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Theorem algextdeglem4 33761
Description: Lemma for algextdeg 33766. By lmhmqusker 33445, the surjective module homomorphism 𝐺 described in algextdeglem2 33759 induces an isomorphism with the quotient space. Therefore, the dimension of that quotient space 𝑃 / 𝑍 is the degree of the algebraic field extension. (Contributed by Thierry Arnoux, 2-Apr-2025.)
Hypotheses
Ref Expression
algextdeg.k 𝐾 = (𝐸s 𝐹)
algextdeg.l 𝐿 = (𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴})))
algextdeg.d 𝐷 = (deg1𝐸)
algextdeg.m 𝑀 = (𝐸 minPoly 𝐹)
algextdeg.f (𝜑𝐸 ∈ Field)
algextdeg.e (𝜑𝐹 ∈ (SubDRing‘𝐸))
algextdeg.a (𝜑𝐴 ∈ (𝐸 IntgRing 𝐹))
algextdeglem.o 𝑂 = (𝐸 evalSub1 𝐹)
algextdeglem.y 𝑃 = (Poly1𝐾)
algextdeglem.u 𝑈 = (Base‘𝑃)
algextdeglem.g 𝐺 = (𝑝𝑈 ↦ ((𝑂𝑝)‘𝐴))
algextdeglem.n 𝑁 = (𝑥𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
algextdeglem.z 𝑍 = (𝐺 “ {(0g𝐿)})
algextdeglem.q 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
algextdeglem.j 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ (𝐺𝑝))
Assertion
Ref Expression
algextdeglem4 (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))
Distinct variable groups:   𝐴,𝑝   𝐸,𝑝   𝐹,𝑝,𝑥   𝐺,𝑝,𝑥   𝐽,𝑝,𝑥   𝐾,𝑝   𝐿,𝑝,𝑥   𝑥,𝑁   𝑂,𝑝   𝑃,𝑝,𝑥   𝑄,𝑝,𝑥   𝑈,𝑝,𝑥   𝑍,𝑝,𝑥   𝜑,𝑝,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑝)   𝐸(𝑥)   𝐾(𝑥)   𝑀(𝑥,𝑝)   𝑁(𝑝)   𝑂(𝑥)

Proof of Theorem algextdeglem4
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 algextdeg.e . . . . . . . 8 (𝜑𝐹 ∈ (SubDRing‘𝐸))
2 issdrg 20789 . . . . . . . 8 (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
31, 2sylib 218 . . . . . . 7 (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
43simp2d 1144 . . . . . 6 (𝜑𝐹 ∈ (SubRing‘𝐸))
5 subrgsubg 20577 . . . . . 6 (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6 eqid 2737 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
76subgss 19145 . . . . . 6 (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
84, 5, 73syl 18 . . . . 5 (𝜑𝐹 ⊆ (Base‘𝐸))
9 algextdeg.k . . . . . 6 𝐾 = (𝐸s 𝐹)
109, 6ressbas2 17283 . . . . 5 (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
118, 10syl 17 . . . 4 (𝜑𝐹 = (Base‘𝐾))
1211fveq2d 6910 . . 3 (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
1312fveq2d 6910 . 2 (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14 eqid 2737 . . . . 5 (0g‘((subringAlg ‘𝐿)‘𝐹)) = (0g‘((subringAlg ‘𝐿)‘𝐹))
15 algextdeg.l . . . . . 6 𝐿 = (𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴})))
16 algextdeg.d . . . . . 6 𝐷 = (deg1𝐸)
17 algextdeg.m . . . . . 6 𝑀 = (𝐸 minPoly 𝐹)
18 algextdeg.f . . . . . 6 (𝜑𝐸 ∈ Field)
19 algextdeg.a . . . . . 6 (𝜑𝐴 ∈ (𝐸 IntgRing 𝐹))
20 algextdeglem.o . . . . . 6 𝑂 = (𝐸 evalSub1 𝐹)
21 algextdeglem.y . . . . . 6 𝑃 = (Poly1𝐾)
22 algextdeglem.u . . . . . 6 𝑈 = (Base‘𝑃)
23 algextdeglem.g . . . . . 6 𝐺 = (𝑝𝑈 ↦ ((𝑂𝑝)‘𝐴))
24 algextdeglem.n . . . . . 6 𝑁 = (𝑥𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
25 algextdeglem.z . . . . . 6 𝑍 = (𝐺 “ {(0g𝐿)})
26 algextdeglem.q . . . . . 6 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
27 algextdeglem.j . . . . . 6 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ (𝐺𝑝))
289, 15, 16, 17, 18, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27algextdeglem2 33759 . . . . 5 (𝜑𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29 eqid 2737 . . . . 5 (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30 eqid 2737 . . . . 5 (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
319fveq2i 6909 . . . . . . . . . . 11 (Poly1𝐾) = (Poly1‘(𝐸s 𝐹))
3221, 31eqtri 2765 . . . . . . . . . 10 𝑃 = (Poly1‘(𝐸s 𝐹))
