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Theorem irngss 33737
Description: All elements of a subring 𝑆 are integral over 𝑆. This is only true in the case of a nonzero ring, since there are no integral elements in a zero ring (see 0ringirng 33739). (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)
elirng.r (𝜑𝑅 ∈ CRing)
elirng.s (𝜑𝑆 ∈ (SubRing‘𝑅))
irngss.1 (𝜑𝑅 ∈ NzRing)
Assertion
Ref Expression
irngss (𝜑𝑆 ⊆ (𝑅 IntgRing 𝑆))

Proof of Theorem irngss
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . 4 ((𝜑𝑥𝑆) → 𝜑)
2 elirng.s . . . . . 6 (𝜑𝑆 ∈ (SubRing‘𝑅))
3 irngval.b . . . . . . 7 𝐵 = (Base‘𝑅)
43subrgss 20572 . . . . . 6 (𝑆 ∈ (SubRing‘𝑅) → 𝑆𝐵)
52, 4syl 17 . . . . 5 (𝜑𝑆𝐵)
65sselda 3983 . . . 4 ((𝜑𝑥𝑆) → 𝑥𝐵)
7 eqid 2737 . . . . . . . . . 10 (Poly1𝑅) = (Poly1𝑅)
8 irngval.u . . . . . . . . . 10 𝑈 = (𝑅s 𝑆)
9 eqid 2737 . . . . . . . . . 10 (Poly1𝑈) = (Poly1𝑈)
10 eqid 2737 . . . . . . . . . 10 (Base‘(Poly1𝑈)) = (Base‘(Poly1𝑈))
112adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12 eqid 2737 . . . . . . . . . 10 ((Poly1𝑅) ↾s (Base‘(Poly1𝑈))) = ((Poly1𝑅) ↾s (Base‘(Poly1𝑈)))
13 eqid 2737 . . . . . . . . . . 11 (var1𝑅) = (var1𝑅)
1413, 11, 8, 9, 10subrgvr1cl 22265 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (var1𝑅) ∈ (Base‘(Poly1𝑈)))
15 eqid 2737 . . . . . . . . . . 11 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
16 simpr 484 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥𝑆)
1715, 8, 7, 9, 10, 11, 16asclply1subcl 22378 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((algSc‘(Poly1𝑅))‘𝑥) ∈ (Base‘(Poly1𝑈)))
187, 8, 9, 10, 11, 12, 14, 17ressply1sub 33595 . . . . . . . . 9 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))((algSc‘(Poly1𝑅))‘𝑥)))
197, 8, 9, 10subrgply1 22234 . . . . . . . . . . . 12 (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1𝑈)) ∈ (SubRing‘(Poly1𝑅)))
20 subrgsubg 20577 . . . . . . . . . . . 12 ((Base‘(Poly1𝑈)) ∈ (SubRing‘(Poly1𝑅)) → (Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)))
212, 19, 203syl 18 . . . . . . . . . . 11 (𝜑 → (Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)))
2221adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)))
23 eqid 2737 . . . . . . . . . . 11 (-g‘(Poly1𝑅)) = (-g‘(Poly1𝑅))
24 eqid 2737 . . . . . . . . . . 11 (-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈)))) = (-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))
2523, 12, 24subgsub 19156 . . . . . . . . . 10 (((Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)) ∧ (var1𝑅) ∈ (Base‘(Poly1𝑈)) ∧ ((algSc‘(Poly1𝑅))‘𝑥) ∈ (Base‘(Poly1𝑈))) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))((algSc‘(Poly1𝑅))‘𝑥)))
2622, 14, 17, 25syl3anc 1373 . . . . . . . . 9 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))((algSc‘(Poly1𝑅))‘𝑥)))
2718, 26eqtr4d 2780 . . . . . . . 8 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))
28 elirng.r . . . . . . . . . . . . 13 (𝜑𝑅 ∈ CRing)
298subrgcrng 20575 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
3028, 2, 29syl2anc 584 . . . . . . . . . . . 12 (𝜑𝑈 ∈ CRing)
319ply1crng 22200 . . . . . . . . . . . 12 (𝑈 ∈ CRing → (Poly1𝑈) ∈ CRing)
3230, 31syl 17 . . . . . . . . . . 11 (𝜑 → (Poly1𝑈) ∈ CRing)
3332adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (Poly1𝑈) ∈ CRing)
3433crnggrpd 20244 . . . . . . . . 9 ((𝜑𝑥𝑆) → (Poly1𝑈) ∈ Grp)
35 eqid 2737 . . . . . . . . . 10 (-g‘(Poly1𝑈)) = (-g‘(Poly1𝑈))
