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Theorem irngnzply1 33705
Description: In the case of a field 𝐸, the roots of nonzero polynomials 𝑝 with coefficients in a subfield 𝐹 are exactly the integral elements over 𝐹. Roots of nonzero polynomials are called algebraic numbers, so this shows that in the case of a field, elements integral over 𝐹 are exactly the algebraic numbers. In this formula, dom 𝑂 represents the polynomials, and 𝑍 the zero polynomial. (Contributed by Thierry Arnoux, 5-Feb-2025.)
Hypotheses
Ref Expression
irngnzply1.o 𝑂 = (𝐸 evalSub1 𝐹)
irngnzply1.z 𝑍 = (0g‘(Poly1𝐸))
irngnzply1.1 0 = (0g𝐸)
irngnzply1.e (𝜑𝐸 ∈ Field)
irngnzply1.f (𝜑𝐹 ∈ (SubDRing‘𝐸))
Assertion
Ref Expression
irngnzply1 (𝜑 → (𝐸 IntgRing 𝐹) = 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
Distinct variable groups:   𝐸,𝑝   𝐹,𝑝   𝑂,𝑝   𝜑,𝑝
Allowed substitution hints:   0 (𝑝)   𝑍(𝑝)

Proof of Theorem irngnzply1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 irngnzply1.o . . . . . . . 8 𝑂 = (𝐸 evalSub1 𝐹)
2 eqid 2734 . . . . . . . 8 (𝐸s 𝐹) = (𝐸s 𝐹)
3 eqid 2734 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
4 irngnzply1.1 . . . . . . . 8 0 = (0g𝐸)
5 irngnzply1.e . . . . . . . . 9 (𝜑𝐸 ∈ Field)
65fldcrngd 20758 . . . . . . . 8 (𝜑𝐸 ∈ CRing)
7 irngnzply1.f . . . . . . . . . 10 (𝜑𝐹 ∈ (SubDRing‘𝐸))
8 issdrg 20805 . . . . . . . . . 10 (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
97, 8sylib 218 . . . . . . . . 9 (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
109simp2d 1142 . . . . . . . 8 (𝜑𝐹 ∈ (SubRing‘𝐸))
111, 2, 3, 4, 6, 10elirng 33700 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸s 𝐹))((𝑂𝑝)‘𝑥) = 0 )))
1211biimpa 476 . . . . . 6 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸s 𝐹))((𝑂𝑝)‘𝑥) = 0 ))
1312simprd 495 . . . . 5 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (Monic1p‘(𝐸s 𝐹))((𝑂𝑝)‘𝑥) = 0 )
14 eqid 2734 . . . . . . . . . 10 (Poly1‘(𝐸s 𝐹)) = (Poly1‘(𝐸s 𝐹))
15 eqid 2734 . . . . . . . . . 10 (Base‘(Poly1‘(𝐸s 𝐹))) = (Base‘(Poly1‘(𝐸s 𝐹)))
16 eqid 2734 . . . . . . . . . 10 (Monic1p‘(𝐸s 𝐹)) = (Monic1p‘(𝐸s 𝐹))
1714, 15, 16mon1pcl 26198 . . . . . . . . 9 (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
1817adantl 481 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
19 eqid 2734 . . . . . . . . . . . . 13 (𝐸s (Base‘𝐸)) = (𝐸s (Base‘𝐸))
201, 3, 19, 2, 14evls1rhm 22341 . . . . . . . . . . . 12 ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))))
216, 10, 20syl2anc 584 . . . . . . . . . . 11 (𝜑𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))))
22 eqid 2734 . . . . . . . . . . . 12 (Base‘(𝐸s (Base‘𝐸))) = (Base‘(𝐸s (Base‘𝐸)))
2315, 22rhmf 20501 . . . . . . . . . . 11 (𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
2421, 23syl 17 . . . . . . . . . 10 (𝜑𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
2524fdmd 6746 . . . . . . . . 9 (𝜑 → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
2625adantr 480 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
2718, 26eleqtrrd 2841 . . . . . . 7 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ∈ dom 𝑂)
28 eqid 2734 . . . . . . . . . 10 (0g‘(Poly1‘(𝐸s 𝐹))) = (0g‘(Poly1‘(𝐸s 𝐹)))
2914, 28, 16mon1pn0 26200 . . . . . . . . 9 (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) → 𝑝 ≠ (0g‘(Poly1‘(𝐸s 𝐹))))
3029adantl 481 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ≠ (0g‘(Poly1‘(𝐸s 𝐹))))
31 eqid 2734 . . . . . . . . . 10 (Poly1𝐸) = (Poly1𝐸)
32 irngnzply1.z . . . . . . . . . 10 𝑍 = (0g‘(Poly1𝐸))
3331, 2, 14, 15, 10, 32ressply10g 33571 . . . . . . . . 9 (𝜑𝑍 = (0g‘(Poly1‘(𝐸s 𝐹))))
3433adantr 480 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑍 = (0g‘(Poly1‘(𝐸s 𝐹))))
3530, 34neeqtrrd 3012 . . . . . . 7 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝𝑍)
36 eldifsn 4790 . . . . . . 7 (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) ↔ (𝑝 ∈ dom 𝑂𝑝𝑍))
3727, 35, 36sylanbrc 583 . . . . . 6 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
3837ad2ant2r 747 . . . . 5 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
395ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝐸 ∈ Field)
40 fvexd 6921 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (Base‘𝐸) ∈ V)
4124ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
4217ad2antrl 728 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
4341, 42ffvelcdmd 7104 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (𝑂𝑝) ∈ (Base‘(𝐸s (Base‘𝐸))))
4419, 3, 22, 39, 40, 43pwselbas 17535 . . . . . . 7 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (𝑂𝑝):(Base‘𝐸)⟶(Base‘𝐸))
4544ffnd 6737 . . . . . 6 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (𝑂𝑝) Fn (Base‘𝐸))