3318adantr 480 . . . . . . . . . 10 ((𝜑𝑝𝑈) → 𝐸 ∈ Field)
341adantr 480 . . . . . . . . . 10 ((𝜑𝑝𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
35 eqid 2737 . . . . . . . . . . . . 13 (0g𝐸) = (0g𝐸)
3618fldcrngd 20742 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ CRing)
3720, 9, 6, 35, 36, 4irngssv 33738 . . . . . . . . . . . 12 (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
3837, 19sseldd 3984 . . . . . . . . . . 11 (𝜑𝐴 ∈ (Base‘𝐸))
3938adantr 480 . . . . . . . . . 10 ((𝜑𝑝𝑈) → 𝐴 ∈ (Base‘𝐸))
40 simpr 484 . . . . . . . . . 10 ((𝜑𝑝𝑈) → 𝑝𝑈)
416, 20, 32, 22, 33, 34, 39, 40evls1fldgencl 33720 . . . . . . . . 9 ((𝜑𝑝𝑈) → ((𝑂𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
4241ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑝𝑈 ((𝑂𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
4323rnmptss 7143 . . . . . . . 8 (∀𝑝𝑈 ((𝑂𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
4442, 43syl 17 . . . . . . 7 (𝜑 → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
4518flddrngd 20741 . . . . . . . 8 (𝜑𝐸 ∈ DivRing)
4620, 32, 6, 22, 36, 4, 38, 23evls1maprhm 22380 . . . . . . . . . 10 (𝜑𝐺 ∈ (𝑃 RingHom 𝐸))
47 rnrhmsubrg 20605 . . . . . . . . . 10 (𝐺 ∈ (𝑃 RingHom 𝐸) → ran 𝐺 ∈ (SubRing‘𝐸))
4846, 47syl 17 . . . . . . . . 9 (𝜑 → ran 𝐺 ∈ (SubRing‘𝐸))
4915oveq1i 7441 . . . . . . . . . . 11 (𝐿s ran 𝐺) = ((𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺)
50 ovex 7464 . . . . . . . . . . . 12 (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V
51 ressabs 17294 . . . . . . . . . . . 12 (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → ((𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺) = (𝐸s ran 𝐺))
5250, 44, 51sylancr 587 . . . . . . . . . . 11 (𝜑 → ((𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺) = (𝐸s ran 𝐺))
5349, 52eqtrid 2789 . . . . . . . . . 10 (𝜑 → (𝐿s ran 𝐺) = (𝐸s ran 𝐺))
54 eqid 2737 . . . . . . . . . . . . . . 15 (0g𝐿) = (0g𝐿)
5538snssd 4809 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → {𝐴} ⊆ (Base‘𝐸))
568, 55unssd 4192 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
576, 45, 56fldgensdrg 33316 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58 issdrg 20789 . . . . . . . . . . . . . . . . . . 19 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
5957, 58sylib 218 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
6059simp2d 1144 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
6115resrhm2b 20602 . . . . . . . . . . . . . . . . . 18 (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → (𝐺 ∈ (𝑃 RingHom 𝐸) ↔ 𝐺 ∈ (𝑃 RingHom 𝐿)))
6261biimpa 476 . . . . . . . . . . . . . . . . 17 ((((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∧ 𝐺 ∈ (𝑃 RingHom 𝐸)) → 𝐺 ∈ (𝑃 RingHom 𝐿))
6360, 44, 46, 62syl21anc 838 . . . . . . . . . . . . . . . 16 (𝜑𝐺 ∈ (𝑃 RingHom 𝐿))
64 rhmghm 20484 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝑃 RingHom 𝐿) → 𝐺 ∈ (𝑃 GrpHom 𝐿))
6563, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ (𝑃 GrpHom 𝐿))
6654, 65, 25, 26, 27, 22, 24ghmquskerco 19302 . . . . . . . . . . . . . 14 (𝜑𝐺 = (𝐽𝑁))
6766rneqd 5949 . . . . . . . . . . . . 13 (𝜑 → ran 𝐺 = ran (𝐽𝑁))
6826a1i 11 . . . . . . . . . . . . . . . 16 (𝜑𝑄 = (𝑃 /s (𝑃 ~QG 𝑍)))
6922a1i 11 . . . . . . . . . . . . . . . 16 (𝜑𝑈 = (Base‘𝑃))
70 ovexd 7466 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 ~QG 𝑍) ∈ V)
713simp3d 1145 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸s 𝐹) ∈ DivRing)
7232, 71ply1lvec 33585 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LVec)
7368, 69, 70, 72qusbas 17590 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
