Theorem extopgrp 14980

Description: The trivial topological group.

Hypothesis

Ref	Expression
extopgrp.1	|- A e. B

Assertion

Ref	Expression
extopgrp	|- <.{<.<.A, A>., A>.}, {(/), {A}}>. e. TopGrp

Proof of Theorem extopgrp

Step	Hyp	Ref	Expression
1		prex 3526	. . 3 |- {(/), {A}} e. _V
2		istopgrp 14971	. . 3 |- ({(/), {A}} e. _V -> (<.{<.<.A, A>., A>.}, {(/), {A}}>. e. TopGrp <-> (({<.<.A, A>., A>.} e. Grp /\ {(/), {A}} e. Top) /\ {<.<.A, A>., A>.} e. (({(/), {A}} X.t {(/), {A}}) Cn {(/), {A}}) /\ (inv` {<.<.A, A>., A>.}) e. ({(/), {A}} Cn {(/), {A}}))))
3	1, 2	ax-mp 7	. 2 |- (<.{<.<.A, A>., A>.}, {(/), {A}}>. e. TopGrp <-> (({<.<.A, A>., A>.} e. Grp /\ {(/), {A}} e. Top) /\ {<.<.A, A>., A>.} e. (({(/), {A}} X.t {(/), {A}}) Cn {(/), {A}}) /\ (inv` {<.<.A, A>., A>.}) e. ({(/), {A}} Cn {(/), {A}})))
4		extopgrp.1	. . . . 5 |- A e. B
5	4	elisseti 2301	. . . 4 |- A e. _V
6	5	grpsn 9340	. . 3 |- {<.<.A, A>., A>.} e. Grp
7		indistop 8918	. . 3 |- {(/), {A}} e. Top
8	6, 7	pm3.2i 307	. 2 |- ({<.<.A, A>., A>.} e. Grp /\ {(/), {A}} e. Top)
9		eqid 1884	. . . . 5 |- ({(/), {A}} X.t {(/), {A}}) = ({(/), {A}} X.t {(/), {A}})
10	9	txtop 8934	. . . 4 |- (({(/), {A}} e. Top /\ {(/), {A}} e. Top) -> ({(/), {A}} X.t {(/), {A}}) e. Top)
11	7, 7, 10	mp2an 761	. . 3 |- ({(/), {A}} X.t {(/), {A}}) e. Top
12		opex 3527	. . . . . . . 8 |- <.A, A>. e. _V
13	12, 5	rnsnop 4375	. . . . . . 7 |- ran {<.<.A, A>., A>.} = {A}
14	13	eqcomi 1888	. . . . . 6 |- {A} = ran {<.<.A, A>., A>.}
15	14	grpfo 9323	. . . . 5 |- ({<.<.A, A>., A>.} e. Grp -> {<.<.A, A>., A>.}:({A} X. {A})-onto->{A})
16		fof 4617	. . . . 5 |- ({<.<.A, A>., A>.}:({A} X. {A})-onto->{A} -> {<.<.A, A>., A>.}:({A} X. {A})-->{A})
17	15, 16	syl 12	. . . 4 |- ({<.<.A, A>., A>.} e. Grp -> {<.<.A, A>., A>.}:({A} X. {A})-->{A})
18	6, 17	ax-mp 7	. . 3 |- {<.<.A, A>., A>.}:({A} X. {A})-->{A}
19		0ex 3446	. . . . . . . . 9 |- (/) e. _V
20		snex 3492	. . . . . . . . 9 |- {A} e. _V
21	19, 20	unipr 3191	. . . . . . . 8 |- U.{(/), {A}} = ((/) u. {A})
22		uncom 2744	. . . . . . . 8 |- ((/) u. {A}) = ({A} u. (/))
23		un0 2896	. . . . . . . 8 |- ({A} u. (/)) = {A}
24	21, 22, 23	3eqtrri 1913	. . . . . . 7 |- {A} = U.{(/), {A}}
25	9, 24, 24	txuni 8935	. . . . . 6 |- (({(/), {A}} e. Top /\ {(/), {A}} e. Top) -> U.({(/), {A}} X.t {(/), {A}}) = ({A} X. {A}))
26	7, 7, 25	mp2an 761	. . . . 5 |- U.({(/), {A}} X.t {(/), {A}}) = ({A} X. {A})
27	26	eqcomi 1888	. . . 4 |- ({A} X. {A}) = U.({(/), {A}} X.t {(/), {A}})
28	27, 20	mapudiscn 14872	. . 3 |- ((({(/), {A}} X.t {(/), {A}}) e. Top /\ {<.<.A, A>., A>.}:({A} X. {A})-->{A}) -> {<.<.A, A>., A>.} e. (({(/), {A}} X.t {(/), {A}}) Cn {(/), {A}}))
29	11, 18, 28	mp2an 761	. 2 |- {<.<.A, A>., A>.} e. (({(/), {A}} X.t {(/), {A}}) Cn {(/), {A}})
30	4	invtrgrp 14979	. . 3 |- (inv` {<.<.A, A>., A>.}) = ( _I |` {A})
31	24	idcn 9042	. . . 4 |- ({(/), {A}} e. Top -> ( _I |` {A}) e. ({(/), {A}} Cn {(/), {A}}))
32	7, 31	ax-mp 7	. . 3 |- ( _I |` {A}) e. ({(/), {A}} Cn {(/), {A}})
33	30, 32	eqeltri 1967	. 2 |- (inv` {<.<.A, A>., A>.}) e. ({(/), {A}} Cn {(/), {A}})
34	3, 8, 29, 33	mpbir3an 1052	1 |- <.{<.<.A, A>., A>.}, {(/), {A}}>. e. TopGrp

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591  (/)c0 2875  {csn 3044  {cpr 3045  <.cop 3046  U.cuni 3177   _I cid 3582   X. cxp 3984  ran crn 3987   |` cres 3988  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Topctop 8857   X.t ctx 8930   Cn ccn 9028  Grpcgr 9311  invcgn 9313  TopGrpctopgrp 14969

This theorem is referenced by:  topgrpsubcn 14982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030  df-grp 9316  df-gid 9317  df-ginv 9318  df-topgrp 14970

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