Theorem mhmhmeotmd 28807

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 16664	. . 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 16663	. . . . 5 |- ( F e. ( S MndHom T ) -> S e. Mnd )
7	1, 6	ax-mp 5	. . . 4 |- S e. Mnd
8		eqid 2471	. . . . 5 |- ( Base ` S ) = ( Base ` S )
9		eqid 2471	. . . . 5 |- ( +f ` S ) = ( +f ` S )
10	8, 9	mndplusf 16633	. . . 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 2471	. . . . 5 |- ( Base ` T ) = ( Base ` T )
13		eqid 2471	. . . . 5 |- ( +f ` T ) = ( +f ` T )
14	12, 13	mndplusf 16633	. . . 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 2471	. . . . 5 |- ( TopOpen ` S ) = ( TopOpen ` S )
18	17, 8	tmdtopon 21174	. . . 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 2471	. . . . 5 |- ( TopOpen ` T ) = ( TopOpen ` T )
21	12, 20	istps 20028	. . . 4 |- ( T e. TopSp <-> ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) ) )
22	4, 21	mpbi 213	. . 3 |- ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) )
23		eqid 2471	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
24		eqid 2471	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
25	8, 23, 24	mhmlin 16667	. . . . 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 1377	. . . 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 16572	. . . . 5 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +f ` S ) y ) = ( x ( +g ` S ) y ) )
28	27	fveq2d 5883	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
29	8, 12	mhmf 16665	. . . . . . 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 6036	. . . . 5 |- ( x e. ( Base ` S ) -> ( F ` x ) e. ( Base ` T ) )
32	30	ffvelrni 6036	. . . . 5 |- ( y e. ( Base ` S ) -> ( F ` y ) e. ( Base ` T ) )
33	12, 24, 13	plusfval 16572	. . . . 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 485	. . . 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 2515	. . 3 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( ( F ` x ) ( +f ` T ) ( F ` y ) ) )
36	17, 9	tmdcn 21176	. . . 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 28806	. 2 |- ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) )
39	13, 20	istmd 21167	. 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 1212	1 |- T e. TopMnd

Syntax hints:   /\ wa 376   = wceq 1452   e. wcel 1904   X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   TopOpenctopn 15398   +fcplusf 16563   Mndcmnd 16613   MndHom cmhm 16658  TopOnctopon 19995   TopSpctps 19996   Cn ccn 20317   tX ctx 20652   Homeochmeo 20845  TopMndctmd 21163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-topgen 15420  df-plusf 16565  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-tx 20654  df-hmeo 20847  df-tmd 21165

This theorem is referenced by:  xrge0tmd  28826