Metamath Proof Explorer


Theorem zarcls0

Description: The closure of the identity ideal in the Zariski topology. Proposition 1.1.2(i) of EGA p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024)

Ref Expression
Hypotheses zarclsx.1 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖𝑗 } )
zarcls0.1 𝑃 = ( PrmIdeal ‘ 𝑅 )
zarcls0.2 0 = ( 0g𝑅 )
Assertion zarcls0 ( 𝑅 ∈ Ring → ( 𝑉 ‘ { 0 } ) = 𝑃 )

Proof

Step Hyp Ref Expression
1 zarclsx.1 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖𝑗 } )
2 zarcls0.1 𝑃 = ( PrmIdeal ‘ 𝑅 )
3 zarcls0.2 0 = ( 0g𝑅 )
4 1 a1i ( 𝑅 ∈ Ring → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖𝑗 } ) )
5 simplr ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑖 = { 0 } )
6 simpll ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑅 ∈ Ring )
7 prmidlidl ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) )
8 6 7 sylancom ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) )
9 eqid ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 )
10 9 3 lidl0cl ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) → 0𝑗 )
11 6 8 10 syl2anc ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 0𝑗 )
12 11 snssd ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → { 0 } ⊆ 𝑗 )
13 5 12 eqsstrd ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑖𝑗 )
14 13 ralrimiva ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) → ∀ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) 𝑖𝑗 )
15 rabid2 ( ( PrmIdeal ‘ 𝑅 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖𝑗 } ↔ ∀ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) 𝑖𝑗 )
16 14 15 sylibr ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) → ( PrmIdeal ‘ 𝑅 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖𝑗 } )
17 2 16 eqtr2id ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖𝑗 } = 𝑃 )
18 9 3 lidl0 ( 𝑅 ∈ Ring → { 0 } ∈ ( LIdeal ‘ 𝑅 ) )
19 2 fvexi 𝑃 ∈ V
20 19 a1i ( 𝑅 ∈ Ring → 𝑃 ∈ V )
21 4 17 18 20 fvmptd ( 𝑅 ∈ Ring → ( 𝑉 ‘ { 0 } ) = 𝑃 )