imastset

Description: The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	imasbas.u	\|- ( ph -> U = ( F "s R ) )
	imasbas.v	\|- ( ph -> V = ( Base ` R ) )
	imasbas.f	\|- ( ph -> F : V -onto-> B )
	imasbas.r	\|- ( ph -> R e. Z )
	imastset.j	\|- J = ( TopOpen ` R )
	imastset.o	\|- O = ( TopSet ` U )
Assertion	imastset	\|- ( ph -> O = ( J qTop F ) )

Proof

Step	Hyp	Ref	Expression
1		imasbas.u	\|- ( ph -> U = ( F "s R ) )
2		imasbas.v	\|- ( ph -> V = ( Base ` R ) )
3		imasbas.f	\|- ( ph -> F : V -onto-> B )
4		imasbas.r	\|- ( ph -> R e. Z )
5		imastset.j	\|- J = ( TopOpen ` R )
6		imastset.o	\|- O = ( TopSet ` U )
7		eqid	\|- ( +g ` R ) = ( +g ` R )
8		eqid	\|- ( .r ` R ) = ( .r ` R )
9		eqid	\|- ( Scalar ` R ) = ( Scalar ` R )
10		eqid	\|- ( Base ` ( Scalar ` R ) ) = ( Base ` ( Scalar ` R ) )
11		eqid	\|- ( .s ` R ) = ( .s ` R )
12		eqid	\|- ( .i ` R ) = ( .i ` R )
13		eqid	\|- ( dist ` R ) = ( dist ` R )
14		eqid	\|- ( le ` R ) = ( le ` R )
15		eqid	\|- ( +g ` U ) = ( +g ` U )
16	1 2 3 4 7 15	imasplusg	\|- ( ph -> ( +g ` U ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
17		eqid	\|- ( .r ` U ) = ( .r ` U )
18	1 2 3 4 8 17	imasmulr	\|- ( ph -> ( .r ` U ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
19		eqid	\|- ( .s ` U ) = ( .s ` U )
20	1 2 3 4 9 10 11 19	imasvsca	\|- ( ph -> ( .s ` U ) = U_ q e. V ( p e. ( Base ` ( Scalar ` R ) ) , x e. { ( F ` q ) } \|-> ( F ` ( p ( .s ` R ) q ) ) ) )
21		eqidd	\|- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } )
22		eqidd	\|- ( ph -> ( J qTop F ) = ( J qTop F ) )
23		eqid	\|- ( dist ` U ) = ( dist ` U )
24	1 2 3 4 13 23	imasds	\|- ( ph -> ( dist ` U ) = ( x e. B , y e. B \|-> inf ( U_ u e. NN ran ( z e. { w e. ( ( V X. V ) ^m ( 1 ... u ) ) \| ( ( F ` ( 1st ` ( w ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( w ` u ) ) ) = y /\ A. v e. ( 1 ... ( u - 1 ) ) ( F ` ( 2nd ` ( w ` v ) ) ) = ( F ` ( 1st ` ( w ` ( v + 1 ) ) ) ) ) } \|-> ( RRs gsum ( ( dist ` R ) o. z ) ) ) , RR , < ) ) )
25		eqidd	\|- ( ph -> ( ( F o. ( le ` R ) ) o. `' F ) = ( ( F o. ( le ` R ) ) o. `' F ) )
26	1 2 7 8 9 10 11 12 5 13 14 16 18 20 21 22 24 25 3 4	imasval	\|- ( ph -> U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) )
27	26	fveq2d	\|- ( ph -> ( TopSet ` U ) = ( TopSet ` ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) ) )
28		ovex	\|- ( J qTop F ) e. _V
29		eqid	\|- ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
30	29	imasvalstr	\|- ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) Struct <. 1 , ; 1 2 >.
31		tsetid	\|- TopSet = Slot ( TopSet ` ndx )
32		snsstp1	\|- { <. ( TopSet ` ndx ) , ( J qTop F ) >. } C_ { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. }
33		ssun2	\|- { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
34	32 33	sstri	\|- { <. ( TopSet ` ndx ) , ( J qTop F ) >. } C_ ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } )
35	30 31 34	strfv	\|- ( ( J qTop F ) e. _V -> ( J qTop F ) = ( TopSet ` ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) ) )
36	28 35	ax-mp	\|- ( J qTop F ) = ( TopSet ` ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , ( +g ` U ) >. , <. ( .r ` ndx ) , ( .r ` U ) >. } u. { <. ( Scalar ` ndx ) , ( Scalar ` R ) >. , <. ( .s ` ndx ) , ( .s ` U ) >. , <. ( .i ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ( .i ` R ) q ) >. } >. } ) u. { <. ( TopSet ` ndx ) , ( J qTop F ) >. , <. ( le ` ndx ) , ( ( F o. ( le ` R ) ) o. `' F ) >. , <. ( dist ` ndx ) , ( dist ` U ) >. } ) )
37	27 6 36	3eqtr4g	\|- ( ph -> O = ( J qTop F ) )

Metamath Proof Explorer

Theorem imastset

Proof