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Theorem irngss 32653
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 32655). (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 483 . . . 4 ((𝜑𝑥𝑆) → 𝜑)
2 elirng.s . . . . . 6 (𝜑𝑆 ∈ (SubRing‘𝑅))
3 irngval.b . . . . . . 7 𝐵 = (Base‘𝑅)
43subrgss 20315 . . . . . 6 (𝑆 ∈ (SubRing‘𝑅) → 𝑆𝐵)
52, 4syl 17 . . . . 5 (𝜑𝑆𝐵)
65sselda 3979 . . . 4 ((𝜑𝑥𝑆) → 𝑥𝐵)
7 eqid 2732 . . . . . . . . . 10 (Poly1𝑅) = (Poly1𝑅)
8 irngval.u . . . . . . . . . 10 𝑈 = (𝑅s 𝑆)
9 eqid 2732 . . . . . . . . . 10 (Poly1𝑈) = (Poly1𝑈)
10 eqid 2732 . . . . . . . . . 10 (Base‘(Poly1𝑈)) = (Base‘(Poly1𝑈))
112adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑆) → 𝑆 ∈ (SubRing‘𝑅))
12 eqid 2732 . . . . . . . . . 10 ((Poly1𝑅) ↾s (Base‘(Poly1𝑈))) = ((Poly1𝑅) ↾s (Base‘(Poly1𝑈)))
13 eqid 2732 . . . . . . . . . . 11 (var1𝑅) = (var1𝑅)
1413, 11, 8, 9, 10subrgvr1cl 21717 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (var1𝑅) ∈ (Base‘(Poly1𝑈)))
15 eqid 2732 . . . . . . . . . . 11 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
16 simpr 485 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥𝑆)
1715, 8, 7, 9, 10, 11, 16asclply1subcl 32564 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((algSc‘(Poly1𝑅))‘𝑥) ∈ (Base‘(Poly1𝑈)))
187, 8, 9, 10, 11, 12, 14, 17ressply1sub 32563 . . . . . . . . 9 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))((algSc‘(Poly1𝑅))‘𝑥)))
197, 8, 9, 10subrgply1 21688 . . . . . . . . . . . 12 (𝑆 ∈ (SubRing‘𝑅) → (Base‘(Poly1𝑈)) ∈ (SubRing‘(Poly1𝑅)))
20 subrgsubg 20320 . . . . . . . . . . . 12 ((Base‘(Poly1𝑈)) ∈ (SubRing‘(Poly1𝑅)) → (Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)))
212, 19, 203syl 18 . . . . . . . . . . 11 (𝜑 → (Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)))
2221adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (Base‘(Poly1𝑈)) ∈ (SubGrp‘(Poly1𝑅)))
23 eqid 2732 . . . . . . . . . . 11 (-g‘(Poly1𝑅)) = (-g‘(Poly1𝑅))
24 eqid 2732 . . . . . . . . . . 11 (-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈)))) = (-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))
2523, 12, 24subgsub 18992 . . . . . . . . . 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 1371 . . . . . . . . 9 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘((Poly1𝑅) ↾s (Base‘(Poly1𝑈))))((algSc‘(Poly1𝑅))‘𝑥)))
2718, 26eqtr4d 2775 . . . . . . . 8 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))
28 elirng.r . . . . . . . . . . . . 13 (𝜑𝑅 ∈ CRing)
298subrgcrng 20318 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑈 ∈ CRing)
3028, 2, 29syl2anc 584 . . . . . . . . . . . 12 (𝜑𝑈 ∈ CRing)
319ply1crng 21653 . . . . . . . . . . . 12 (𝑈 ∈ CRing → (Poly1𝑈) ∈ CRing)
3230, 31syl 17 . . . . . . . . . . 11 (𝜑 → (Poly1𝑈) ∈ CRing)
3332adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (Poly1𝑈) ∈ CRing)
3433crnggrpd 20030 . . . . . . . . 9 ((𝜑𝑥𝑆) → (Poly1𝑈) ∈ Grp)
35 eqid 2732 . . . . . . . . . 10 (-g‘(Poly1𝑈)) = (-g‘(Poly1𝑈))
