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Theorem algextdeglem4 33726
Description: Lemma for algextdeg 33731. By lmhmqusker 33425, the surjective module homomorphism 𝐺 described in algextdeglem2 33724 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 20806 . . . . . . . 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 1142 . . . . . 6 (𝜑𝐹 ∈ (SubRing‘𝐸))
5 subrgsubg 20594 . . . . . 6 (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6 eqid 2735 . . . . . . 7 (Base‘𝐸) = (Base‘𝐸)
76subgss 19158 . . . . . 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 6911 . . 3 (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
1312fveq2d 6911 . 2 (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14 eqid 2735 . . . . 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 33724 . . . . 5 (𝜑𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29 eqid 2735 . . . . 5 (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30 eqid 2735 . . . . 5 (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
319fveq2i 6910 . . . . . . . . . . 11 (Poly1𝐾) = (Poly1‘(𝐸s 𝐹))
3221, 31eqtri 2763 . . . . . . . . . 10 𝑃 = (Poly1‘(𝐸s 𝐹))
3318adantr 480 . . . . . . . . . 10 ((𝜑𝑝𝑈) → 𝐸 ∈ Field)
341adantr 480 . . . . . . . . . 10 ((𝜑𝑝𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
35 eqid 2735 . . . . . . . . . . . . 13 (0g𝐸) = (0g𝐸)
3618fldcrngd 20759 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ CRing)
3720, 9, 6, 35, 36, 4irngssv 33703 . . . . . . . . . . . 12 (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
3837, 19sseldd 3996 . . . . . . . . . . 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 33695 . . . . . . . . 9 ((𝜑𝑝𝑈) → ((𝑂𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
4241ralrimiva 3144 . . . . . . . 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 20758 . . . . . . . 8 (𝜑𝐸 ∈ DivRing)
4620, 32, 6, 22, 36, 4, 38, 23evls1maprhm 22396 . . . . . . . . . 10 (𝜑𝐺 ∈ (𝑃 RingHom 𝐸))
47 rnrhmsubrg 20622 . . . . . . . . . 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 17295 . . . . . . . . . . . 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 2787 . . . . . . . . . 10 (𝜑 → (𝐿s ran 𝐺) = (𝐸s ran 𝐺))
54 eqid 2735 . . . . . . . . . . . . . . 15 (0g𝐿) = (0g𝐿)
5538snssd 4814 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → {𝐴} ⊆ (Base‘𝐸))
568, 55unssd 4202 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
576, 45, 56fldgensdrg 33296 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58 issdrg 20806 . . . . . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
6115resrhm2b 20619 . . . . . . . . . . . . . . . . . 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 20501 . . . . . . . . . . . . . . . 16 (𝐺 ∈ (𝑃 RingHom 𝐿) → 𝐺 ∈ (𝑃 GrpHom 𝐿))
6563, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ (𝑃 GrpHom 𝐿))
6654, 65, 25, 26, 27, 22, 24ghmquskerco 19315 . . . . . . . . . . . . . 14 (𝜑𝐺 = (𝐽𝑁))
6766rneqd 5952 . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸s 𝐹) ∈ DivRing)
7232, 71ply1lvec 33565 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ LVec)
7368, 69, 70, 72qusbas 17592 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74 eqid 2735 . . . . . . . . . . . . . . . 16 (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
7554ghmker 19273 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ (𝑃 GrpHom 𝐿) → (𝐺 “ {(0g𝐿)}) ∈ (NrmSGrp‘𝑃))
7665, 75syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺 “ {(0g𝐿)}) ∈ (NrmSGrp‘𝑃))
7725, 76eqeltrid 2843 . . . . . . . . . . . . . . . 16 (𝜑𝑍 ∈ (NrmSGrp‘𝑃))
7822, 74, 24, 77qusrn 33417 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79 eqid 2735 . . . . . . . . . . . . . . . . . . . . 21 ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
8020, 32, 6, 22, 36, 4, 38, 23, 79evls1maplmhm 22397 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
8180elexd 3502 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺 ∈ V)
8281adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
8382imaexd 7939 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ (Base‘𝑄)) → (𝐺𝑝) ∈ V)
