Theorem istgp2 24440

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 24427	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2		tgptps 24429	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3		tgpsubcn.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
4		tgpsubcn.3	. . . 4 ⊢ − = (-g‘𝐺)
5	3, 4	tgpsubcn 24439	. . 3 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6	1, 2, 5	3jca 1146	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7		simp1 1154	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8		grpmnd 19180	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
9	8	3ad2ant1 1151	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10		simp2 1155	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
12		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
13		eqid 2763	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
14	7	3ad2ant1 1151	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15		simp2 1155	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16		simp3 1156	. . . . . . . 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 19251	. . . . . . 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 7520	. . . . . 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 2763	. . . . . . 7 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
20	11, 12, 19	plusffval 18851	. . . . . 6 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
21	18, 20	eqtr4di 2816	. . . . 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 23282	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
23	10, 22	sylib 221	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
24	23, 23	cnmpt1st 24017	. . . . . 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 24018	. . . . . . 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 19226	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27	26	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
28	27	feqmptd 6976	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
29		eqid 2763	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	11, 4, 13, 29	grpinvval2 19262	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ((0g‘𝐺) − 𝑥))
31	7, 30	sylan 592	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ((0g‘𝐺) − 𝑥))
32	31	mpteq2dva 5201	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)))
34	11, 29	grpidcl 19205	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
35	34	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g‘𝐺) ∈ (Base‘𝐺))
36	23, 23, 35	cnmptc 24011	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g‘𝐺)) ∈ (𝐽 Cn 𝐽))
37	23	cnmptid 24010	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38		simp3 1156	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
39	23, 36, 37, 38	cnmpt12f 24015	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)) ∈ (𝐽 Cn 𝐽))
40	33, 39	eqeltrd 2864	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
41	23, 23, 25, 40	cnmpt21f 24021	. . . . . 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 24024	. . . . 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 2865	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
44	19, 3	istmd 24423	. . . 4 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
45	9, 10, 43, 44	syl3anbrc 1362	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
46	3, 13	istgp 24426	. . 3 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ (𝐽 Cn 𝐽)))
47	7, 45, 40, 46	syl3anbrc 1362	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
48	6, 47	impbii 212	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ↦ cmpt 5189  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  Basecbs 17416  +_gcplusg 17457  TopOpenctopn 17621  0_gc0g 17639  +_𝑓cplusf 18842  Mndcmnd 18952  Grpcgrp 19173  inv_gcminusg 19174  -_gcsg 19175  TopOnctopon 23258  TopSpctps 23280   Cn ccn 23572   ×_t ctx 23909  TopMndctmd 24419  TopGrpctgp 24420

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877  df-0g 17641  df-topgen 17643  df-plusf 18844  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-grp 19176  df-minusg 19177  df-sbg 19178  df-top 23242  df-topon 23259  df-topsp 23281  df-bases 23294  df-cn 23575  df-cnp 23576  df-tx 23911  df-tmd 24421  df-tgp 24422

This theorem is used by:  distgp  24448  indistgp  24449  qustgplem  24470  ngptgp  24985  cnfldtgp  25220