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Theorem irngval 33700
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 3502 . 2 (𝜑𝑅 ∈ V)
3 irngval.b . . . . 5 𝐵 = (Base‘𝑅)
43fvexi 6921 . . . 4 𝐵 ∈ V
54a1i 11 . . 3 (𝜑𝐵 ∈ V)
6 irngval.s . . 3 (𝜑𝑆𝐵)
75, 6ssexd 5330 . 2 (𝜑𝑆 ∈ V)
8 fvexd 6922 . . 3 (𝜑 → (Monic1p𝑈) ∈ V)
9 fvex 6920 . . . . . 6 (𝑂𝑓) ∈ V
109cnvex 7948 . . . . 5 (𝑂𝑓) ∈ V
1110imaex 7937 . . . 4 ((𝑂𝑓) “ { 0 }) ∈ V
1211rgenw 3063 . . 3 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V
13 iunexg 7987 . . 3 (((Monic1p𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V) → 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V)
148, 12, 13sylancl 586 . 2 (𝜑 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V)
15 oveq12 7440 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟s 𝑠) = (𝑅s 𝑆))
16 irngval.u . . . . . 6 𝑈 = (𝑅s 𝑆)
1715, 16eqtr4di 2793 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟s 𝑠) = 𝑈)
1817fveq2d 6911 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (Monic1p‘(𝑟s 𝑠)) = (Monic1p𝑈))
19 oveq12 7440 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = (𝑅 evalSub1 𝑆))
20 irngval.o . . . . . . . 8 𝑂 = (𝑅 evalSub1 𝑆)
2119, 20eqtr4di 2793 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂)
2221fveq1d 6909 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂𝑓))
2322cnveqd 5889 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂𝑓))
24 simpl 482 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
2524fveq2d 6911 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (0g𝑟) = (0g𝑅))
26 irngval.0 . . . . . . 7 0 = (0g𝑅)
2725, 26eqtr4di 2793 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (0g𝑟) = 0 )
2827sneqd 4643 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {(0g𝑟)} = { 0 })
2923, 28imaeq12d 6081 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g𝑟)}) = ((𝑂𝑓) “ { 0 }))
3018, 29iuneq12d 5026 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g𝑟)}) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
31 df-irng 33699 . . 3 IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ 𝑓 ∈ (Monic1p‘(𝑟s 𝑠))(((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g𝑟)}))
3230, 31ovmpoga 7587 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }) ∈ V) → (𝑅 IntgRing 𝑆) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
332, 7, 14, 32syl3anc 1370 1 (𝜑 → (𝑅 IntgRing 𝑆) = 𝑓 ∈ (Monic1p𝑈)((𝑂𝑓) “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  {csn 4631   ciun 4996  ccnv 5688  cima 5692  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  0gc0g 17486  Ringcrg 20251   evalSub1 ces1 22333  Monic1pcmn1 26180   IntgRing cirng 33698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-irng 33699
This theorem is referenced by:  elirng  33701
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