8483uniexd 7761 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ (Base‘𝑄)) → (𝐺𝑝) ∈ V)
8527, 84dmmptd 6714 . . . . . . . . . . . . . . 15 (𝜑 → dom 𝐽 = (Base‘𝑄))
8673, 78, 853eqtr4rd 2786 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐽 = ran 𝑁)
87 rncoeq 5993 . . . . . . . . . . . . . 14 (dom 𝐽 = ran 𝑁 → ran (𝐽𝑁) = ran 𝐽)
8886, 87syl 17 . . . . . . . . . . . . 13 (𝜑 → ran (𝐽𝑁) = ran 𝐽)
8967, 88eqtrd 2775 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 = ran 𝐽)
9089oveq2d 7447 . . . . . . . . . . 11 (𝜑 → (𝐿s ran 𝐺) = (𝐿s ran 𝐽))
91 eqid 2735 . . . . . . . . . . . 12 (𝐿s ran 𝐽) = (𝐿s ran 𝐽)
929subrgcrng 20592 . . . . . . . . . . . . . . 15 ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
9336, 4, 92syl2anc 584 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ CRing)
9421ply1crng 22216 . . . . . . . . . . . . . 14 (𝐾 ∈ CRing → 𝑃 ∈ CRing)
9593, 94syl 17 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ CRing)
9654, 63, 25, 26, 27, 95rhmquskerlem 33433 . . . . . . . . . . . 12 (𝜑𝐽 ∈ (𝑄 RingHom 𝐿))
9720, 32, 6, 22, 36, 4, 38, 23evls1maprnss 22398 . . . . . . . . . . . . . . 15 (𝜑𝐹 ⊆ ran 𝐺)
98 eqid 2735 . . . . . . . . . . . . . . . . . 18 (1r𝐸) = (1r𝐸)
999, 98subrg1 20599 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) = (1r𝐾))
1004, 99syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1r𝐸) = (1r𝐾))
10198subrg1cl 20597 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) ∈ 𝐹)
1024, 101syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1r𝐸) ∈ 𝐹)
103100, 102eqeltrrd 2840 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝐾) ∈ 𝐹)
10497, 103sseldd 3996 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐾) ∈ ran 𝐺)
105 drngnzr 20765 . . . . . . . . . . . . . . . . 17 (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
10698, 35nzrnz 20532 . . . . . . . . . . . . . . . . 17 (𝐸 ∈ NzRing → (1r𝐸) ≠ (0g𝐸))
10745, 105, 1063syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (1r𝐸) ≠ (0g𝐸))
10836crnggrpd 20265 . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ Grp)
109108grpmndd 18977 . . . . . . . . . . . . . . . . 17 (𝜑𝐸 ∈ Mnd)
110 sdrgsubrg 20809 . . . . . . . . . . . . . . . . . . 19 ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111 subrgsubg 20594 . . . . . . . . . . . . . . . . . . 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 19165 . . . . . . . . . . . . . . . . . 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 33297 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
11615, 6, 35ress0g 18788 . . . . . . . . . . . . . . . . 17 ((𝐸 ∈ Mnd ∧ (0g𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) → (0g𝐸) = (0g𝐿))
117109, 114, 115, 116syl3anc 1370 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝐸) = (0g𝐿))
118107, 100, 1173netr3d 3015 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝐾) ≠ (0g𝐿))
119 nelsn 4671 . . . . . . . . . . . . . . 15 ((1r𝐾) ≠ (0g𝐿) → ¬ (1r𝐾) ∈ {(0g𝐿)})
120118, 119syl 17 . . . . . . . . . . . . . 14 (𝜑 → ¬ (1r𝐾) ∈ {(0g𝐿)})
121 nelne1 3037 . . . . . . . . . . . . . 14 (((1r𝐾) ∈ ran 𝐺 ∧ ¬ (1r𝐾) ∈ {(0g𝐿)}) → ran 𝐺 ≠ {(0g𝐿)})
122104, 120, 121syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ran 𝐺 ≠ {(0g𝐿)})
12389, 122eqnetrrd 3007 . . . . . . . . . . . 12 (𝜑 → ran 𝐽 ≠ {(0g𝐿)})
124 eqid 2735 . . . . . . . . . . . . 13 (oppr𝑃) = (oppr𝑃)
1259sdrgdrng 20808 . . . . . . . . . . . . . . 15 (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126 drngnzr 20765 . . . . . . . . . . . . . . 15 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
1271, 125, 1263syl 18 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ NzRing)
12821ply1nz 26176 . . . . . . . . . . . . . 14 (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129127, 128syl 17 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ NzRing)
130 eqid 2735 . . . . . . . . . . . . . . . 16 {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)}
131 eqid 2735 . . . . . . . . . . . . . . . 16 (RSpan‘𝑃) = (RSpan‘𝑃)
1329fveq2i 6910 . . . . . . . . . . . . . . . 16 (idlGen1p𝐾) = (idlGen1p‘(𝐸s 𝐹))
