Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rspecval Structured version   Visualization version   GIF version

Theorem rspecval 33401
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 6891 . . 3 (𝑟 = 𝑅 → (IDLsrg‘𝑟) = (IDLsrg‘𝑅))
2 fveq2 6891 . . 3 (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
31, 2oveq12d 7432 . 2 (𝑟 = 𝑅 → ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
4 df-rspec 33400 . 2 Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
5 ovex 7447 . 2 ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)) ∈ V
63, 4, 5fvmpt 6999 1 (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6542  (class class class)co 7414  s cress 17200  Ringcrg 20164  PrmIdealcprmidl 33086  IDLsrgcidlsrg 33147  Speccrspec 33399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-rspec 33400
This theorem is referenced by:  rspecbas  33402  rspectset  33403  rspectopn  33404
  Copyright terms: Public domain W3C validator