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Theorem ngptgp 23803
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.
StepHypRef Expression
1 ngpgrp 23766 . . 3 (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
21adantr 481 . 2 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ Grp)
3 ngpms 23767 . . . 4 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
43adantr 481 . . 3 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ MetSp)
5 mstps 23619 . . 3 (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
64, 5syl 17 . 2 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopSp)
7 eqid 2740 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
8 eqid 2740 . . . . . 6 (-g𝐺) = (-g𝐺)
97, 8grpsubf 18665 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
102, 9syl 17 . . . 4 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g𝐺):((Base‘𝐺) × (Base‘𝐺))⟶(Base‘𝐺))
11 rphalfcl 12768 . . . . . . 7 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
12 simplll 772 . . . . . . . . . . . . 13 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel))
1312, 4syl 17 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ MetSp)
14 simpllr 773 . . . . . . . . . . . . 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)))
1514simpld 495 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑥 ∈ (Base‘𝐺))
16 simprl 768 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑢 ∈ (Base‘𝐺))
17 eqid 2740 . . . . . . . . . . . . 13 (dist‘𝐺) = (dist‘𝐺)
187, 17mscl 23625 . . . . . . . . . . . 12 ((𝐺 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺)) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
1913, 15, 16, 18syl3anc 1370 . . . . . . . . . . 11 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥(dist‘𝐺)𝑢) ∈ ℝ)
2014simprd 496 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
21 simprr 770 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑣 ∈ (Base‘𝐺))
227, 17mscl 23625 . . . . . . . . . . . 12 ((𝐺 ∈ MetSp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑦(dist‘𝐺)𝑣) ∈ ℝ)
2313, 20, 21, 22syl3anc 1370 . . . . . . . . . . 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 12749 . . . . . . . . . . . 12 (𝑧 ∈ ℝ+𝑧 ∈ ℝ)
2524ad2antlr 724 . . . . . . . . . . 11 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝑧 ∈ ℝ)
26 lt2halves 12219 . . . . . . . . . . 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))
2719, 23, 25, 26syl3anc 1370 . . . . . . . . . 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))
2812, 2syl 17 . . . . . . . . . . . . . 14 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
297, 8grpsubcl 18666 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(-g𝐺)𝑦) ∈ (Base‘𝐺))
3028, 15, 20, 29syl3anc 1370 . . . . . . . . . . . . 13 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑥(-g𝐺)𝑦) ∈ (Base‘𝐺))
317, 8grpsubcl 18666 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g𝐺)𝑣) ∈ (Base‘𝐺))
3228, 16, 21, 31syl3anc 1370 . . . . . . . . . . . . 13 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g𝐺)𝑣) ∈ (Base‘𝐺))
337, 8grpsubcl 18666 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g𝐺)𝑦) ∈ (Base‘𝐺))
3428, 16, 20, 33syl3anc 1370 . . . . . . . . . . . . 13 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g𝐺)𝑦) ∈ (Base‘𝐺))
357, 17mstri 23633 . . . . . . . . . . . . 13 ((𝐺 ∈ MetSp ∧ ((𝑥(-g𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑢(-g𝐺)𝑣) ∈ (Base‘𝐺) ∧ (𝑢(-g𝐺)𝑦) ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ≤ (((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑦)) + ((𝑢(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣))))
3613, 30, 32, 34, 35syl13anc 1371 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ≤ (((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑦)) + ((𝑢(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣))))
3712simpld 495 . . . . . . . . . . . . . 14 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → 𝐺 ∈ NrmGrp)
387, 8, 17ngpsubcan 23781 . . . . . . . . . . . . . 14 ((𝐺 ∈ NrmGrp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑦)) = (𝑥(dist‘𝐺)𝑢))
3937, 15, 16, 20, 38syl13anc 1371 . . . . . . . . . . . . 13 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑦)) = (𝑥(dist‘𝐺)𝑢))
40 eqid 2740 . . . . . . . . . . . . . . . . 17 (+g𝐺) = (+g𝐺)
41 eqid 2740 . . . . . . . . . . . . . . . . 17 (invg𝐺) = (invg𝐺)
427, 40, 41, 8grpsubval 18636 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑢(-g𝐺)𝑦) = (𝑢(+g𝐺)((invg𝐺)‘𝑦)))
4316, 20, 42syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g𝐺)𝑦) = (𝑢(+g𝐺)((invg𝐺)‘𝑦)))
447, 40, 41, 8grpsubval 18636 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺)) → (𝑢(-g𝐺)𝑣) = (𝑢(+g𝐺)((invg𝐺)‘𝑣)))
4544adantl 482 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (𝑢(-g𝐺)𝑣) = (𝑢(+g𝐺)((invg𝐺)‘𝑣)))
4643, 45oveq12d 7290 . . . . . . . . . . . . . 14 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑢(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) = ((𝑢(+g𝐺)((invg𝐺)‘𝑦))(dist‘𝐺)(𝑢(+g𝐺)((invg𝐺)‘𝑣))))
