊢ (𝐺 ∈ ∞MetSp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
87	4, 84, 86	3syl 18	. . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
88		eqid 2622	. . . . . 6 ⊢ (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))
89	88, 88, 88	txmetcn 22353	. . . . 5 ⊢ ((((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)) ∧ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)) ∧ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺))) → ((-g‘𝐺) ∈ (((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) Cn (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) ↔ ((-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ ℝ+ ∃𝑟 ∈ ℝ+ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))))
90	87, 87, 87, 89	syl3anc 1326	. . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((-g‘𝐺) ∈ (((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) Cn (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) ↔ ((-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ ℝ+ ∃𝑟 ∈ ℝ+ ∀𝑢 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐺)(((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g‘𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g‘𝐺)𝑣)) < 𝑧))))
91	10, 83, 90	mpbir2and 957	. . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g‘𝐺) ∈ (((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) Cn (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))))
92		eqid 2622	. . . . . . 7 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
93	92, 7, 85	mstopn 22257	. . . . . 6 ⊢ (𝐺 ∈ MetSp → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
94	4, 93	syl 17	. . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
95	94, 94	oveq12d 6668	. . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) = ((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))))
96	95, 94	oveq12d 6668	. . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)) = (((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) Cn (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))))
97	91, 96	eleqtrrd 2704	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
98	92, 8	istgp2 21895	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
99	2, 6, 97, 98	syl3anbrc 1246	1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653   × cxp 5112   ↾ cres 5116  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935   + caddc 9939   < clt 10074   ≤ cle 10075   / cdiv 10684  2c2 11070  ℝ⁺crp 11832  Basecbs 15857  +_gcplusg 15941  distcds 15950  TopOpenctopn 16082  Grpcgrp 17422  inv_gcminusg 17423  -_gcsg 17424  Abelcabl 18194  ∞Metcxmt 19731  MetOpencmopn 19736  TopSpctps 20736   Cn ccn 21028   ×_t ctx 21363  TopGrpctgp 21875  ∞MetSpcxme 22122  MetSpcmt 22123  NrmGrpcngp 22382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-abl 18196  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-tmd 21876  df-tgp 21877  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388

This theorem is referenced by:  nrgtgp  22476  nlmtlm  22498