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Theorem irngnzply1 32657
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 2732 . . . . . . . 8 (𝐸s 𝐹) = (𝐸s 𝐹)
3 eqid 2732 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
4 irngnzply1.1 . . . . . . . 8 0 = (0g𝐸)
5 irngnzply1.e . . . . . . . . 9 (𝜑𝐸 ∈ Field)
65fldcrngd 20279 . . . . . . . 8 (𝜑𝐸 ∈ CRing)
7 irngnzply1.f . . . . . . . . . 10 (𝜑𝐹 ∈ (SubDRing‘𝐸))
8 issdrg 20355 . . . . . . . . . 10 (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
97, 8sylib 217 . . . . . . . . 9 (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
109simp2d 1143 . . . . . . . 8 (𝜑𝐹 ∈ (SubRing‘𝐸))
111, 2, 3, 4, 6, 10elirng 32652 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸s 𝐹))((𝑂𝑝)‘𝑥) = 0 )))
1211biimpa 477 . . . . . 6 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → (𝑥 ∈ (Base‘𝐸) ∧ ∃𝑝 ∈ (Monic1p‘(𝐸s 𝐹))((𝑂𝑝)‘𝑥) = 0 ))
1312simprd 496 . . . . 5 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (Monic1p‘(𝐸s 𝐹))((𝑂𝑝)‘𝑥) = 0 )
14 eqid 2732 . . . . . . . . . 10 (Poly1‘(𝐸s 𝐹)) = (Poly1‘(𝐸s 𝐹))
15 eqid 2732 . . . . . . . . . 10 (Base‘(Poly1‘(𝐸s 𝐹))) = (Base‘(Poly1‘(𝐸s 𝐹)))
16 eqid 2732 . . . . . . . . . 10 (Monic1p‘(𝐸s 𝐹)) = (Monic1p‘(𝐸s 𝐹))
1714, 15, 16mon1pcl 25593 . . . . . . . . 9 (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
1817adantl 482 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
19 eqid 2732 . . . . . . . . . . . . 13 (𝐸s (Base‘𝐸)) = (𝐸s (Base‘𝐸))
201, 3, 19, 2, 14evls1rhm 21772 . . . . . . . . . . . 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 2732 . . . . . . . . . . . 12 (Base‘(𝐸s (Base‘𝐸))) = (Base‘(𝐸s (Base‘𝐸)))
2315, 22rhmf 20215 . . . . . . . . . . 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 6716 . . . . . . . . 9 (𝜑 → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
2625adantr 481 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
2718, 26eleqtrrd 2836 . . . . . . 7 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ∈ dom 𝑂)
28 eqid 2732 . . . . . . . . . 10 (0g‘(Poly1‘(𝐸s 𝐹))) = (0g‘(Poly1‘(𝐸s 𝐹)))
2914, 28, 16mon1pn0 25595 . . . . . . . . 9 (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) → 𝑝 ≠ (0g‘(Poly1‘(𝐸s 𝐹))))
3029adantl 482 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ≠ (0g‘(Poly1‘(𝐸s 𝐹))))
31 eqid 2732 . . . . . . . . . 10 (Poly1𝐸) = (Poly1𝐸)
32 irngnzply1.z . . . . . . . . . 10 𝑍 = (0g‘(Poly1𝐸))
3331, 2, 14, 15, 10, 32ressply10g 32560 . . . . . . . . 9 (𝜑𝑍 = (0g‘(Poly1‘(𝐸s 𝐹))))
3433adantr 481 . . . . . . . 8 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑍 = (0g‘(Poly1‘(𝐸s 𝐹))))
3530, 34neeqtrrd 3015 . . . . . . 7 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝𝑍)
36 eldifsn 4784 . . . . . . 7 (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) ↔ (𝑝 ∈ dom 𝑂𝑝𝑍))
3727, 35, 36sylanbrc 583 . . . . . 6 ((𝜑𝑝 ∈ (Monic1p‘(𝐸s 𝐹))) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
3837ad2ant2r 745 . . . . 5 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (dom 𝑂 ∖ {𝑍}))
395ad2antrr 724 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝐸 ∈ Field)
40 fvexd 6894 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (Base‘𝐸) ∈ V)
4124ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
4217ad2antrl 726 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
4341, 42ffvelcdmd 7073 . . . . . . . 8 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (𝑂𝑝) ∈ (Base‘(𝐸s (Base‘𝐸))))
4419, 3, 22, 39, 40, 43pwselbas 17419 . . . . . . 7 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (𝑂𝑝):(Base‘𝐸)⟶(Base‘𝐸))
4544ffnd 6706 . . . . . 6 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → (𝑂𝑝) Fn (Base‘𝐸))
4612simpld 495 . . . . . . 7 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 ∈ (Base‘𝐸))
4746adantr 481 . . . . . 6 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑥 ∈ (Base‘𝐸))
48 simprr 771 . . . . . 6 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → ((𝑂𝑝)‘𝑥) = 0 )
49 fniniseg 7047 . . . . . . 7 ((𝑂𝑝) Fn (Base‘𝐸) → (𝑥 ∈ ((𝑂𝑝) “ { 0 }) ↔ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂𝑝)‘𝑥) = 0 )))
5049biimpar 478 . . . . . 6 (((𝑂𝑝) Fn (Base‘𝐸) ∧ (𝑥 ∈ (Base‘𝐸) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑥 ∈ ((𝑂𝑝) “ { 0 }))
5145, 47, 48, 50syl12anc 835 . . . . 5 (((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) ∧ (𝑝 ∈ (Monic1p‘(𝐸s 𝐹)) ∧ ((𝑂𝑝)‘𝑥) = 0 )) → 𝑥 ∈ ((𝑂𝑝) “ { 0 }))
5213, 38, 51reximssdv 3172 . . . 4 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
53 eliun 4995 . . . 4 (𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }) ↔ ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
5452, 53sylibr 233 . . 3 ((𝜑𝑥 ∈ (𝐸 IntgRing 𝐹)) → 𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
55 nfv 1917 . . . . 5 𝑝𝜑
