Theorem istgp2 22303

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 22290	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2		tgptps 22292	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3		tgpsubcn.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
4		tgpsubcn.3	. . . 4 ⊢ − = (-g‘𝐺)
5	3, 4	tgpsubcn 22302	. . 3 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6	1, 2, 5	3jca 1119	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7		simp1 1127	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8		grpmnd 17816	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
9	8	3ad2ant1 1124	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10		simp2 1128	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11		eqid 2778	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
12		eqid 2778	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
13		eqid 2778	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
14	7	3ad2ant1 1124	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15		simp2 1128	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16		simp3 1129	. . . . . . . 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 17875	. . . . . . 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	mpt2eq3dva 6996	. . . . . 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 2778	. . . . . . 7 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
20	11, 12, 19	plusffval 17633	. . . . . 6 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
21	18, 20	syl6eqr 2832	. . . . 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 21146	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
23	10, 22	sylib 210	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
24	23, 23	cnmpt1st 21880	. . . . . 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 21881	. . . . . . 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 17853	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27	26	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
28	27	feqmptd 6509	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
29		eqid 2778	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	11, 4, 13, 29	grpinvval2 17885	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ((0g‘𝐺) − 𝑥))
31	7, 30	sylan 575	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ((0g‘𝐺) − 𝑥))
32	31	mpteq2dva 4979	. . . . . . . . 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 2814	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)))
34	11, 29	grpidcl 17837	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
35	34	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g‘𝐺) ∈ (Base‘𝐺))
36	23, 23, 35	cnmptc 21874	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g‘𝐺)) ∈ (𝐽 Cn 𝐽))
37	23	cnmptid 21873	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38		simp3 1129	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
39	23, 36, 37, 38	cnmpt12f 21878	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)) ∈ (𝐽 Cn 𝐽))
40	33, 39	eqeltrd 2859	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
41	23, 23, 25, 40	cnmpt21f 21884	. . . . . 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 21887	. . . . 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 2860	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
44	19, 3	istmd 22286	. . . 4 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
45	9, 10, 43, 44	syl3anbrc 1400	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
46	3, 13	istgp 22289	. . 3 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ (𝐽 Cn 𝐽)))
47	7, 45, 40, 46	syl3anbrc 1400	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
48	6, 47	impbii 201	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 198   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107   ↦ cmpt 4965  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  Basecbs 16255  +_gcplusg 16338  TopOpenctopn 16468  0_gc0g 16486  +_𝑓cplusf 17625  Mndcmnd 17680  Grpcgrp 17809  inv_gcminusg 17810  -_gcsg 17811  TopOnctopon 21122  TopSpctps 21144   Cn ccn 21436   ×_t ctx 21772  TopMndctmd 22282  TopGrpctgp 22283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-0g 16488  df-topgen 16490  df-plusf 17627  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-sbg 17814  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cn 21439  df-cnp 21440  df-tx 21774  df-tmd 22284  df-tgp 22285

This theorem is referenced by:  distgp  22311  indistgp  22312  qustgplem  22332  ngptgp  22848  cnfldtgp  23080