7554ghmker 19260 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ (𝑃 GrpHom 𝐿) → (𝐺 “ {(0g𝐿)}) ∈ (NrmSGrp‘𝑃))
7665, 75syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺 “ {(0g𝐿)}) ∈ (NrmSGrp‘𝑃))
7725, 76eqeltrid 2845 . . . . . . . . . . . . . . . 16 (𝜑𝑍 ∈ (NrmSGrp‘𝑃))
7822, 74, 24, 77qusrn 33437 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
8020, 32, 6, 22, 36, 4, 38, 23, 79evls1maplmhm 22381 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
8180elexd 3504 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 ∈ V)
8281adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
8382imaexd 7938 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (Base‘𝑄)) → (𝐺𝑝) ∈ V)
8483uniexd 7762 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (Base‘𝑄)) → (𝐺𝑝) ∈ V)
8527, 84dmmptd 6713 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐽 = (Base‘𝑄))
8673, 78, 853eqtr4rd 2788 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐽 = ran 𝑁)
87 rncoeq 5990 . . . . . . . . . . . . . 14 (dom 𝐽 = ran 𝑁 → ran (𝐽𝑁) = ran 𝐽)
8886, 87syl 17 . . . . . . . . . . . . 13 (𝜑 → ran (𝐽𝑁) = ran 𝐽)
8967, 88eqtrd 2777 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 = ran 𝐽)
9089oveq2d 7447 . . . . . . . . . . 11 (𝜑 → (𝐿s ran 𝐺) = (𝐿s ran 𝐽))
91 eqid 2737 . . . . . . . . . . . 12 (𝐿s ran 𝐽) = (𝐿s ran 𝐽)
929subrgcrng 20575 . . . . . . . . . . . . . . 15 ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
9336, 4, 92syl2anc 584 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ CRing)
9421ply1crng 22200 . . . . . . . . . . . . . 14 (𝐾 ∈ CRing → 𝑃 ∈ CRing)
9593, 94syl 17 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ CRing)
9654, 63, 25, 26, 27, 95rhmquskerlem 33453 . . . . . . . . . . . 12 (𝜑𝐽 ∈ (𝑄 RingHom 𝐿))
9720, 32, 6, 22, 36, 4, 38, 23evls1maprnss 22382 . . . . . . . . . . . . . . 15 (𝜑𝐹 ⊆ ran 𝐺)
98 eqid 2737 . . . . . . . . . . . . . . . . . 18 (1r𝐸) = (1r𝐸)
999, 98subrg1 20582 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) = (1r𝐾))
1004, 99syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1r𝐸) = (1r𝐾))
10198subrg1cl 20580 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) ∈ 𝐹)
1024, 101syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1r𝐸) ∈ 𝐹)
103100, 102eqeltrrd 2842 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝐾) ∈ 𝐹)
10497, 103sseldd 3984 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐾) ∈ ran 𝐺)
105 drngnzr 20748 . . . . . . . . . . . . . . . . 17 (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
10698, 35nzrnz 20515 . . . . . . . . . . . . . . . . 17 (𝐸 ∈ NzRing → (1r𝐸) ≠ (0g𝐸))
10745, 105, 1063syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (1r𝐸) ≠ (0g𝐸))
10836crnggrpd 20244 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ Grp)
109108grpmndd 18964 . . . . . . . . . . . . . . . . 17 (𝜑𝐸 ∈ Mnd)
110 sdrgsubrg 20792 . . . . . . . . . . . . . . . . . . 19 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111 subrgsubg 20577 . . . . . . . . . . . . . . . . . . 19 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸))
11257, 110, 1113syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸))
11335subg0cl 19152 . . . . . . . . . . . . . . . . . 18 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸) → (0g𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
114112, 113syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
1156, 45, 56fldgenssv 33317 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
11615, 6, 35ress0g 18775 . . . . . . . . . . . . . . . . 17 ((𝐸 ∈ Mnd ∧ (0g𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) → (0g𝐸) = (0g𝐿))
117109, 114, 115, 116syl3anc 1373 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝐸) = (0g𝐿))
