Theorem mhmhmeotmd 27545

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 15781	. . 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 15780	. . . . 5 |- ( F e. ( S MndHom T ) -> S e. Mnd )
7	1, 6	ax-mp 5	. . . 4 |- S e. Mnd
8		eqid 2467	. . . . 5 |- ( Base ` S ) = ( Base ` S )
9		eqid 2467	. . . . 5 |- ( +f ` S ) = ( +f ` S )
10	8, 9	mndplusf 15744	. . . 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 2467	. . . . 5 |- ( Base ` T ) = ( Base ` T )
13		eqid 2467	. . . . 5 |- ( +f ` T ) = ( +f ` T )
14	12, 13	mndplusf 15744	. . . 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 2467	. . . . 5 |- ( TopOpen ` S ) = ( TopOpen ` S )
18	17, 8	tmdtopon 20315	. . . 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 2467	. . . . 5 |- ( TopOpen ` T ) = ( TopOpen ` T )
21	12, 20	istps 19204	. . . 4 |- ( T e. TopSp <-> ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) ) )
22	4, 21	mpbi 208	. . 3 |- ( TopOpen ` T ) e. (TopOn ` ( Base ` T ) )
23		eqid 2467	. . . . . 6 |- ( +g ` S ) = ( +g ` S )
24		eqid 2467	. . . . . 6 |- ( +g ` T ) = ( +g ` T )
25	8, 23, 24	mhmlin 15784	. . . . 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 1311	. . . 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 15741	. . . . 5 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +f ` S ) y ) = ( x ( +g ` S ) y ) )
28	27	fveq2d 5868	. . . 4 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
29	8, 12	mhmf 15782	. . . . . . 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 6018	. . . . 5 |- ( x e. ( Base ` S ) -> ( F ` x ) e. ( Base ` T ) )
32	30	ffvelrni 6018	. . . . 5 |- ( y e. ( Base ` S ) -> ( F ` y ) e. ( Base ` T ) )
33	12, 24, 13	plusfval 15741	. . . . 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 2518	. . 3 |- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( ( F ` x ) ( +f ` T ) ( F ` y ) ) )
36	17, 9	tmdcn 20317	. . . 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 27544	. 2 |- ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) )
39	13, 20	istmd 20308	. 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 1178	1 |- T e. TopMnd

Syntax hints:   /\ wa 369   = wceq 1379   e. wcel 1767   X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   TopOpenctopn 14673   Mndcmnd 15722   +fcplusf 15725   MndHom cmhm 15775  TopOnctopon 19162   TopSpctps 19164   Cn ccn 19491   tX ctx 19796   Homeochmeo 19989  TopMndctmd 20304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-topgen 14695  df-mnd 15728  df-plusf 15729  df-mhm 15777  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cn 19494  df-tx 19798  df-hmeo 19991  df-tmd 20306

This theorem is referenced by:  xrge0tmd  27564