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 
|- S = ( Spec ` R )
rspectset.1 
|- I = ( LIdeal ` R )
rspectset.2 
|- J = ran ( i e. I |-> { j e. I | -. i C_ j } )
Assertion rspectset 
|- ( R e. Ring -> J = ( TopSet ` S ) )

Proof

Step Hyp Ref Expression
1 rspecbas.1 
 |-  S = ( Spec ` R )
2 rspectset.1 
 |-  I = ( LIdeal ` R )
3 rspectset.2 
 |-  J = ran ( i e. I |-> { j e. I | -. i C_ j } )
4 fvex 
 |-  ( PrmIdeal ` R ) e. _V
5 eqid 
 |-  ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) )
6 eqid 
 |-  ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( IDLsrg ` R ) )
7 5 6 resstset 
 |-  ( ( PrmIdeal ` R ) e. _V -> ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) )
8 4 7 ax-mp 
 |-  ( TopSet ` ( IDLsrg ` R ) ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) )
9 eqid 
 |-  ( IDLsrg ` R ) = ( IDLsrg ` R )
10 9 2 3 idlsrgtset 
 |-  ( R e. Ring -> J = ( TopSet ` ( IDLsrg ` R ) ) )
11 rspecval 
 |-  ( R e. Ring -> ( Spec ` R ) = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) )
12 1 11 syl5eq 
 |-  ( R e. Ring -> S = ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) )
13 12 fveq2d 
 |-  ( R e. Ring -> ( TopSet ` S ) = ( TopSet ` ( ( IDLsrg ` R ) |`s ( PrmIdeal ` R ) ) ) )
14 8 10 13 3eqtr4a 
 |-  ( R e. Ring -> J = ( TopSet ` S ) )