Theorem istgp2 23442

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 23429	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2		tgptps 23431	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
3		tgpsubcn.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
4		tgpsubcn.3	. . . 4 ⊢ − = (-g‘𝐺)
5	3, 4	tgpsubcn 23441	. . 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 18755	. . . . 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 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 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 18820	. . . . . . 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 7434	. . . . . 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 18503	. . . . . 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 22283	. . . . . . 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 23019	. . . . . 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 23020	. . . . . . 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 18797	. . . . . . . . . . 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 6910	. . . . . . . . 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 18830	. . . . . . . . . . 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 5205	. . . . . . . . 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 18778	. . . . . . . . . . 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 23013	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ (0g‘𝐺)) ∈ (𝐽 Cn 𝐽))
37	23	cnmptid 23012	. . . . . . . . 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 23017	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (𝑥 ∈ (Base‘𝐺) ↦ ((0g‘𝐺) − 𝑥)) ∈ (𝐽 Cn 𝐽))
40	33, 39	eqeltrd 2838	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (invg‘𝐺) ∈ (𝐽 Cn 𝐽))
41	23, 23, 25, 40	cnmpt21f 23023	. . . . . 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 23026	. . . . 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 2839	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
44	19, 3	istmd 23425	. . . 4 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
45	9, 10, 43, 44	syl3anbrc 1343	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) → 𝐺 ∈ TopMnd)
46	3, 13	istgp 23428	. . 3 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ (𝐽 Cn 𝐽)))
47	7, 45, 40, 46	syl3anbrc 1343	. 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 1087   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  Basecbs 17083  +_gcplusg 17133  TopOpenctopn 17303  0_gc0g 17321  +_𝑓cplusf 18494  Mndcmnd 18556  Grpcgrp 18748  inv_gcminusg 18749  -_gcsg 18750  TopOnctopon 22259  TopSpctps 22281   Cn ccn 22575   ×_t ctx 22911  TopMndctmd 23421  TopGrpctgp 23422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-0g 17323  df-topgen 17325  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cn 22578  df-cnp 22579  df-tx 22913  df-tmd 23423  df-tgp 23424

This theorem is referenced by:  distgp  23450  indistgp  23451  qustgplem  23472  ngptgp  23992  cnfldtgp  24232