Theorem ngptgp 24580

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 24543	. . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
2	1	adantr 480	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ Grp)
3		ngpms 24544	. . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
4	3	adantr 480	. . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ MetSp)
5		mstps 24399	. . 3 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
6	4, 5	syl 17	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopSp)
7		eqid 2736	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
8		eqid 2736	. . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺)
9	7, 8	grpsubf 18949	. . . . 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 12934	. . . . . . 7 ⊢ (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
12		simplll 774	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ MetSp)
14		simpllr 775	. . . . . . . . . . . . 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 494	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑥 ∈ (Base‘𝐺))
16		simprl 770	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
17		eqid 2736	. . . . . . . . . . . . 13 ⊢ (dist‘𝐺) = (dist‘𝐺)
18	7, 17	mscl 24405	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
19	13, 15, 16, 18	syl3anc 1373	. . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
21		simprr 772	. . . . . . . . . . . 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 24405	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ MetSp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ)
23	13, 20, 21, 22	syl3anc 1373	. . . . . . . . . . 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 12914	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ+ → 𝑧 ∈ ℝ)
25	24	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑧 ∈ ℝ)
26		lt2halves 12376	. . . . . . . . . . 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 1373	. . . . . . . . . 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 17	. . . . . . . . . . . . . 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 18950	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
30	28, 15, 20, 29	syl3anc 1373	. . . . . . . . . . . . 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 18950	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) ∈ (Base‘𝐺))
32	28, 16, 21, 31	syl3anc 1373	. . . . . . . . . . . . 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 18950	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) ∈ (Base‘𝐺))
34	28, 16, 20, 33	syl3anc 1373	. . . . . . . . . . . . 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 24413	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 494	. . . . . . . . . . . . . 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 24558	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐺) = (+g‘𝐺)
41		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (invg‘𝐺) = (invg‘𝐺)
42	7, 40, 41, 8	grpsubval 18915	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑦) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑦)))
43	16, 20, 42	syl2anc 584	. . . . . . . . . . . . . . 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 18915	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g‘𝐺)𝑣) = (𝑢(+g‘𝐺)((invg‘𝐺)‘𝑣)))
45	44	adantl 481	. . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . 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 18917	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑦) ∈ (Base‘𝐺))
48	28, 20, 47	syl2anc 584	. . . . . . . . . . . . . . 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 18917	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑣) ∈ (Base‘𝐺))
50	28, 21, 49	syl2anc 584	. . . . . . . . . . . . . . 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 24555	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 24557	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 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 7376	. . . . . . . . . . . 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 5124	. . . . . . . . . . 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 24405	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 11161	. . . . . . . . . . . 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 11223	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 695	. . . . . . . . . 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 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))
65	15, 16	ovresd 7525	. . . . . . . . . . 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 5108	. . . . . . . . . 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 7525	. . . . . . . . . . 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 5108	. . . . . . . . . 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 632	. . . . . . . . 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 7525	. . . . . . . . . 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 5108	. . . . . . . . 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 294	. . . . . . . 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 3179	. . . . . . 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 5102	. . . . . . . . . . 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 5102	. . . . . . . . . . 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 632	. . . . . . . . . 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 341	. . . . . . . . 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 3200	. . . . . . . 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 3576	. . . . . . 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 686	. . . . . 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 3128	. . . . 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 3179	. . . 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 24398	. . . . . 6 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
84		eqid 2736	. . . . . . 7 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
85	7, 84	xmsxmet 24400	. . . . . 6 ⊢ (𝐺 ∈ ∞MetSp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
86	4, 83, 85	3syl 18	. . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
87		eqid 2736	. . . . . 6 ⊢ (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))
88	87, 87, 87	txmetcn 24492	. . . . 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 1373	. . . 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 713	. . 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 2736	. . . . . . 7 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
92	91, 7, 84	mstopn 24396	. . . . . 6 ⊢ (𝐺 ∈ MetSp → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
93	4, 92	syl 17	. . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
94	93, 93	oveq12d 7376	. . . 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 7376	. . 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 2839	. 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
97	91, 8	istgp2 24035	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
98	2, 6, 96, 97	syl3anbrc 1344	1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   class class class wbr 5098   × cxp 5622   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ℝcr 11025   + caddc 11029   < clt 11166   ≤ cle 11167   / cdiv 11794  2c2 12200  ℝ⁺crp 12905  Basecbs 17136  +_gcplusg 17177  distcds 17186  TopOpenctopn 17341  Grpcgrp 18863  inv_gcminusg 18864  -_gcsg 18865  Abelcabl 19710  ∞Metcxmet 21294  MetOpencmopn 21299  TopSpctps 22876   Cn ccn 23168   ×_t ctx 23504  TopGrpctgp 24015  ∞MetSpcxms 24261  MetSpcms 24262  NrmGrpcngp 24521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-icc 13268  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-rest 17342  df-topn 17343  df-0g 17361  df-gsum 17362  df-topgen 17363  df-pt 17364  df-prds 17367  df-xrs 17423  df-qtop 17428  df-imas 17429  df-xps 17431  df-mre 17505  df-mrc 17506  df-acs 17508  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-abl 19712  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-cn 23171  df-cnp 23172  df-tx 23506  df-hmeo 23699  df-tmd 24016  df-tgp 24017  df-xms 24264  df-ms 24265  df-tms 24266  df-nm 24526  df-ngp 24527

This theorem is referenced by:  nrgtgp  24616  nlmtlm  24638