118107, 100, 1173netr3d 3017 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝐾) ≠ (0g𝐿))
119 nelsn 4666 . . . . . . . . . . . . . . 15 ((1r𝐾) ≠ (0g𝐿) → ¬ (1r𝐾) ∈ {(0g𝐿)})
120118, 119syl 17 . . . . . . . . . . . . . 14 (𝜑 → ¬ (1r𝐾) ∈ {(0g𝐿)})
121 nelne1 3039 . . . . . . . . . . . . . 14 (((1r𝐾) ∈ ran 𝐺 ∧ ¬ (1r𝐾) ∈ {(0g𝐿)}) → ran 𝐺 ≠ {(0g𝐿)})
122104, 120, 121syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ran 𝐺 ≠ {(0g𝐿)})
12389, 122eqnetrrd 3009 . . . . . . . . . . . 12 (𝜑 → ran 𝐽 ≠ {(0g𝐿)})
124 eqid 2737 . . . . . . . . . . . . 13 (oppr𝑃) = (oppr𝑃)
1259sdrgdrng 20791 . . . . . . . . . . . . . . 15 (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126 drngnzr 20748 . . . . . . . . . . . . . . 15 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
1271, 125, 1263syl 18 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ NzRing)
12821ply1nz 26161 . . . . . . . . . . . . . 14 (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129127, 128syl 17 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ NzRing)
130 eqid 2737 . . . . . . . . . . . . . . . 16 {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)}
131 eqid 2737 . . . . . . . . . . . . . . . 16 (RSpan‘𝑃) = (RSpan‘𝑃)
1329fveq2i 6909 . . . . . . . . . . . . . . . 16 (idlGen1p𝐾) = (idlGen1p‘(𝐸s 𝐹))
13320, 32, 6, 18, 1, 38, 35, 130, 131, 132ply1annig1p 33747 . . . . . . . . . . . . . . 15 (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)} = ((RSpan‘𝑃)‘{((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})}))
134117sneqd 4638 . . . . . . . . . . . . . . . . . 18 (𝜑 → {(0g𝐸)} = {(0g𝐿)})
135134imaeq2d 6078 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺 “ {(0g𝐸)}) = (𝐺 “ {(0g𝐿)}))
13625, 135eqtr4id 2796 . . . . . . . . . . . . . . . 16 (𝜑𝑍 = (𝐺 “ {(0g𝐸)}))
13722mpteq1i 5238 . . . . . . . . . . . . . . . . . 18 (𝑝𝑈 ↦ ((𝑂𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂𝑝)‘𝐴))
13823, 137eqtri 2765 . . . . . . . . . . . . . . . . 17 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂𝑝)‘𝐴))
13920, 32, 6, 36, 4, 38, 35, 130, 138ply1annidllem 33744 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)} = (𝐺 “ {(0g𝐸)}))
140136, 139eqtr4d 2780 . . . . . . . . . . . . . . 15 (𝜑𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})
141 eqid 2737 . . . . . . . . . . . . . . . . . 18 (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
14220, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141minplyval 33748 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) = ((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)}))
143142sneqd 4638 . . . . . . . . . . . . . . . 16 (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} = {((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})})
144143fveq2d 6910 . . . . . . . . . . . . . . 15 (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})}))
145133, 140, 1443eqtr4d 2787 . . . . . . . . . . . . . 14 (𝜑𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146 eqid 2737 . . . . . . . . . . . . . . . 16 (0g𝑃) = (0g𝑃)
147 eqid 2737 . . . . . . . . . . . . . . . . . 18 (0g‘(Poly1𝐸)) = (0g‘(Poly1𝐸))
148147, 18, 1, 141, 19irngnminplynz 33755 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1𝐸)))
149 eqid 2737 . . . . . . . . . . . . . . . . . 18 (Poly1𝐸) = (Poly1𝐸)
150149, 9, 21, 22, 4, 147ressply10g 33592 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g‘(Poly1𝐸)) = (0g𝑃))
151148, 150neeqtrd 3010 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g𝑃))