3610, 35grpsubcl 18879 . . . . . . . . 9 (((Poly1𝑈) ∈ Grp ∧ (var1𝑅) ∈ (Base‘(Poly1𝑈)) ∧ ((algSc‘(Poly1𝑅))‘𝑥) ∈ (Base‘(Poly1𝑈))) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑈)))
3734, 14, 17, 36syl3anc 1371 . . . . . . . 8 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑈))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑈)))
3827, 37eqeltrrd 2834 . . . . . . 7 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑈)))
39 eqid 2732 . . . . . . . . 9 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
40 eqid 2732 . . . . . . . . 9 ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))
41 eqid 2732 . . . . . . . . 9 (eval1𝑅) = (eval1𝑅)
42 irngss.1 . . . . . . . . . 10 (𝜑𝑅 ∈ NzRing)
4342adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑅 ∈ NzRing)
4428adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑅 ∈ CRing)
45 eqid 2732 . . . . . . . . 9 (Monic1p𝑅) = (Monic1p𝑅)
46 eqid 2732 . . . . . . . . 9 ( deg1𝑅) = ( deg1𝑅)
47 irngval.0 . . . . . . . . 9 0 = (0g𝑅)
487, 39, 3, 13, 23, 15, 40, 41, 43, 44, 6, 45, 46, 47ply1remlem 25611 . . . . . . . 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 1142 . . . . . . 7 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Monic1p𝑅))
5038, 49elind 4191 . . . . . 6 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ ((Base‘(Poly1𝑈)) ∩ (Monic1p𝑅)))
51 eqid 2732 . . . . . . . 8 (Monic1p𝑈) = (Monic1p𝑈)
527, 8, 9, 10, 2, 45, 51ressply1mon1p 32561 . . . . . . 7 (𝜑 → (Monic1p𝑈) = ((Base‘(Poly1𝑈)) ∩ (Monic1p𝑅)))
5352adantr 481 . . . . . 6 ((𝜑𝑥𝑆) → (Monic1p𝑈) = ((Base‘(Poly1𝑈)) ∩ (Monic1p𝑅)))
5450, 53eleqtrrd 2836 . . . . 5 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Monic1p𝑈))
55 fveq2 6879 . . . . . . . 8 (𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) → (𝑂𝑓) = (𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
5655fveq1d 6881 . . . . . . 7 (𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) → ((𝑂𝑓)‘𝑥) = ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥))
5756eqeq1d 2734 . . . . . 6 (𝑓 = ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) → (((𝑂𝑓)‘𝑥) = 0 ↔ ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 ))
5857adantl 482 . . . . 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 32554 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑂 = ((eval1𝑅) ↾ (Base‘(Poly1𝑈))))
6160fveq1d 6881 . . . . . . . 8 ((𝜑𝑥𝑆) → (𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = (((eval1𝑅) ↾ (Base‘(Poly1𝑈)))‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
6238fvresd 6899 . . . . . . . 8 ((𝜑𝑥𝑆) → (((eval1𝑅) ↾ (Base‘(Poly1𝑈)))‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
6361, 62eqtrd 2772 . . . . . . 7 ((𝜑𝑥𝑆) → (𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) = ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))))
6463fveq1d 6881 . . . . . 6 ((𝜑𝑥𝑆) → ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥))
65 eqid 2732 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
66 eqid 2732 . . . . . . . . 9 (Base‘(𝑅s 𝐵)) = (Base‘(𝑅s 𝐵))
673fvexi 6893 . . . . . . . . . 10 𝐵 ∈ V