477, 41grpinvcl 18638 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑦) ∈ (Base‘𝐺))
4828, 20, 47syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((invg𝐺)‘𝑦) ∈ (Base‘𝐺))
497, 41grpinvcl 18638 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ 𝑣 ∈ (Base‘𝐺)) → ((invg𝐺)‘𝑣) ∈ (Base‘𝐺))
5028, 21, 49syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((invg𝐺)‘𝑣) ∈ (Base‘𝐺))
517, 40, 17ngplcan 23778 . . . . . . . . . . . . . . 15 (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (((invg𝐺)‘𝑦) ∈ (Base‘𝐺) ∧ ((invg𝐺)‘𝑣) ∈ (Base‘𝐺) ∧ 𝑢 ∈ (Base‘𝐺))) → ((𝑢(+g𝐺)((invg𝐺)‘𝑦))(dist‘𝐺)(𝑢(+g𝐺)((invg𝐺)‘𝑣))) = (((invg𝐺)‘𝑦)(dist‘𝐺)((invg𝐺)‘𝑣)))
5212, 48, 50, 16, 51syl13anc 1371 . . . . . . . . . . . . . 14 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑢(+g𝐺)((invg𝐺)‘𝑦))(dist‘𝐺)(𝑢(+g𝐺)((invg𝐺)‘𝑣))) = (((invg𝐺)‘𝑦)(dist‘𝐺)((invg𝐺)‘𝑣)))
537, 41, 17ngpinvds 23780 . . . . . . . . . . . . . . 15 (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((invg𝐺)‘𝑦)(dist‘𝐺)((invg𝐺)‘𝑣)) = (𝑦(dist‘𝐺)𝑣))
5412, 20, 21, 53syl12anc 834 . . . . . . . . . . . . . 14 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((invg𝐺)‘𝑦)(dist‘𝐺)((invg𝐺)‘𝑣)) = (𝑦(dist‘𝐺)𝑣))
5546, 52, 543eqtrd 2784 . . . . . . . . . . . . 13 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑢(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) = (𝑦(dist‘𝐺)𝑣))
5639, 55oveq12d 7290 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑦)) + ((𝑢(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣))) = ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)))
5736, 56breqtrd 5105 . . . . . . . . . . 11 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ≤ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)))
587, 17mscl 23625 . . . . . . . . . . . . 13 ((𝐺 ∈ MetSp ∧ (𝑥(-g𝐺)𝑦) ∈ (Base‘𝐺) ∧ (𝑢(-g𝐺)𝑣) ∈ (Base‘𝐺)) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ∈ ℝ)
5913, 30, 32, 58syl3anc 1370 . . . . . . . . . . . 12 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ∈ ℝ)
6019, 23readdcld 11015 . . . . . . . . . . . 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 11076 . . . . . . . . . . . 12 ((((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ∈ ℝ ∧ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ≤ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∧ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) < 𝑧))
6259, 60, 25, 61syl3anc 1370 . . . . . . . . . . 11 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) ≤ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) ∧ ((𝑥(dist‘𝐺)𝑢) + (𝑦(dist‘𝐺)𝑣)) < 𝑧) → ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) < 𝑧))
6357, 62mpand 692 . . . . . . . . . 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(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) < 𝑧))
6427, 63syld 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(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) < 𝑧))
6515, 16ovresd 7434 . . . . . . . . . . 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))
6665breq1d 5089 . . . . . . . . . 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)))
6720, 21ovresd 7434 . . . . . . . . . . 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))
6867breq1d 5089 . . . . . . . . . 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)))
6966, 68anbi12d 631 . . . . . . . . 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))))
7030, 32ovresd 7434 . . . . . . . . . 10 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → ((𝑥(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) = ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)))
7170breq1d 5089 . . . . . . . . 9 (((((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ ℝ+) ∧ (𝑢 ∈ (Base‘𝐺) ∧ 𝑣 ∈ (Base‘𝐺))) → (((𝑥(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧 ↔ ((𝑥(-g𝐺)𝑦)(dist‘𝐺)(𝑢(-g𝐺)𝑣)) < 𝑧))
7264, 69, 713imtr4d 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))
7372ralrimivva 3117 . . . . . . 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))
74 breq2 5083 . . . . . . . . . . 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 5083 . . . . . . . . . . 11 (𝑟 = (𝑧 / 2) → ((𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟 ↔ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)))
7674, 75anbi12d 631 . . . . . . . . . 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))))
7776imbi1d 342 . . . . . . . . 9 (𝑟 = (𝑧 / 2) → ((((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < 𝑟 ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < 𝑟) → ((𝑥(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧) ↔ (((𝑥((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑢) < (𝑧 / 2) ∧ (𝑦((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))𝑣) < (𝑧 / 2)) → ((𝑥(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧)))
78772ralbidv 3125 . . . . . . . 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧)))