15220, 32, 6, 18, 1, 38, 141, 146, 151minplyirred 33754 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153 eqid 2737 . . . . . . . . . . . . . . . 16 ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154 fldsdrgfld 20799 . . . . . . . . . . . . . . . . . . 19 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
15518, 1, 154syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸s 𝐹) ∈ Field)
1569, 155eqeltrid 2845 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ Field)
15721ply1pid 26222 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ Field → 𝑃 ∈ PID)
158156, 157syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ PID)
15920, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141minplycl 33749 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160159, 22eleqtrrdi 2852 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
16195crngringd 20243 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ Ring)
162160snssd 4809 . . . . . . . . . . . . . . . . 17 (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163 eqid 2737 . . . . . . . . . . . . . . . . . 18 (LIdeal‘𝑃) = (LIdeal‘𝑃)
164131, 22, 163rspcl 21245 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ Ring ∧ {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈) → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
165161, 162, 164syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
16622, 131, 146, 153, 158, 160, 151, 165mxidlirred 33500 . . . . . . . . . . . . . . 15 (𝜑 → (((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (MaxIdeal‘𝑃) ↔ ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃)))
167152, 166mpbird 257 . . . . . . . . . . . . . 14 (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (MaxIdeal‘𝑃))
168145, 167eqeltrd 2841 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ (MaxIdeal‘𝑃))
169 eqid 2737 . . . . . . . . . . . . . . . 16 (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170169, 124crngmxidl 33497 . . . . . . . . . . . . . . 15 (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr𝑃)))
17195, 170syl 17 . . . . . . . . . . . . . 14 (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr𝑃)))
172168, 171eleqtrd 2843 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ (MaxIdeal‘(oppr𝑃)))
173124, 26, 129, 168, 172qsdrngi 33523 . . . . . . . . . . . 12 (𝜑𝑄 ∈ DivRing)
17491, 54, 96, 123, 173rndrhmcl 33299 . . . . . . . . . . 11 (𝜑 → (𝐿s ran 𝐽) ∈ DivRing)
17590, 174eqeltrd 2841 . . . . . . . . . 10 (𝜑 → (𝐿s ran 𝐺) ∈ DivRing)
17653, 175eqeltrrd 2842 . . . . . . . . 9 (𝜑 → (𝐸s ran 𝐺) ∈ DivRing)
177 issdrg 20789 . . . . . . . . 9 (ran 𝐺 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ ran 𝐺 ∈ (SubRing‘𝐸) ∧ (𝐸s ran 𝐺) ∈ DivRing))
17845, 48, 176, 177syl3anbrc 1344 . . . . . . . 8 (𝜑 → ran 𝐺 ∈ (SubDRing‘𝐸))
179 fveq2 6906 . . . . . . . . . . . . . 14 (𝑝 = (var1𝐾) → (𝑂𝑝) = (𝑂‘(var1𝐾)))
180179fveq1d 6908 . . . . . . . . . . . . 13 (𝑝 = (var1𝐾) → ((𝑂𝑝)‘𝐴) = ((𝑂‘(var1𝐾))‘𝐴))
181180eqeq2d 2748 . . . . . . . . . . . 12 (𝑝 = (var1𝐾) → (𝐴 = ((𝑂𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1𝐾))‘𝐴)))
1829, 71eqeltrid 2845 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ DivRing)
183182drngringd 20737 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Ring)
184 eqid 2737 . . . . . . . . . . . . . 14 (var1𝐾) = (var1𝐾)
185184, 21, 22vr1cl 22219 . . . . . . . . . . . . 13 (𝐾 ∈ Ring → (var1𝐾) ∈ 𝑈)
186183, 185syl 17 . . . . . . . . . . . 12 (𝜑 → (var1𝐾) ∈ 𝑈)
18720, 184, 9, 6, 36, 4evls1var 22342 . . . . . . . . . . . . . 14 (𝜑 → (𝑂‘(var1𝐾)) = ( I ↾ (Base‘𝐸)))
188187fveq1d 6908 . . . . . . . . . . . . 13 (𝜑 → ((𝑂‘(var1𝐾))‘𝐴) = (( I ↾ (Base‘𝐸))‘𝐴))
189 fvresi 7193 . . . . . . . . . . . . . 14 (𝐴 ∈ (Base‘𝐸) → (( I ↾ (Base‘𝐸))‘𝐴) = 𝐴)