56 nfiu1 5025 . . . . . 6 𝑝 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })
5756nfcri 2890 . . . . 5 𝑝 𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })
5855, 57nfan 1902 . . . 4 𝑝(𝜑𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
595ad2antrr 724 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝐸 ∈ Field)
607ad2antrr 724 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝐹 ∈ (SubDRing‘𝐸))
61 eldifi 4123 . . . . . . . 8 (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝 ∈ dom 𝑂)
6261adantl 482 . . . . . . 7 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ dom 𝑂)
6362adantr 481 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑝 ∈ dom 𝑂)
64 eldifsni 4787 . . . . . . . 8 (𝑝 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑝𝑍)
6564adantl 482 . . . . . . 7 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝𝑍)
6665adantr 481 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑝𝑍)
675adantr 481 . . . . . . . . . 10 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝐸 ∈ Field)
68 fvexd 6894 . . . . . . . . . 10 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (Base‘𝐸) ∈ V)
6924adantr 481 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
7025adantr 481 . . . . . . . . . . . 12 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
7162, 70eleqtrd 2835 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → 𝑝 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
7269, 71ffvelcdmd 7073 . . . . . . . . . 10 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂𝑝) ∈ (Base‘(𝐸s (Base‘𝐸))))
7319, 3, 22, 67, 68, 72pwselbas 17419 . . . . . . . . 9 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂𝑝):(Base‘𝐸)⟶(Base‘𝐸))
7473ffnd 6706 . . . . . . . 8 ((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) → (𝑂𝑝) Fn (Base‘𝐸))
7549biimpa 477 . . . . . . . 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 496 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → ((𝑂𝑝)‘𝑥) = 0 )
7876simpld 495 . . . . . 6 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (Base‘𝐸))
791, 32, 4, 59, 60, 3, 63, 66, 77, 78irngnzply1lem 32656 . . . . 5 (((𝜑𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
8079adantllr 717 . . . 4 ((((𝜑𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })) ∧ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})) ∧ 𝑥 ∈ ((𝑂𝑝) “ { 0 })) → 𝑥 ∈ (𝐸 IntgRing 𝐹))
8153biimpi 215 . . . . 5 (𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }) → ∃𝑝 ∈ (dom 𝑂 ∖ {𝑍})𝑥 ∈ ((𝑂𝑝) “ { 0 }))
8281adantl 482 . . . 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 799 . 2 (𝜑 → (𝑥 ∈ (𝐸 IntgRing 𝐹) ↔ 𝑥 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 })))
8584eqrdv 2730 1 (𝜑 → (𝐸 IntgRing 𝐹) = 𝑝 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑝) “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070  Vcvv 3474  cdif 3942  {csn 4623   ciun 4991  ccnv 5669  dom cdm 5670  cima 5673   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7394  Basecbs 17128  s cress 17157  0gc0g 17369  s cpws 17376  CRingccrg 20017   RingHom crh 20200  DivRingcdr 20267  Fieldcfield 20268  SubRingcsubrg 20310  SubDRingcsdrg 20353  Poly1cpl1 21632   evalSub1 ces1 21763  Monic1pcmn1 25574   IntgRing cirng 32649
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171  ax-addf 11173  ax-mulf 11174
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-ofr 7655  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-tpos 8195  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-er 8688  df-map 8807  df-pm 8808  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fsupp 9347  df-sup 9421  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-fzo 13612  df-seq 13951  df-hash 14275  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-0g 17371  df-gsum 17372  df-prds 17377  df-pws 17379  df-mre 17514  df-mrc 17515  df-acs 17517  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-mhm 18649  df-submnd 18650  df-grp 18799  df-minusg 18800  df-sbg 18801  df-mulg 18925  df-subg 18977  df-ghm 19058  df-cntz 19149  df-cmn 19616  df-abl 19617  df-mgp 19949  df-ur 19966  df-srg 19970  df-ring 20018  df-cring 20019  df-oppr 20104  df-dvdsr 20125  df-unit 20126  df-invr 20156  df-rnghom 20203  df-drng 20269  df-field 20270  df-subrg 20312  df-sdrg 20354  df-lmod 20424  df-lss 20494  df-lsp 20534  df-rlreg 20837  df-cnfld 20881  df-assa 21343  df-asp 21344  df-ascl 21345  df-psr 21395  df-mvr 21396  df-mpl 21397  df-opsr 21399  df-evls 21566  df-evl 21567  df-psr1 21635  df-vr1 21636  df-ply1 21637  df-coe1 21638  df-evls1 21765  df-evl1 21766  df-mdeg 25501  df-deg1 25502  df-mon1 25579  df-uc1p 25580  df-irng 32650
This theorem is referenced by: (None)
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