3610, 35grpsubcl 19038 . . . . . . . . 9 (((Poly1𝑈) ∈ Grp ∧ (var1𝑅) ∈ (Base‘(Poly1𝑈)) ∧ ((algSc‘(Poly1𝑅))‘𝑥) ∈ (Base‘(Poly1𝑈))) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑈)))
3734, 14, 17, 36syl3anc 1373 . . . . . . . 8 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑈)))
3827, 37eqeltrrd 2842 . . . . . . 7 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑈)))
39 eqid 2737 . . . . . . . . 9 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
40 eqid 2737 . . . . . . . . 9 ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))
41 eqid 2737 . . . . . . . . 9 (eval1𝑅) = (eval1𝑅)
42 irngss.1 . . . . . . . . . 10 (𝜑𝑅 ∈ NzRing)
4342adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑅 ∈ NzRing)
4428adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑅 ∈ CRing)
45 eqid 2737 . . . . . . . . 9 (Monic1p𝑅) = (Monic1p𝑅)
46 eqid 2737 . . . . . . . . 9 (deg1𝑅) = (deg1𝑅)
47 irngval.0 . . . . . . . . 9 0 = (0g𝑅)
487, 39, 3, 13, 23, 15, 40, 41, 43, 44, 6, 45, 46, 47ply1remlem 26204 . . . . . . . 8 ((𝜑𝑥𝑆) → (((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Monic1p𝑅) ∧ ((deg1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = 1 ∧ (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 }) = {𝑥}))
4948simp1d 1143 . . . . . . 7 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Monic1p𝑅))
5038, 49elind 4200 . . . . . 6 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ ((Base‘(Poly1𝑈)) ∩ (Monic1p𝑅)))
51 eqid 2737 . . . . . . . 8 (Monic1p𝑈) = (Monic1p𝑈)
527, 8, 9, 10, 2, 45, 51ressply1mon1p 33593 . . . . . . 7 (𝜑 → (Monic1p𝑈) = ((Base‘(Poly1𝑈)) ∩ (Monic1p𝑅)))
5352adantr 480 . . . . . 6 ((𝜑𝑥𝑆) → (Monic1p𝑈) = ((Base‘(Poly1𝑈)) ∩ (Monic1p𝑅)))
5450, 53eleqtrrd 2844 . . . . 5 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Monic1p𝑈))
55 fveq2 6906 . . . . . . . 8 (𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) → (𝑂𝑓) = (𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
5655fveq1d 6908 . . . . . . 7 (𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) → ((𝑂𝑓)‘𝑥) = ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥))
5756eqeq1d 2739 . . . . . 6 (𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) → (((𝑂𝑓)‘𝑥) = 0 ↔ ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 ))
5857adantl 481 . . . . 5 (((𝜑𝑥𝑆) ∧ 𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) → (((𝑂𝑓)‘𝑥) = 0 ↔ ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 ))
59 irngval.o . . . . . . . . . 10 𝑂 = (𝑅 evalSub1 𝑆)
6059, 3, 9, 8, 10, 41, 44, 11ressply1evl 22374 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑂 = ((eval1𝑅) ↾ (Base‘(Poly1𝑈))))
6160fveq1d 6908 . . . . . . . 8 ((𝜑𝑥𝑆) → (𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = (((eval1𝑅) ↾ (Base‘(Poly1𝑈)))‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
6238fvresd 6926 . . . . . . . 8 ((𝜑𝑥𝑆) → (((eval1𝑅) ↾ (Base‘(Poly1𝑈)))‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
6361, 62eqtrd 2777 . . . . . . 7 ((𝜑𝑥𝑆) → (𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
6463fveq1d 6908 . . . . . 6 ((𝜑𝑥𝑆) → ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥))
65 eqid 2737 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
66 eqid 2737 . . . . . . . . 9 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
673fvexi 6920 . . . . . . . . . 10 𝐵 ∈ V
6867a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝐵 ∈ V)
6941, 7, 65, 3evl1rhm 22336 . . . . . . . . . . . 12 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