13320, 32, 6, 18, 1, 38, 35, 130, 131, 132ply1annig1p 33712 . . . . . . . . . . . . . . 15 (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)} = ((RSpan‘𝑃)‘{((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})}))
134117sneqd 4643 . . . . . . . . . . . . . . . . . 18 (𝜑 → {(0g𝐸)} = {(0g𝐿)})
135134imaeq2d 6080 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺 “ {(0g𝐸)}) = (𝐺 “ {(0g𝐿)}))
13625, 135eqtr4id 2794 . . . . . . . . . . . . . . . 16 (𝜑𝑍 = (𝐺 “ {(0g𝐸)}))
13722mpteq1i 5244 . . . . . . . . . . . . . . . . . 18 (𝑝𝑈 ↦ ((𝑂𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂𝑝)‘𝐴))
13823, 137eqtri 2763 . . . . . . . . . . . . . . . . 17 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂𝑝)‘𝐴))
13920, 32, 6, 36, 4, 38, 35, 130, 138ply1annidllem 33709 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)} = (𝐺 “ {(0g𝐸)}))
140136, 139eqtr4d 2778 . . . . . . . . . . . . . . 15 (𝜑𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})
141 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
14220, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141minplyval 33713 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) = ((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)}))
143142sneqd 4643 . . . . . . . . . . . . . . . 16 (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} = {((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})})
144143fveq2d 6911 . . . . . . . . . . . . . . 15 (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((idlGen1p𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = (0g𝐸)})}))
145133, 140, 1443eqtr4d 2785 . . . . . . . . . . . . . 14 (𝜑𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146 eqid 2735 . . . . . . . . . . . . . . . 16 (0g𝑃) = (0g𝑃)
147 eqid 2735 . . . . . . . . . . . . . . . . . 18 (0g‘(Poly1𝐸)) = (0g‘(Poly1𝐸))
148147, 18, 1, 141, 19irngnminplynz 33720 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1𝐸)))
149 eqid 2735 . . . . . . . . . . . . . . . . . 18 (Poly1𝐸) = (Poly1𝐸)
150149, 9, 21, 22, 4, 147ressply10g 33572 . . . . . . . . . . . . . . . . 17 (𝜑 → (0g‘(Poly1𝐸)) = (0g𝑃))
151148, 150neeqtrd 3008 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g𝑃))
15220, 32, 6, 18, 1, 38, 141, 146, 151minplyirred 33719 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153 eqid 2735 . . . . . . . . . . . . . . . 16 ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154 fldsdrgfld 20816 . . . . . . . . . . . . . . . . . . 19 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
15518, 1, 154syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐸s 𝐹) ∈ Field)
1569, 155eqeltrid 2843 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ Field)
15721ply1pid 26237 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ Field → 𝑃 ∈ PID)
158156, 157syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ PID)
15920, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141minplycl 33714 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160159, 22eleqtrrdi 2850 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
16195crngringd 20264 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ Ring)
162160snssd 4814 . . . . . . . . . . . . . . . . 17 (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163 eqid 2735 . . . . . . . . . . . . . . . . . 18 (LIdeal‘𝑃) = (LIdeal‘𝑃)
164131, 22, 163rspcl 21263 . . . . . . . . . . . . . . . . 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 33480 . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ (MaxIdeal‘𝑃))
169 eqid 2735 . . . . . . . . . . . . . . . 16 (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170169, 124crngmxidl 33477 . . . . . . . . . . . . . . 15 (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr𝑃)))
17195, 170syl 17 . . . . . . . . . . . . . 14 (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr𝑃)))
172168, 171eleqtrd 2841 . . . . . . . . . . . . 13 (𝜑𝑍 ∈ (MaxIdeal‘(oppr𝑃)))
173124, 26, 129, 168, 172qsdrngi 33503 . . . . . . . . . . . 12 (𝜑𝑄 ∈ DivRing)
