Step |
Hyp |
Ref |
Expression |
1 |
|
rspecbas.1 |
⊢ 𝑆 = ( Spec ‘ 𝑅 ) |
2 |
|
prmidlssidl |
⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) ⊆ ( LIdeal ‘ 𝑅 ) ) |
3 |
|
eqid |
⊢ ( IDLsrg ‘ 𝑅 ) = ( IDLsrg ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
5 |
3 4
|
idlsrgbas |
⊢ ( 𝑅 ∈ Ring → ( LIdeal ‘ 𝑅 ) = ( Base ‘ ( IDLsrg ‘ 𝑅 ) ) ) |
6 |
2 5
|
sseqtrd |
⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) ⊆ ( Base ‘ ( IDLsrg ‘ 𝑅 ) ) ) |
7 |
|
eqid |
⊢ ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) = ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ ( IDLsrg ‘ 𝑅 ) ) = ( Base ‘ ( IDLsrg ‘ 𝑅 ) ) |
9 |
7 8
|
ressbas2 |
⊢ ( ( PrmIdeal ‘ 𝑅 ) ⊆ ( Base ‘ ( IDLsrg ‘ 𝑅 ) ) → ( PrmIdeal ‘ 𝑅 ) = ( Base ‘ ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) ) ) |
10 |
6 9
|
syl |
⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) = ( Base ‘ ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) ) ) |
11 |
|
rspecval |
⊢ ( 𝑅 ∈ Ring → ( Spec ‘ 𝑅 ) = ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) ) |
12 |
1 11
|
syl5eq |
⊢ ( 𝑅 ∈ Ring → 𝑆 = ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) ) |
13 |
12
|
fveq2d |
⊢ ( 𝑅 ∈ Ring → ( Base ‘ 𝑆 ) = ( Base ‘ ( ( IDLsrg ‘ 𝑅 ) ↾s ( PrmIdeal ‘ 𝑅 ) ) ) ) |
14 |
10 13
|
eqtr4d |
⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) = ( Base ‘ 𝑆 ) ) |