7039, 66rhmf 20485 . . . . . . . . . . . 12 ((eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
7128, 69, 703syl 18 . . . . . . . . . . 11 (𝜑 → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
7271adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
73 eqid 2737 . . . . . . . . . . . . . 14 (PwSer1𝑈) = (PwSer1𝑈)
74 eqid 2737 . . . . . . . . . . . . . 14 (Base‘(PwSer1𝑈)) = (Base‘(PwSer1𝑈))
757, 8, 9, 10, 2, 73, 74, 39ressply1bas2 22229 . . . . . . . . . . . . 13 (𝜑 → (Base‘(Poly1𝑈)) = ((Base‘(PwSer1𝑈)) ∩ (Base‘(Poly1𝑅))))
7675adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → (Base‘(Poly1𝑈)) = ((Base‘(PwSer1𝑈)) ∩ (Base‘(Poly1𝑅))))
7738, 76eleqtrd 2843 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ ((Base‘(PwSer1𝑈)) ∩ (Base‘(Poly1𝑅))))
7877elin2d 4205 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑅)))
7972, 78ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑥𝑆) → ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) ∈ (Base‘(𝑅s 𝐵)))
8065, 3, 66, 43, 68, 79pwselbas 17534 . . . . . . . 8 ((𝜑𝑥𝑆) → ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))):𝐵𝐵)
8180ffnd 6737 . . . . . . 7 ((𝜑𝑥𝑆) → ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) Fn 𝐵)
82 vsnid 4663 . . . . . . . 8 𝑥 ∈ {𝑥}
8348simp3d 1145 . . . . . . . 8 ((𝜑𝑥𝑆) → (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 }) = {𝑥})
8482, 83eleqtrrid 2848 . . . . . . 7 ((𝜑𝑥𝑆) → 𝑥 ∈ (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 }))
85 fniniseg 7080 . . . . . . . 8 (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) Fn 𝐵 → (𝑥 ∈ (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 }) ↔ (𝑥𝐵 ∧ (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 )))
8685simplbda 499 . . . . . . 7 ((((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) Fn 𝐵𝑥 ∈ (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 })) → (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 )
8781, 84, 86syl2anc 584 . . . . . 6 ((𝜑𝑥𝑆) → (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 )
8864, 87eqtrd 2777 . . . . 5 ((𝜑𝑥𝑆) → ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 )
8954, 58, 88rspcedvd 3624 . . . 4 ((𝜑𝑥𝑆) → ∃𝑓 ∈ (Monic1p𝑈)((𝑂𝑓)‘𝑥) = 0 )
9059, 8, 3, 47, 28, 2elirng 33736 . . . . 5 (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥𝐵 ∧ ∃𝑓 ∈ (Monic1p𝑈)((𝑂𝑓)‘𝑥) = 0 )))
9190biimpar 477 . . . 4 ((𝜑 ∧ (𝑥𝐵 ∧ ∃𝑓 ∈ (Monic1p𝑈)((𝑂𝑓)‘𝑥) = 0 )) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
921, 6, 89, 91syl12anc 837 . . 3 ((𝜑𝑥𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
9392ex 412 . 2 (𝜑 → (𝑥𝑆𝑥 ∈ (𝑅 IntgRing 𝑆)))
9493ssrdv 3989 1 (𝜑𝑆 ⊆ (𝑅 IntgRing 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951  {csn 4626  ccnv 5684  cres 5687  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1c1 11156  Basecbs 17247  s cress 17274  0gc0g 17484  s cpws 17491  Grpcgrp 18951  -gcsg 18953  SubGrpcsubg 19138  CRingccrg 20231   RingHom crh 20469  NzRingcnzr 20512  SubRingcsubrg 20569  algSccascl 21872  PwSer1cps1 22176  var1cv1 22177  Poly1cpl1 22178   evalSub1 ces1 22317  eval1ce1 22318  deg1cdg1 26093  Monic1pcmn1 26165   IntgRing cirng 33733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-rhm 20472  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-lmod 20860  df-lss 20930  df-lsp 20970  df-cnfld 21365  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evls1 22319  df-evl1 22320  df-mdeg 26094  df-deg1 26095  df-mon1 26170  df-irng 33734
This theorem is referenced by: (None)
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