Theorem mhmhmeotmd 28375

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 16296	. . 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 16295	. . . . 5 |- ( F e. ( S MndHom T ) -> S e. Mnd )
7	1, 6	ax-mp 5	. . . 4 |- S e. Mnd
8		eqid 2404	. . . . 5 |- ( Base ` S ) = ( Base ` S )
9		eqid 2404	. . . . 5 |- ( +f ` S ) = ( +f ` S )
10	8, 9	mndplusf 16265	. . . 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 2404	. . . . 5 |- ( Base ` T ) = ( Base ` T )
13		eqid 2404	. . . . 5 |- ( +f ` T ) = ( +f ` T )
14	12, 13	mndplusf 16265	. . . 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 2404	. . . . 5 |- ( TopOpen ` S ) = ( TopOpen ` S )
18	17, 8	tmdtopon 20874	. . . 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 2404	. . . . 5 |- ( TopOpen ` T ) = ( TopOpen ` T )
21	12, 20	istps 19731	. . . 4 |- ( T e. TopSp <-> ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) ) )
22	4, 21	mpbi 210	. . 3 |- ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) )
23		eqid 2404	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
24		eqid 2404	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
25	8, 23, 24	mhmlin 16299	. . . . 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 1315	. . . 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 16204	. . . . 5 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +f ` S ) y ) = ( x ( +g ` S ) y ) )
28	27	fveq2d 5855	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
29	8, 12	mhmf 16297	. . . . . . 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 6010	. . . . 5 |- ( x e. ( Base ` S ) -> ( F ` x ) e. ( Base ` T ) )
32	30	ffvelrni 6010	. . . . 5 |- ( y e. ( Base ` S ) -> ( F ` y ) e. ( Base ` T ) )
33	12, 24, 13	plusfval 16204	. . . . 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 477	. . . 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 2455	. . 3 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( ( F ` x ) ( +f ` T ) ( F ` y ) ) )
36	17, 9	tmdcn 20876	. . . 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 28374	. 2 |- ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) )
39	13, 20	istmd 20867	. 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 1181	1 |- T e. TopMnd

Syntax hints:   /\ wa 369   = wceq 1407   e. wcel 1844   X. cxp 4823   -->wf 5567   ` cfv 5571  (class class class)co 6280   Basecbs 14843   +g cplusg 14911   TopOpenctopn 15038   +fcplusf 16195   Mndcmnd 16245   MndHom cmhm 16290  TopOnctopon 19689   TopSpctps 19691   Cn ccn 20020   tX ctx 20355   Homeochmeo 20548  TopMndctmd 20863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-map 7461  df-topgen 15060  df-plusf 16197  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mhm 16292  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cn 20023  df-tx 20357  df-hmeo 20550  df-tmd 20865

This theorem is referenced by:  xrge0tmd  28394