Theorem ngptgp 24844

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 24807	. . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
2	1	adantr 486	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ Grp)
3		ngpms 24808	. . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
4	3	adantr 486	. . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ MetSp)
5		mstps 24663	. . 3 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
6	4, 5	syl 18	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopSp)
7		eqid 2765	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
8		eqid 2765	. . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺)
9	7, 8	grpsubf 19129	. . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
10	2, 9	syl 18	. . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g‘𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
11		rphalfcl 13061	. . . . . . 7 ⊢ (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
12		simplll 787	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel))
13	12, 4	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ MetSp)
14		simpllr 788	. . . . . . . . . . . . 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)))
15	14	simpld 500	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑥 ∈ (Base‘𝐺))
16		simprl 783	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
17		eqid 2765	. . . . . . . . . . . . 13 ⊢ (dist‘𝐺) = (dist‘𝐺)
18	7, 17	mscl 24669	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
19	13, 15, 16, 18	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
20	14	simprd 501	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
21		simprr 785	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑣 ∈ (Base‘𝐺))
22	7, 17	mscl 24669	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ MetSp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ)
23	13, 20, 21, 22	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ)
24		rpre 13041	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
25	24	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑧 ∈ ℝ)
26		lt2halves 12494	. . . . . . . . . . 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))
27	19, 23, 25, 26	syl3anc 1398	. . . . . . . . . 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))
28	12, 2	syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
29	7, 8	grpsubcl 19130	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
30	28, 15, 20, 29	syl3anc 1398	. . . . . . . . . . . . 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))
31	7, 8	grpsubcl 19130	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺))
32	28, 16, 21, 31	syl3anc 1398	. . . . . . . . . . . . 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))
33	7, 8	grpsubcl 19130	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
34	28, 16, 20, 33	syl3anc 1398	. . . . . . . . . . . . 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))
35	7, 17	mstri 24677	. . . . . . . . . . . . 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))))
36	13, 30, 32, 34, 35	syl13anc 1399	. . . . . . . . . . . 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))))
37	12	simpld 500	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ NrmGrp)
38	7, 8, 17	ngpsubcan 24822	. . . . . . . . . . . . . 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))
39	37, 15, 16, 20, 38	syl13anc 1399	. . . . . . . . . . . . 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))
40		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐺) = (+g‘𝐺)
41		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (invg‘𝐺) = (invg‘𝐺)
42	7, 40, 41, 8	grpsubval 19096	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦)))
43	16, 20, 42	syl2anc 596	. . . . . . . . . . . . . . 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)))
44	7, 40, 41, 8	grpsubval 19096	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣)))
45	44	adantl 487	. . . . . . . . . . . . . . 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)))
46	43, 45	oveq12d 7437	. . . . . . . . . . . . . 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))))
47	7, 41	grpinvcl 19098	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺))
48	28, 20, 47	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺))
49	7, 41	grpinvcl 19098	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺))
50	28, 21, 49	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺))
51	7, 40, 17	ngplcan 24819	. . . . . . . . . . . . . . 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)))
52	12, 48, 50, 16, 51	syl13anc 1399	. . . . . . . . . . . . . 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)))
53	7, 41, 17	ngpinvds 24821	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((invg‘𝐺)‘𝑦)(dist‘𝐺)((invg‘𝐺)‘𝑣)) = (𝑦(dist‘𝐺)𝑣))
54	12, 20, 21, 53	syl12anc 850	. . . . . . . . . . . . . 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))
55	46, 52, 54	3eqtrd 2804	. . . . . . . . . . . . 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))
56	39, 55	oveq12d 7437	. . . . . . . . . . . 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)))
57	36, 56	breqtrd 5139	. . . . . . . . . . 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)))
58	7, 17	mscl 24669	. . . . . . . . . . . . 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)
59	13, 30, 32, 58	syl3anc 1398	. . . . . . . . . . . 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)
