Theorem istgp2 23340

Description: A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
tgpsubcn.2	⊢ 𝐽 = (TopOpen‘𝐺)
tgpsubcn.3	⊢ − = (-g‘𝐺)

Assertion

Ref	Expression
istgp2	⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Proof of Theorem istgp2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgpgrp 23327	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2		tgptps 23329	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3		tgpsubcn.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
4		tgpsubcn.3	. . . 4 ⊢ − = (-g‘𝐺)
5	3, 4	tgpsubcn 23339	. . 3 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6	1, 2, 5	3jca 1127	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7		simp1 1135	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8		grpmnd 18672	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
9	8	3ad2ant1 1132	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10		simp2 1136	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
12		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
13		eqid 2736	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
14	7	3ad2ant1 1132	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15		simp2 1136	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16		simp3 1137	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑦 ∈ (Base‘𝐺))
17	11, 12, 4, 13, 14, 15, 16	grpsubinv 18736	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥 − ((invg‘𝐺)‘𝑦)) = (𝑥(+g‘𝐺)𝑦))
18	17	mpoeq3dva 7406	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 − ((invg‘𝐺)‘𝑦))) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦)))
19		eqid 2736	. . . . . . 7 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
20	11, 12, 19	plusffval 18421	. . . . . 6 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
21	18, 20	eqtr4di 2794	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 − ((invg‘𝐺)‘𝑦))) = (+𝑓‘𝐺))
22	11, 3	istps 22181	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
23	10, 22	sylib 217	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
24	23, 23	cnmpt1st 22917	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
25	23, 23	cnmpt2nd 22918	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
26	11, 13	grpinvf 18714	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27	26	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
28	27	feqmptd 6887	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
29		eqid 2736	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	11, 4, 13, 29	grpinvval2 18746	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ((0g‘𝐺) − 𝑥))
31	7, 30	sylan 580	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ((0g‘𝐺) − 𝑥))
32	31	mpteq2dva 5189	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)))
33	28, 32	eqtrd 2776	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)))
34	11, 29	grpidcl 18695	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
35	34	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g‘𝐺) ∈ (Base‘𝐺))
36	23, 23, 35	cnmptc 22911	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g‘𝐺)) ∈ (𝐽 Cn 𝐽))
37	23	cnmptid 22910	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38		simp3 1137	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
39	23, 36, 37, 38	cnmpt12f 22915	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)) ∈ (𝐽 Cn 𝐽))
40	33, 39	eqeltrd 2837	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
41	23, 23, 25, 40	cnmpt21f 22921	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
42	23, 23, 24, 41, 38	cnmpt22f 22924	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥 − ((invg‘𝐺)‘𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
43	21, 42	eqeltrrd 2838	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
44	19, 3	istmd 23323	. . . 4 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
45	9, 10, 43, 44	syl3anbrc 1342	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
46	3, 13	istgp 23326	. . 3 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ (𝐽 Cn 𝐽)))
47	7, 45, 40, 46	syl3anbrc 1342	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
48	6, 47	impbii 208	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 205   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ↦ cmpt 5172  ⟶wf 6469  ‘cfv 6473  (class class class)co 7329   ∈ cmpo 7331  Basecbs 17001  +_gcplusg 17051  TopOpenctopn 17221  0_gc0g 17239  +_𝑓cplusf 18412  Mndcmnd 18474  Grpcgrp 18665  inv_gcminusg 18666  -_gcsg 18667  TopOnctopon 22157  TopSpctps 22179   Cn ccn 22473   ×_t ctx 22809  TopMndctmd 23319  TopGrpctgp 23320

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-1st 7891  df-2nd 7892  df-map 8680  df-0g 17241  df-topgen 17243  df-plusf 18414  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-grp 18668  df-minusg 18669  df-sbg 18670  df-top 22141  df-topon 22158  df-topsp 22180  df-bases 22194  df-cn 22476  df-cnp 22477  df-tx 22811  df-tmd 23321  df-tgp 23322

This theorem is referenced by:  distgp  23348  indistgp  23349  qustgplem  23370  ngptgp  23890  cnfldtgp  24130