4612simpld 494 . . . . . . 7 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 ∈ (Base‘𝐸))
4746adantr 480 . . . . . 6 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (Base‘𝐸))
48 simprr 773 . . . . . 6 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → ((𝑂𝑝)‘𝑥) = 0 )
49 fniniseg 7079 . . . . . . 7 ((𝑂𝑝) Fn (Base‘𝐸) → (𝑥 ∈ ((𝑂𝑝) “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂𝑝)‘𝑥) = 0 )))
5049biimpar 477 . . . . . 6 (((𝑂𝑝) Fn (Base‘𝐸) ∧ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑥 ∈ ((𝑂𝑝) “ { 0 }))
5145, 47, 48, 50syl12anc 837 . . . . 5 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑥 ∈ ((𝑂𝑝) “ { 0 }))
5213, 38, 51reximssdv 3170 . . . 4 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
53 eliun 4999 . . . 4 (𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
5452, 53sylibr 234 . . 3 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
55 nfv 1911 . . . . 5 𝑝𝜑
56 nfiu1 5031 . . . . . 6 𝑝 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })
5756nfcri 2894 . . . . 5 𝑝 𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })
5855, 57nfan 1896 . . . 4 𝑝(𝜑𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
595ad2antrr 726 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝐸 ∈ Field)
607ad2antrr 726 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝐹 ∈ (SubDRing‘𝐸))
61 eldifi 4140 . . . . . . . 8 (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝 ∈ dom 𝑂)
6261adantl 481 . . . . . . 7 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ dom 𝑂)
6362adantr 480 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑝 ∈ dom 𝑂)
64 eldifsni 4794 . . . . . . . 8 (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝𝑍)
6564adantl 481 . . . . . . 7 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝𝑍)
6665adantr 480 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑝𝑍)
675adantr 480 . . . . . . . . . 10 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝐸 ∈ Field)
68 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (Base‘𝐸) ∈ V)
6924adantr 480 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
7025adantr 480 . . . . . . . . . . . 12 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
7162, 70eleqtrd 2840 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
7269, 71ffvelcdmd 7104 . . . . . . . . . 10 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂𝑝) ∈ (Base‘(𝐸s (Base‘𝐸))))
7319, 3, 22, 67, 68, 72pwselbas 17535 . . . . . . . . 9 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂𝑝):(Base‘𝐸)⟶(Base‘𝐸))
7473ffnd 6737 . . . . . . . 8 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂𝑝) Fn (Base‘𝐸))
7549biimpa 476 . . . . . . . 8 (((𝑂𝑝) Fn (Base‘𝐸) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂𝑝)‘𝑥) = 0 ))
7674, 75sylan 580 . . . . . . 7 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂𝑝)‘𝑥) = 0 ))
7776simprd 495 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → ((𝑂𝑝)‘𝑥) = 0 )
7876simpld 494 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (Base‘𝐸))
791, 32, 4, 59, 60, 3, 63, 66, 77, 78irngnzply1lem 33704 . . . . 5 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
8079adantllr 719 . . . 4 ((((𝜑𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })) ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
8153biimpi 216 . . . . 5 (𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
8281adantl 481 . . . 4 ((𝜑𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
8358, 80, 82r19.29af 3265 . . 3 ((𝜑𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
8454, 83impbida 801 . 2 (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ 𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })))
8584eqrdv 2732 1 (𝜑 → (𝐸 IntgRing 𝐹) = 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cdif 3959  {csn 4630   ciun 4995  ccnv 5687  dom cdm 5688  cima 5691   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  Basecbs 17244  s cress 17273  0gc0g 17485  s cpws 17492  CRingccrg 20251   RingHom crh 20485  SubRingcsubrg 20585  DivRingcdr 20745  Fieldcfield 20746  SubDRingcsdrg 20803  Poly1cpl1 22193   evalSub1 ces1 22332  Monic1pcmn1 26179   IntgRing cirng 33697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-rlreg 20710  df-drng 20747  df-field 20748  df-sdrg 20804  df-lmod 20876  df-lss 20947  df-lsp 20987  df-cnfld 21382  df-assa 21890  df-asp 21891  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-evls 22115  df-evl 22116  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-evls1 22334  df-evl1 22335  df-mdeg 26108  df-deg1 26109  df-mon1 26184  df-uc1p 26185  df-irng 33698
This theorem is referenced by:  irngnminplynz  33719
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