19038, 189syl 17 . . . . . . . . . . . . 13 (𝜑 → (( I ↾ (Base‘𝐸))‘𝐴) = 𝐴)
191188, 190eqtr2d 2778 . . . . . . . . . . . 12 (𝜑𝐴 = ((𝑂‘(var1𝐾))‘𝐴))
192181, 186, 191rspcedvdw 3625 . . . . . . . . . . 11 (𝜑 → ∃𝑝𝑈 𝐴 = ((𝑂𝑝)‘𝐴))
19323, 192, 19elrnmptd 5974 . . . . . . . . . 10 (𝜑𝐴 ∈ ran 𝐺)
194193snssd 4809 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ ran 𝐺)
19597, 194unssd 4192 . . . . . . . 8 (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
1966, 45, 178, 195fldgenssp 33320 . . . . . . 7 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
19744, 196eqssd 4001 . . . . . 6 (𝜑 → ran 𝐺 = (𝐸 fldGen (𝐹 ∪ {𝐴})))
19815, 6ressbas2 17283 . . . . . . 7 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿))
199115, 198syl 17 . . . . . 6 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿))
200 eqidd 2738 . . . . . . 7 (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
2016, 45, 56fldgenssid 33315 . . . . . . . . 9 (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202201unssad 4193 . . . . . . . 8 (𝜑𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203202, 199sseqtrd 4020 . . . . . . 7 (𝜑𝐹 ⊆ (Base‘𝐿))
204200, 203srabase 21177 . . . . . 6 (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205197, 199, 2043eqtrd 2781 . . . . 5 (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206 imaeq2 6074 . . . . . . 7 (𝑞 = 𝑝 → (𝐺𝑞) = (𝐺𝑝))
207206unieqd 4920 . . . . . 6 (𝑞 = 𝑝 (𝐺𝑞) = (𝐺𝑝))
208207cbvmptv 5255 . . . . 5 (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ (𝐺𝑞)) = (𝑝 ∈ (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ (𝐺𝑝))
20914, 28, 29, 30, 205, 208lmhmqusker 33445 . . . 4 (𝜑 → (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ (𝐺𝑞)) ∈ ((𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) LMIso ((subringAlg ‘𝐿)‘𝐹)))
210 eqidd 2738 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐿) = (0g𝐿))
211200, 210, 203sralmod0 21195 . . . . . . . . . . . . 13 (𝜑 → (0g𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212211sneqd 4638 . . . . . . . . . . . 12 (𝜑 → {(0g𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213212imaeq2d 6078 . . . . . . . . . . 11 (𝜑 → (𝐺 “ {(0g𝐿)}) = (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
21425, 213eqtrid 2789 . . . . . . . . . 10 (𝜑𝑍 = (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
215214oveq2d 7447 . . . . . . . . 9 (𝜑 → (𝑃 ~QG 𝑍) = (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
216215oveq2d 7447 . . . . . . . 8 (𝜑 → (𝑃 /s (𝑃 ~QG 𝑍)) = (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
21726, 216eqtrid 2789 . . . . . . 7 (𝜑𝑄 = (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218217fveq2d 6910 . . . . . 6 (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219218mpteq1d 5237 . . . . 5 (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ (𝐺𝑝)) = (𝑝 ∈ (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ (𝐺𝑝)))
220219, 27, 2083eqtr4g 2802 . . . 4 (𝜑𝐽 = (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ (𝐺𝑞)))
221217oveq1d 7446 . . . 4 (𝜑 → (𝑄 LMIso ((subringAlg ‘𝐿)‘𝐹)) = ((𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) LMIso ((subringAlg ‘𝐿)‘𝐹)))
222209, 220, 2213eltr4d 2856 . . 3 (𝜑𝐽 ∈ (𝑄 LMIso ((subringAlg ‘𝐿)‘𝐹)))
2239, 15, 16, 17, 18, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27algextdeglem3 33760 . . 3 (𝜑𝑄 ∈ LVec)
224222, 223lmimdim 33654 . 2 (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
2256, 18, 56fldgenfld 33322 . . . . 5 (𝜑 → (𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
22615, 225eqeltrid 2845 . . . 4 (𝜑𝐿 ∈ Field)
2279, 15, 16, 17, 18, 1, 19algextdeglem1 33758 . . . . 5 (𝜑 → (𝐿s 𝐹) = 𝐾)
