mhmhmeotmd

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

	Ref	Expression
Hypotheses	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	mhmhmeotmd	\|- T e. TopMnd

Proof

Step	Hyp	Ref	Expression
1		mhmhmeotmd.m	\|- F e. ( S MndHom T )
2		mhmhmeotmd.h	\|- F e. ( ( TopOpen ` S ) Homeo ( TopOpen ` T ) )
3		mhmhmeotmd.t	\|- S e. TopMnd
4		mhmhmeotmd.s	\|- T e. TopSp
5		mhmrcl2	\|- ( F e. ( S MndHom T ) -> T e. Mnd )
6	1 5	ax-mp	\|- T e. Mnd
7		mhmrcl1	\|- ( F e. ( S MndHom T ) -> S e. Mnd )
8	1 7	ax-mp	\|- S e. Mnd
9		eqid	\|- ( Base ` S ) = ( Base ` S )
10		eqid	\|- ( +f ` S ) = ( +f ` S )
11	9 10	mndplusf	\|- ( S e. Mnd -> ( +f ` S ) : ( ( Base ` S ) X. ( Base ` S ) ) --> ( Base ` S ) )
12	8 11	ax-mp	\|- ( +f ` S ) : ( ( Base ` S ) X. ( Base ` S ) ) --> ( Base ` S )
13		eqid	\|- ( Base ` T ) = ( Base ` T )
14		eqid	\|- ( +f ` T ) = ( +f ` T )
15	13 14	mndplusf	\|- ( T e. Mnd -> ( +f ` T ) : ( ( Base ` T ) X. ( Base ` T ) ) --> ( Base ` T ) )
16	6 15	ax-mp	\|- ( +f ` T ) : ( ( Base ` T ) X. ( Base ` T ) ) --> ( Base ` T )
17		eqid	\|- ( TopOpen ` S ) = ( TopOpen ` S )
18	17 9	tmdtopon	\|- ( S e. TopMnd -> ( TopOpen ` S ) e. ( TopOn ` ( Base ` S ) ) )
19	3 18	ax-mp	\|- ( TopOpen ` S ) e. ( TopOn ` ( Base ` S ) )
20		eqid	\|- ( TopOpen ` T ) = ( TopOpen ` T )
21	13 20	istps	\|- ( T e. TopSp <-> ( TopOpen ` T ) e. ( TopOn ` ( Base ` T ) ) )
22	4 21	mpbi	\|- ( TopOpen ` T ) e. ( TopOn ` ( Base ` T ) )
23		eqid	\|- ( +g ` S ) = ( +g ` S )
24		eqid	\|- ( +g ` T ) = ( +g ` T )
25	9 23 24	mhmlin	\|- ( ( 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	\|- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
27	9 23 10	plusfval	\|- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +f ` S ) y ) = ( x ( +g ` S ) y ) )
28	27	fveq2d	\|- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
29	9 13	mhmf	\|- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
30	1 29	ax-mp	\|- F : ( Base ` S ) --> ( Base ` T )
31	30	ffvelrni	\|- ( x e. ( Base ` S ) -> ( F ` x ) e. ( Base ` T ) )
32	30	ffvelrni	\|- ( y e. ( Base ` S ) -> ( F ` y ) e. ( Base ` T ) )
33	13 24 14	plusfval	\|- ( ( ( 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	\|- ( ( 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	\|- ( ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +f ` S ) y ) ) = ( ( F ` x ) ( +f ` T ) ( F ` y ) ) )
36	17 10	tmdcn	\|- ( S e. TopMnd -> ( +f ` S ) e. ( ( ( TopOpen ` S ) tX ( TopOpen ` S ) ) Cn ( TopOpen ` S ) ) )
37	3 36	ax-mp	\|- ( +f ` S ) e. ( ( ( TopOpen ` S ) tX ( TopOpen ` S ) ) Cn ( TopOpen ` S ) )
38	2 12 16 19 22 35 37	mndpluscn	\|- ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) )
39	14 20	istmd	\|- ( T e. TopMnd <-> ( T e. Mnd /\ T e. TopSp /\ ( +f ` T ) e. ( ( ( TopOpen ` T ) tX ( TopOpen ` T ) ) Cn ( TopOpen ` T ) ) ) )
40	6 4 38 39	mpbir3an	\|- T e. TopMnd

Metamath Proof Explorer

Theorem mhmhmeotmd

Proof