Theorem mhmhmeotmd 26491

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 15570	. . 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 15569	. . . . 5 |- ( F e. ( S MndHom T ) -> S e. Mnd )
7	1, 6	ax-mp 5	. . . 4 |- S e. Mnd
8		eqid 2451	. . . . 5 |- ( Base ` S ) = ( Base ` S )
9		eqid 2451	. . . . 5 |- ( +f ` S ) = ( +f ` S )
10	8, 9	mndplusf 15533	. . . 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 2451	. . . . 5 |- ( Base ` T ) = ( Base ` T )
13		eqid 2451	. . . . 5 |- ( +f ` T ) = ( +f ` T )
14	12, 13	mndplusf 15533	. . . 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 2451	. . . . 5 |- ( TopOpen ` S ) = ( TopOpen ` S )
18	17, 8	tmdtopon 19768	. . . 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 2451	. . . . 5 |- ( TopOpen ` T ) = ( TopOpen ` T )
21	12, 20	istps 18657	. . . 4 |- ( T e. TopSp <-> ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) ) )
22	4, 21	mpbi 208	. . 3 |- ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) )
23		eqid 2451	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
24		eqid 2451	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
25	8, 23, 24	mhmlin 15573	. . . . 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 1302	. . . 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 15530	. . . . 5 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +f ` S ) y ) = ( x ( +g ` S ) y ) )
28	27	fveq2d 5793	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
29	8, 12	mhmf 15571	. . . . . . 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 5941	. . . . 5 |- ( x e. ( Base ` S ) -> ( F ` x ) e. ( Base ` T ) )
32	30	ffvelrni 5941	. . . . 5 |- ( y e. ( Base ` S ) -> ( F ` y ) e. ( Base ` T ) )
33	12, 24, 13	plusfval 15530	. . . . 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 2502	. . 3 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( ( F ` x ) ( +f ` T ) ( F ` y ) ) )
36	17, 9	tmdcn 19770	. . . 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 26490	. 2 |- ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) )
39	13, 20	istmd 19761	. 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 1170	1 |- T e. TopMnd

Syntax hints:   /\ wa 369   = wceq 1370   e. wcel 1758   X. cxp 4936   -->wf 5512   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   TopOpenctopn 14462   Mndcmnd 15511   +fcplusf 15514   MndHom cmhm 15564  TopOnctopon 18615   TopSpctps 18617   Cn ccn 18944   tX ctx 19249   Homeochmeo 19442  TopMndctmd 19757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-map 7316  df-topgen 14484  df-mnd 15517  df-plusf 15518  df-mhm 15566  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cn 18947  df-tx 19251  df-hmeo 19444  df-tmd 19759

This theorem is referenced by:  xrge0tmd  26510