Theorem mhmhmeotmd 29973

Description: Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)

Hypotheses

Ref	Expression
mhmhmeotmd.m	|- F e. ( S MndHom T )
mhmhmeotmd.h	|- F e. ( ( TopOpen ` S ) Homeo ( TopOpen ` T ) )
mhmhmeotmd.t	|- S e. TopMnd
mhmhmeotmd.s	|- T e. TopSp

Assertion

Ref	Expression
mhmhmeotmd	|- T e. TopMnd

Proof of Theorem mhmhmeotmd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhmhmeotmd.m	. . 3 |- F e. ( S MndHom T )
2		mhmrcl2 17339	. . 3 |- ( F e. ( S MndHom T ) -> T e. Mnd )
3	1, 2	ax-mp 5	. 2 |- T e. Mnd
4		mhmhmeotmd.s	. 2 |- T e. TopSp
5		mhmhmeotmd.h	. . 3 |- F e. ( ( TopOpen ` S ) Homeo ( TopOpen ` T ) )
6		mhmrcl1 17338	. . . . 5 |- ( F e. ( S MndHom T ) -> S e. Mnd )
7	1, 6	ax-mp 5	. . . 4 |- S e. Mnd
8		eqid 2622	. . . . 5 |- ( Base ` S ) = ( Base ` S )
9		eqid 2622	. . . . 5 |- ( +f ` S ) = ( +f ` S )
10	8, 9	mndplusf 17309	. . . 4 |- ( S e. Mnd -> ( +f ` S ) : ( ( Base ` S ) X. ( Base ` S ) ) --> ( Base ` S ) )
11	7, 10	ax-mp 5	. . 3 |- ( +f ` S ) : ( ( Base ` S ) X. ( Base ` S ) ) --> ( Base ` S )
12		eqid 2622	. . . . 5 |- ( Base ` T ) = ( Base ` T )
13		eqid 2622	. . . . 5 |- ( +f ` T ) = ( +f ` T )
14	12, 13	mndplusf 17309	. . . 4 |- ( T e. Mnd -> ( +f ` T ) : ( ( Base ` T ) X. ( Base ` T ) ) --> ( Base ` T ) )
15	3, 14	ax-mp 5	. . 3 |- ( +f ` T ) : ( ( Base ` T ) X. ( Base ` T ) ) --> ( Base ` T )
16		mhmhmeotmd.t	. . . 4 |- S e. TopMnd
17		eqid 2622	. . . . 5 |- ( TopOpen ` S ) = ( TopOpen ` S )
18	17, 8	tmdtopon 21885	. . . 4 |- ( S e. TopMnd -> ( TopOpen ` S ) e. (TopOn ` ( Base ` S ) ) )
19	16, 18	ax-mp 5	. . 3 |- ( TopOpen ` S ) e. (TopOn ` ( Base ` S ) )
20		eqid 2622	. . . . 5 |- ( TopOpen ` T ) = ( TopOpen ` T )
21	12, 20	istps 20738	. . . 4 |- ( T e. TopSp <-> ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) ) )
22	4, 21	mpbi 220	. . 3 |- ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) )
23		eqid 2622	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
24		eqid 2622	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
25	8, 23, 24	mhmlin 17342	. . . . 5 |- ( ( F e. ( S MndHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
26	1, 25	mp3an1 1411	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
27	8, 23, 9	plusfval 17248	. . . . 5 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +f ` S ) y ) = ( x ( +g ` S ) y ) )
28	27	fveq2d 6195	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
29	8, 12	mhmf 17340	. . . . . . 7 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
30	1, 29	ax-mp 5	. . . . . 6 |- F : ( Base ` S ) --> ( Base ` T )
31	30	ffvelrni 6358	. . . . 5 |- ( x e. ( Base ` S ) -> ( F ` x ) e. ( Base ` T ) )
32	30	ffvelrni 6358	. . . . 5 |- ( y e. ( Base ` S ) -> ( F ` y ) e. ( Base ` T ) )
33	12, 24, 13	plusfval 17248	. . . . 5 |- ( ( ( F ` x ) e. ( Base ` T ) /\ ( F ` y ) e. ( Base ` T ) ) -> ( ( F ` x ) ( +f ` T ) ( F ` y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
34	31, 32, 33	syl2an 494	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( ( F ` x ) ( +f ` T ) ( F ` y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
35	26, 28, 34	3eqtr4d 2666	. . 3 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( ( F ` x ) ( +f ` T ) ( F ` y ) ) )
36	17, 9	tmdcn 21887	. . . 4 |- ( S e. TopMnd -> ( +f ` S ) e. ( ( ( TopOpen ` S ) tX ( TopOpen ` S ) ) Cn ( TopOpen ` S ) ) )
37	16, 36	ax-mp 5	. . 3 |- ( +f ` S ) e. ( ( ( TopOpen ` S ) tX ( TopOpen ` S ) ) Cn ( TopOpen ` S ) )
38	5, 11, 15, 19, 22, 35, 37	mndpluscn 29972	. 2 |- ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) )
39	13, 20	istmd 21878	. 2 |- ( T e. TopMnd <-> ( T e. Mnd /\ T e. TopSp /\ ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) ) ) )
40	3, 4, 38, 39	mpbir3an 1244	1 |- T e. TopMnd

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   +fcplusf 17239   Mndcmnd 17294   MndHom cmhm 17333  TopOnctopon 20715   TopSpctps 20736   Cn ccn 21028   tX ctx 21363   Homeochmeo 21556  TopMndctmd 21874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-tx 21365  df-hmeo 21558  df-tmd 21876

This theorem is referenced by:  xrge0tmd  29992