22811oveq2d 7447 . . . . 5 (𝜑 → (𝐿s 𝐹) = (𝐿s (Base‘𝐾)))
229227, 228eqtr3d 2779 . . . 4 (𝜑𝐾 = (𝐿s (Base‘𝐾)))
23015subsubrg 20598 . . . . . . 7 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐹 ∈ (SubRing‘𝐿) ↔ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))))
231230biimpar 477 . . . . . 6 (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))) → 𝐹 ∈ (SubRing‘𝐿))
23260, 4, 202, 231syl12anc 837 . . . . 5 (𝜑𝐹 ∈ (SubRing‘𝐿))
23311, 232eqeltrrd 2842 . . . 4 (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234 brfldext 33698 . . . . 5 ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
235234biimpar 477 . . . 4 (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
236226, 156, 229, 233, 235syl22anc 839 . . 3 (𝜑𝐿/FldExt𝐾)
237 extdgval 33705 . . 3 (𝐿/FldExt𝐾 → (𝐿[:]𝐾) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
238236, 237syl 17 . 2 (𝜑 → (𝐿[:]𝐾) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
23913, 224, 2383eqtr4d 2787 1 (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  cun 3949  wss 3951  {csn 4626   cuni 4907   class class class wbr 5143  cmpt 5225   I cid 5577  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  Basecbs 17247  s cress 17274  0gc0g 17484   /s cqus 17550  Mndcmnd 18747  SubGrpcsubg 19138  NrmSGrpcnsg 19139   ~QG cqg 19140   GrpHom cghm 19230  1rcur 20178  Ringcrg 20230  CRingccrg 20231  opprcoppr 20333  Irredcir 20356   RingHom crh 20469  NzRingcnzr 20512  SubRingcsubrg 20569  DivRingcdr 20729  Fieldcfield 20730  SubDRingcsdrg 20787   LMHom clmhm 21018   LMIso clmim 21019  LVecclvec 21101  subringAlg csra 21170  LIdealclidl 21216  RSpancrsp 21217  PIDcpid 21346  var1cv1 22177  Poly1cpl1 22178   evalSub1 ces1 22317  deg1cdg1 26093  idlGen1pcig1p 26169   fldGen cfldgen 33312  MaxIdealcmxidl 33487  dimcldim 33649  /FldExtcfldext 33689  [:]cextdg 33692   IntgRing cirng 33733   minPoly cminply 33742
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-reg 9632  ax-inf2 9681  ax-ac2 10503  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-rpss 7743  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-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  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-inf 9483  df-oi 9550  df-r1 9804  df-rank 9805  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  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-xnn0 12600  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-ocomp 17318  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-imas 17553  df-qus 17554  df-mre 17629  df-mrc 17630  df-mri 17631  df-acs 17632  df-proset 18340  df-drs 18341  df-poset 18359  df-ipo 18573  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-nsg 19142  df-eqg 19143  df-ghm 19231  df-gim 19277  df-cntz 19335  df-oppg 19364  df-lsm 19654  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-irred 20359  df-invr 20388  df-dvr 20401  df-rhm 20472  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-drng 20731  df-field 20732  df-sdrg 20788  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lmim 21022  df-lbs 21074  df-lvec 21102  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-lpidl 21332  df-lpir 21333  df-pid 21347  df-cnfld 21365  df-dsmm 21752  df-frlm 21767  df-uvc 21803  df-lindf 21826  df-linds 21827  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-uc1p 26171  df-q1p 26172  df-r1p 26173  df-ig1p 26174  df-fldgen 33313  df-mxidl 33488  df-dim 33650  df-fldext 33693  df-extdg 33694  df-irng 33734  df-minply 33743
This theorem is referenced by:  algextdeg  33766
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