Theorem rspectset

Description: Topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024)

	Ref	Expression
Hypotheses	rspecbas.1	No typesetting found for |- S = ( Spec ` R ) with typecode |-
	rspectset.1	$$⊢I=\mathrm{LIdeal}\left(R\right)$$
	rspectset.2	$$⊢J=\mathrm{ran}\left(i\in I⟼\left\{j\in I|\neg i\subseteq j\right\}\right)$$
Assertion	rspectset	$$⊢R\in \mathrm{Ring}\to J=\mathrm{TopSet}\left(S\right)$$

Proof

Step	Hyp	Ref	Expression
1		rspecbas.1	Could not format S = ( Spec ` R ) : No typesetting found for |- S = ( Spec ` R ) with typecode |-
2		rspectset.1	$$⊢I=\mathrm{LIdeal}\left(R\right)$$
3		rspectset.2	$$⊢J=\mathrm{ran}\left(i\in I⟼\left\{j\in I|\neg i\subseteq j\right\}\right)$$
4		fvex	Could not format ( PrmIdeal ` R ) e. _V : No typesetting found for |- ( PrmIdeal ` R ) e. _V with typecode |-
5		eqid	Could not format ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) with typecode |-
6		eqid	Could not format ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( IDLsrg ` R ) ) : No typesetting found for |- ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( IDLsrg ` R ) ) with typecode |-
7	5 6	resstset	Could not format ( ( PrmIdeal ` R ) e. _V -> ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) ) : No typesetting found for |- ( ( PrmIdeal ` R ) e. _V -> ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) ) with typecode |-
8	4 7	ax-mp	Could not format ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) with typecode |-
9		eqid	Could not format ( IDLsrg ` R ) = ( IDLsrg ` R ) : No typesetting found for |- ( IDLsrg ` R ) = ( IDLsrg ` R ) with typecode |-
10	9 2 3	idlsrgtset	Could not format ( R e. Ring -> J = ( TopSet ` ( IDLsrg ` R ) ) ) : No typesetting found for |- ( R e. Ring -> J = ( TopSet ` ( IDLsrg ` R ) ) ) with typecode |-
11		rspecval	Could not format ( R e. Ring -> ( Spec ` R ) = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( R e. Ring -> ( Spec ` R ) = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) with typecode |-
12	1 11	syl5eq	Could not format ( R e. Ring -> S = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( R e. Ring -> S = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) with typecode |-
13	12	fveq2d	Could not format ( R e. Ring -> ( TopSet ` S ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) ) : No typesetting found for |- ( R e. Ring -> ( TopSet ` S ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) ) with typecode |-
14	8 10 13	3eqtr4a	$$⊢R\in \mathrm{Ring}\to J=\mathrm{TopSet}\left(S\right)$$