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Theorem irngval 32687
Description: The elements of a field 𝑅 integral over a subset 𝑆. In the case of a subfield, those are the algebraic numbers over the field 𝑆 within the field 𝑅. That is, the numbers 𝑋 which are roots of monic polynomials 𝑃(𝑋) with coefficients in 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)
Hypotheses
Ref Expression
irngval.o 𝑂 = (𝑅 evalSub1 𝑆)
irngval.u 𝑈 = (𝑅s 𝑆)
irngval.b 𝐵 = (Base‘𝑅)
irngval.0 0 = (0g𝑅)
irngval.r (𝜑𝑅 ∈ Ring)
irngval.s (𝜑𝑆𝐵)
Assertion
Ref Expression
irngval (𝜑 → (𝑅 IntgRing 𝑆) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
Distinct variable groups:   𝐵,𝑓   𝑓,𝑂   𝑅,𝑓   𝑆,𝑓   𝑈,𝑓   𝜑,𝑓
Allowed substitution hint:   0 (𝑓)

Proof of Theorem irngval
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irngval.r . . 3 (𝜑𝑅 ∈ Ring)
21elexd 3495 . 2 (𝜑𝑅 ∈ V)
3 irngval.b . . . . 5 𝐵 = (Base‘𝑅)
43fvexi 6895 . . . 4 𝐵 ∈ V
54a1i 11 . . 3 (𝜑𝐵 ∈ V)
6 irngval.s . . 3 (𝜑𝑆𝐵)
75, 6ssexd 5320 . 2 (𝜑𝑆 ∈ V)
8 fvexd 6896 . . 3 (𝜑 → (Monic1p𝑈) ∈ V)
9 fvex 6894 . . . . . 6 (𝑂𝑓) ∈ V
109cnvex 7903 . . . . 5 (𝑂𝑓) ∈ V
1110imaex 7894 . . . 4 ((𝑂𝑓) “ { 0 }) ∈ V
1211rgenw 3066 . . 3 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V
13 iunexg 7937 . . 3 (((Monic1p𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V) → 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V)
148, 12, 13sylancl 587 . 2 (𝜑 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V)
15 oveq12 7405 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟s 𝑠) = (𝑅s 𝑆))
16 irngval.u . . . . . 6 𝑈 = (𝑅s 𝑆)
1715, 16eqtr4di 2791 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟s 𝑠) = 𝑈)
1817fveq2d 6885 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (Monic1p‘(𝑟s 𝑠)) = (Monic1p𝑈))
19 oveq12 7405 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = (𝑅 evalSub1 𝑆))
20 irngval.o . . . . . . . 8 𝑂 = (𝑅 evalSub1 𝑆)
2119, 20eqtr4di 2791 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂)
2221fveq1d 6883 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂𝑓))
2322cnveqd 5870 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂𝑓))
24 simpl 484 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
2524fveq2d 6885 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (0g𝑟) = (0g𝑅))
26 irngval.0 . . . . . . 7 0 = (0g𝑅)
2725, 26eqtr4di 2791 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (0g𝑟) = 0 )
2827sneqd 4636 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {(0g𝑟)} = { 0 })
2923, 28imaeq12d 6053 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g𝑟)}) = ((𝑂𝑓) “ { 0 }))
3018, 29iuneq12d 5021 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g𝑟)}) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
31 df-irng 32686 . . 3 IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g𝑟)}))
3230, 31ovmpoga 7549 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V) → (𝑅 IntgRing 𝑆) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
332, 7, 14, 32syl3anc 1372 1 (𝜑 → (𝑅 IntgRing 𝑆) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  wss 3946  {csn 4624   ciun 4993  ccnv 5671  cima 5675  cfv 6535  (class class class)co 7396  Basecbs 17131  s cress 17160  0gc0g 17372  Ringcrg 20038   evalSub1 ces1 21801  Monic1pcmn1 25612   IntgRing cirng 32685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-irng 32686
This theorem is referenced by:  elirng  32688
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