Theorem istgp2 24007

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 23994	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2		tgptps 23996	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3		tgpsubcn.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
4		tgpsubcn.3	. . . 4 ⊢ − = (-g‘𝐺)
5	3, 4	tgpsubcn 24006	. . 3 ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
6	1, 2, 5	3jca 1128	. 2 ⊢ (𝐺 ∈ TopGrp → (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
7		simp1 1136	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Grp)
8		grpmnd 18855	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
9	8	3ad2ant1 1133	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ Mnd)
10		simp2 1137	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopSp)
11		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
12		eqid 2733	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
13		eqid 2733	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
14	7	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝐺 ∈ Grp)
15		simp2 1137	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → 𝑥 ∈ (Base‘𝐺))
16		simp3 1138	. . . . . . . 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 18927	. . . . . . 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 7429	. . . . . 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 2733	. . . . . . 7 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
20	11, 12, 19	plusffval 18556	. . . . . 6 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ (Base‘𝐺), 𝑦 ∈ (Base‘𝐺) ↦ (𝑥(+g‘𝐺)𝑦))
21	18, 20	eqtr4di 2786	. . . . 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 22850	. . . . . . 7 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝐺)))
23	10, 22	sylib 218	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
24	23, 23	cnmpt1st 23584	. . . . . 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 23585	. . . . . . 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 18901	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
27	26	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
28	27	feqmptd 6896	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
29		eqid 2733	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
30	11, 4, 13, 29	grpinvval2 18938	. . . . . . . . . . 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 5186	. . . . . . . . 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 2768	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)))
34	11, 29	grpidcl 18880	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺))
35	34	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (0g‘𝐺) ∈ (Base‘𝐺))
36	23, 23, 35	cnmptc 23578	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g‘𝐺)) ∈ (𝐽 Cn 𝐽))
37	23	cnmptid 23577	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
38		simp3 1138	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
39	23, 36, 37, 38	cnmpt12f 23582	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)) ∈ (𝐽 Cn 𝐽))
40	33, 39	eqeltrd 2833	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
41	23, 23, 25, 40	cnmpt21f 23588	. . . . . 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 23591	. . . . 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 2834	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
44	19, 3	istmd 23990	. . . 4 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
45	9, 10, 43, 44	syl3anbrc 1344	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
46	3, 13	istgp 23993	. . 3 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ (𝐽 Cn 𝐽)))
47	7, 45, 40, 46	syl3anbrc 1344	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopGrp)
48	6, 47	impbii 209	1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ↦ cmpt 5174  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  Basecbs 17122  +_gcplusg 17163  TopOpenctopn 17327  0_gc0g 17345  +_𝑓cplusf 18547  Mndcmnd 18644  Grpcgrp 18848  inv_gcminusg 18849  -_gcsg 18850  TopOnctopon 22826  TopSpctps 22848   Cn ccn 23140   ×_t ctx 23476  TopMndctmd 23986  TopGrpctgp 23987

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 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-0g 17347  df-topgen 17349  df-plusf 18549  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-grp 18851  df-minusg 18852  df-sbg 18853  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cn 23143  df-cnp 23144  df-tx 23478  df-tmd 23988  df-tgp 23989

This theorem is referenced by:  distgp  24015  indistgp  24016  qustgplem  24037  ngptgp  24552  cnfldtgp  24788