Metamath Proof Explorer


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 = LIdeal R
rspectset.2 J = ran i I j I | ¬ i j
Assertion rspectset R Ring J = TopSet S

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 = LIdeal R
3 rspectset.2 J = ran i I j I | ¬ i j
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 Ring J = TopSet S