6867a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝐵 ∈ V)
6941, 7, 65, 3evl1rhm 21782 . . . . . . . . . . . 12 (𝑅 ∈ CRing → (eval1𝑅) ∈ ((Poly1𝑅) RingHom (𝑅s 𝐵)))
7039, 66rhmf 20215 . . . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (eval1𝑅):(Base‘(Poly1𝑅))⟶(Base‘(𝑅s 𝐵)))
73 eqid 2732 . . . . . . . . . . . . . 14 (PwSer1𝑈) = (PwSer1𝑈)
74 eqid 2732 . . . . . . . . . . . . . 14 (Base‘(PwSer1𝑈)) = (Base‘(PwSer1𝑈))
757, 8, 9, 10, 2, 73, 74, 39ressply1bas2 21683 . . . . . . . . . . . . 13 (𝜑 → (Base‘(Poly1𝑈)) = ((Base‘(PwSer1𝑈)) ∩ (Base‘(Poly1𝑅))))
7675adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → (Base‘(Poly1𝑈)) = ((Base‘(PwSer1𝑈)) ∩ (Base‘(Poly1𝑅))))
7738, 76eleqtrd 2835 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ ((Base‘(PwSer1𝑈)) ∩ (Base‘(Poly1𝑅))))
7877elin2d 4196 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)) ∈ (Base‘(Poly1𝑅)))
7972, 78ffvelcdmd 7073 . . . . . . . . 9 ((𝜑𝑥𝑆) → ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) ∈ (Base‘(𝑅s 𝐵)))
8065, 3, 66, 43, 68, 79pwselbas 17419 . . . . . . . 8 ((𝜑𝑥𝑆) → ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))):𝐵𝐵)
8180ffnd 6706 . . . . . . 7 ((𝜑𝑥𝑆) → ((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) Fn 𝐵)
82 vsnid 4660 . . . . . . . 8 𝑥 ∈ {𝑥}
8348simp3d 1144 . . . . . . . 8 ((𝜑𝑥𝑆) → (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 }) = {𝑥})
8482, 83eleqtrrid 2840 . . . . . . 7 ((𝜑𝑥𝑆) → 𝑥 ∈ (((eval1𝑅)‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥))) “ { 0 }))
85 fniniseg 7047 . . . . . . . 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 500 . . . . . . 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 2772 . . . . 5 ((𝜑𝑥𝑆) → ((𝑂‘((var1𝑅)(-g‘(Poly1𝑅))((algSc‘(Poly1𝑅))‘𝑥)))‘𝑥) = 0 )
8954, 58, 88rspcedvd 3612 . . . 4 ((𝜑𝑥𝑆) → ∃𝑓 ∈ (Monic1p𝑈)((𝑂𝑓)‘𝑥) = 0 )
9059, 8, 3, 47, 28, 2elirng 32652 . . . . 5 (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥𝐵 ∧ ∃𝑓 ∈ (Monic1p𝑈)((𝑂𝑓)‘𝑥) = 0 )))
9190biimpar 478 . . . 4 ((𝜑 ∧ (𝑥𝐵 ∧ ∃𝑓 ∈ (Monic1p𝑈)((𝑂𝑓)‘𝑥) = 0 )) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
921, 6, 89, 91syl12anc 835 . . 3 ((𝜑𝑥𝑆) → 𝑥 ∈ (𝑅 IntgRing 𝑆))
9392ex 413 . 2 (𝜑 → (𝑥𝑆𝑥 ∈ (𝑅 IntgRing 𝑆)))
9493ssrdv 3985 1 (𝜑𝑆 ⊆ (𝑅 IntgRing 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  cin 3944  wss 3945  {csn 4623  ccnv 5669  cres 5672  cima 5673   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7394  1c1 11095  Basecbs 17128  s cress 17157  0gc0g 17369  s cpws 17376  Grpcgrp 18796  -gcsg 18798  SubGrpcsubg 18974  CRingccrg 20017   RingHom crh 20200  NzRingcnzr 20243  SubRingcsubrg 20310  algSccascl 21342  PwSer1cps1 21630  var1cv1 21631  Poly1cpl1 21632   evalSub1 ces1 21763  eval1ce1 21764   deg1 cdg1 25500  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-pre-sup 11172  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-nzr 20244  df-subrg 20312  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-irng 32650
This theorem is referenced by: (None)
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