60	19, 23	readdcld 11253	. . . . . . . . . . . 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)
61		lelttr 11315	. . . . . . . . . . . 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))
62	59, 60, 25, 61	syl3anc 1398	. . . . . . . . . . 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))
63	57, 62	mpand 708	. . . . . . . . . 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))
64	27, 63	syld 48	. . . . . . . . 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))
65	15, 16	ovresd 7586	. . . . . . . . . . 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))
66	65	breq1d 5121	. . . . . . . . . 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)))
67	20, 21	ovresd 7586	. . . . . . . . . . 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))
68	67	breq1d 5121	. . . . . . . . . 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)))
69	66, 68	anbi12d 644	. . . . . . . . 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))))
70	30, 32	ovresd 7586	. . . . . . . . . 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)))
71	70	breq1d 5121	. . . . . . . . 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))
72	64, 69, 71	3imtr4d 297	. . . . . . . 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))
73	72	ralrimivva 3210	. . . . . . 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))
74		breq2 5115	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑧 / 2) → ((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ↔ (𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2)))
75		breq2 5115	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑧 / 2) → ((𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟 ↔ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)))
76	74, 75	anbi12d 644	. . . . . . . . . 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))))
77	76	imbi1d 344	. . . . . . . . 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)))
78	77	2ralbidv 3231	. . . . . . . 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)))
79	78	rspcev 3583	. . . . . . 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))
80	11, 73, 79	syl2an2 699	. . . . . 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))
81	80	ralrimiva 3159	. . . . 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))
82	81	ralrimivva 3210	. . . 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))
83		msxms 24662	. . . . . 6 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
84		eqid 2765	. . . . . . 7 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
85	7, 84	xmsxmet 24664	. . . . . 6 ⊢ (𝐺 ∈ ∞MetSp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
86	4, 83, 85	3syl 19	. . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
87		eqid 2765	. . . . . 6 ⊢ (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))
88	87, 87, 87	txmetcn 24756	. . . . 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))))
89	86, 86, 86, 88	syl3anc 1398	. . . 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))))
90	10, 82, 89	mpbir2and 726	. . 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))))))
91		eqid 2765	. . . . . . 7 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
92	91, 7, 84	mstopn 24660	. . . . . 6 ⊢ (𝐺 ∈ MetSp → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
93	4, 92	syl 18	. . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
94	93, 93	oveq12d 7437	. . . 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))))))
95	94, 93	oveq12d 7437	. . 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))))))
96	90, 95	eleqtrrd 2868	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
97	91, 8	istgp2 24299	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
98	2, 6, 96, 97	syl3anbrc 1362	1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   class class class wbr 5111   × cxp 5661   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  ℝcr 11114   + caddc 11118   < clt 11258   ≤ cle 11259   / cdiv 11886  2c2 12310  ℝ⁺crp 13032  Basecbs 17291  +_gcplusg 17332  distcds 17341  TopOpenctopn 17496  Grpcgrp 19044  inv_gcminusg 19045  -_gcsg 19046  Abelcabl 19895  ∞Metcxmet 21557  MetOpencmopn 21562  TopSpctps 23139   Cn ccn 23431   ×_t ctx 23768  TopGrpctgp 24279  ∞MetSpcxms 24525  MetSpcms 24526  NrmGrpcngp 24785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-fi 9378  df-sup 9409  df-inf 9410  df-oi 9479  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-q 12989  df-rp 13033  df-xneg 13153  df-xadd 13154  df-xmul 13155  df-icc 13395  df-fz 13552  df-fzo 13700  df-seq 14056  df-hash 14385  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-hom 17356  df-cco 17357  df-rest 17497  df-topn 17498  df-0g 17516  df-gsum 17517  df-topgen 17518  df-pt 17519  df-prds 17522  df-xrs 17578  df-qtop 17583  df-imas 17584  df-xps 17586  df-mre 17660  df-mrc 17661  df-acs 17663  df-plusf 18719  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-submnd 18879  df-grp 19047  df-minusg 19048  df-sbg 19049  df-mulg 19178  df-cntz 19431  df-cmn 19896  df-abl 19897  df-psmet 21564  df-xmet 21565  df-met 21566  df-bl 21567  df-mopn 21568  df-top 23101  df-topon 23118  df-topsp 23140  df-bases 23153  df-cn 23434  df-cnp 23435  df-tx 23770  df-hmeo 23963  df-tmd 24280  df-tgp 24281  df-xms 24528  df-ms 24529  df-tms 24530  df-nm 24790  df-ngp 24791

This theorem is used by:  nrgtgp  24880  nlmtlm  24902