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Theorem rspecval 31348
 Description: Value of the spectrum of the ring 𝑅. Notation 1.1.1 of [EGA] p. 80. (Contributed by Thierry Arnoux, 2-Jun-2024.)
Assertion
Ref Expression
rspecval (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))

Proof of Theorem rspecval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . 3 (𝑟 = 𝑅 → (IDLsrg‘𝑟) = (IDLsrg‘𝑅))
2 fveq2 6663 . . 3 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
31, 2oveq12d 7174 . 2 (𝑟 = 𝑅 → ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
4 df-rspec 31347 . 2 Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
5 ovex 7189 . 2 ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) ∈ V
63, 4, 5fvmpt 6764 1 (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6340  (class class class)co 7156   ↾s cress 16556  Ringcrg 19379  PrmIdealcprmidl 31144  IDLsrgcidlsrg 31179  Speccrspec 31346 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-rspec 31347 This theorem is referenced by:  rspecbas  31349  rspectset  31350  rspectopn  31351
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