7978rspcev 3561 . . . . . . 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))
8011, 73, 79syl2an2 683 . . . . . 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))
8180ralrimiva 3110 . . . . 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))
8281ralrimivva 3117 . . . 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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))
83 msxms 23618 . . . . . 6 (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
84 eqid 2740 . . . . . . 7 ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
857, 84xmsxmet 23620 . . . . . 6 (𝐺 ∈ ∞MetSp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
864, 83, 853syl 18 . . . . 5 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (∞Met‘(Base‘𝐺)))
87 eqid 2740 . . . . . 6 (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))
8887, 87, 87txmetcn 23715 . . . . 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))) → ((-g𝐺) ∈ (((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) Cn (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) ↔ ((-g𝐺):((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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))))
8986, 86, 86, 88syl3anc 1370 . . . 4 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → ((-g𝐺) ∈ (((MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) ×t (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) Cn (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) ↔ ((-g𝐺):((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(-g𝐺)𝑦)((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))(𝑢(-g𝐺)𝑣)) < 𝑧))))
9010, 82, 89mpbir2and 710 . . 3 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g𝐺) ∈ (((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 2740 . . . . . . 7 (TopOpen‘𝐺) = (TopOpen‘𝐺)
9291, 7, 84mstopn 23616 . . . . . 6 (𝐺 ∈ MetSp → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
934, 92syl 17 . . . . 5 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (TopOpen‘𝐺) = (MetOpen‘((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))))
9493, 93oveq12d 7290 . . . 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))))))
9594, 93oveq12d 7290 . . 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))))))
9690, 95eleqtrrd 2844 . 2 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → (-g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺)))
9791, 8istgp2 23253 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ (((TopOpen‘𝐺) ×t (TopOpen‘𝐺)) Cn (TopOpen‘𝐺))))
982, 6, 96, 97syl3anbrc 1342 1 ((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wcel 2110  wral 3066  wrex 3067   class class class wbr 5079   × cxp 5588  cres 5592  wf 6428  cfv 6432  (class class class)co 7272  cr 10881   + caddc 10885   < clt 11020  cle 11021   / cdiv 11643  2c2 12039  +crp 12741  Basecbs 16923  +gcplusg 16973  distcds 16982  TopOpenctopn 17143  Grpcgrp 18588  invgcminusg 18589  -gcsg 18590  Abelcabl 19398  ∞Metcxmet 20593  MetOpencmopn 20598  TopSpctps 22092   Cn ccn 22386   ×t ctx 22722  TopGrpctgp 23233  ∞MetSpcxms 23481  MetSpcms 23482  NrmGrpcngp 23744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959  ax-pre-sup 10960
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-of 7528  df-om 7708  df-1st 7825  df-2nd 7826  df-supp 7970  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-1o 8289  df-2o 8290  df-er 8490  df-map 8609  df-ixp 8678  df-en 8726  df-dom 8727  df-sdom 8728  df-fin 8729  df-fsupp 9117  df-fi 9158  df-sup 9189  df-inf 9190  df-oi 9257  df-card 9708  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-div 11644  df-nn 11985  df-2 12047  df-3 12048  df-4 12049  df-5 12050  df-6 12051  df-7 12052  df-8 12053  df-9 12054  df-n0 12245  df-z 12331  df-dec 12449  df-uz 12594  df-q 12700  df-rp 12742  df-xneg 12859  df-xadd 12860  df-xmul 12861  df-icc 13097  df-fz 13251  df-fzo 13394  df-seq 13733  df-hash 14056  df-struct 16859  df-sets 16876  df-slot 16894  df-ndx 16906  df-base 16924  df-ress 16953  df-plusg 16986  df-mulr 16987  df-sca 16989  df-vsca 16990  df-ip 16991  df-tset 16992  df-ple 16993  df-ds 16995  df-hom 16997  df-cco 16998  df-rest 17144  df-topn 17145  df-0g 17163  df-gsum 17164  df-topgen 17165  df-pt 17166  df-prds 17169  df-xrs 17224  df-qtop 17229  df-imas 17230  df-xps 17232  df-mre 17306  df-mrc 17307  df-acs 17309  df-plusf 18336  df-mgm 18337  df-sgrp 18386  df-mnd 18397  df-submnd 18442  df-grp 18591  df-minusg 18592  df-sbg 18593  df-mulg 18712  df-cntz 18934  df-cmn 19399  df-abl 19400  df-psmet 20600  df-xmet 20601  df-met 20602  df-bl 20603  df-mopn 20604  df-top 22054  df-topon 22071  df-topsp 22093  df-bases 22107  df-cn 22389  df-cnp 22390  df-tx 22724  df-hmeo 22917  df-tmd 23234  df-tgp 23235  df-xms 23484  df-ms 23485  df-tms 23486  df-nm 23749  df-ngp 23750
This theorem is referenced by:  nrgtgp  23847  nlmtlm  23869
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