17491, 54, 96, 123, 173rndrhmcl 33280 . . . . . . . . . . 11 (𝜑 → (𝐿s ran 𝐽) ∈ DivRing)
17590, 174eqeltrd 2839 . . . . . . . . . 10 (𝜑 → (𝐿s ran 𝐺) ∈ DivRing)
17653, 175eqeltrrd 2840 . . . . . . . . 9 (𝜑 → (𝐸s ran 𝐺) ∈ DivRing)
177 issdrg 20806 . . . . . . . . 9 (ran 𝐺 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ ran 𝐺 ∈ (SubRing‘𝐸) ∧ (𝐸s ran 𝐺) ∈ DivRing))
17845, 48, 176, 177syl3anbrc 1342 . . . . . . . 8 (𝜑 → ran 𝐺 ∈ (SubDRing‘𝐸))
179 fveq2 6907 . . . . . . . . . . . . . 14 (𝑝 = (var1𝐾) → (𝑂𝑝) = (𝑂‘(var1𝐾)))
180179fveq1d 6909 . . . . . . . . . . . . 13 (𝑝 = (var1𝐾) → ((𝑂𝑝)‘𝐴) = ((𝑂‘(var1𝐾))‘𝐴))
181180eqeq2d 2746 . . . . . . . . . . . 12 (𝑝 = (var1𝐾) → (𝐴 = ((𝑂𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1𝐾))‘𝐴)))
1829, 71eqeltrid 2843 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ DivRing)
183182drngringd 20754 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Ring)
184 eqid 2735 . . . . . . . . . . . . . 14 (var1𝐾) = (var1𝐾)
185184, 21, 22vr1cl 22235 . . . . . . . . . . . . 13 (𝐾 ∈ Ring → (var1𝐾) ∈ 𝑈)
186183, 185syl 17 . . . . . . . . . . . 12 (𝜑 → (var1𝐾) ∈ 𝑈)
18720, 184, 9, 6, 36, 4evls1var 22358 . . . . . . . . . . . . . 14 (𝜑 → (𝑂‘(var1𝐾)) = ( I ↾ (Base‘𝐸)))
188187fveq1d 6909 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 12 (𝜑𝐴 = ((𝑂‘(var1𝐾))‘𝐴))
192181, 186, 191rspcedvdw 3625 . . . . . . . . . . 11 (𝜑 → ∃𝑝𝑈 𝐴 = ((𝑂𝑝)‘𝐴))
19323, 192, 19elrnmptd 5977 . . . . . . . . . 10 (𝜑𝐴 ∈ ran 𝐺)
194193snssd 4814 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ ran 𝐺)
19597, 194unssd 4202 . . . . . . . 8 (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
1966, 45, 178, 195fldgenssp 33300 . . . . . . 7 (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
19744, 196eqssd 4013 . . . . . 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 2736 . . . . . . 7 (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
2016, 45, 56fldgenssid 33295 . . . . . . . . 9 (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202201unssad 4203 . . . . . . . 8 (𝜑𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203202, 199sseqtrd 4036 . . . . . . 7 (𝜑𝐹 ⊆ (Base‘𝐿))
204200, 203srabase 21195 . . . . . 6 (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205197, 199, 2043eqtrd 2779 . . . . 5 (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206 imaeq2 6076 . . . . . . 7 (𝑞 = 𝑝 → (𝐺𝑞) = (𝐺𝑝))
207206unieqd 4925 . . . . . 6 (𝑞 = 𝑝 (𝐺𝑞) = (𝐺𝑝))
208207cbvmptv 5261 . . . . 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 33425 . . . 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 2736 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐿) = (0g𝐿))
211200, 210, 203sralmod0 21213 . . . . . . . . . . . . 13 (𝜑 → (0g𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212211sneqd 4643 . . . . . . . . . . . 12 (𝜑 → {(0g𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213212imaeq2d 6080 . . . . . . . . . . 11 (𝜑 → (𝐺 “ {(0g𝐿)}) = (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
21425, 213eqtrid 2787 . . . . . . . . . 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 2787 . . . . . . 7 (𝜑𝑄 = (𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218217fveq2d 6911 . . . . . 6 (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219218mpteq1d 5243 . . . . 5 (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ (𝐺𝑝)) = (𝑝 ∈ (Base‘(𝑃 /s (𝑃 ~QG (𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ (𝐺𝑝)))
220219, 27, 2083eqtr4g 2800 . . . 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 2854 . . 3 (𝜑𝐽 ∈ (𝑄 LMIso ((subringAlg ‘𝐿)‘𝐹)))
2239, 15, 16, 17, 18, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27algextdeglem3 33725 . . 3 (𝜑𝑄 ∈ LVec)
224222, 223lmimdim 33631 . 2 (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
2256, 18, 56fldgenfld 33302 . . . . 5 (𝜑 → (𝐸s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
22615, 225eqeltrid 2843 . . . 4 (𝜑𝐿 ∈ Field)
2279, 15, 16, 17, 18, 1, 19algextdeglem1 33723 . . . . 5 (𝜑 → (𝐿s 𝐹) = 𝐾)
22811oveq2d 7447 . . . . 5 (𝜑 → (𝐿s 𝐹) = (𝐿s (Base‘𝐾)))
229227, 228eqtr3d 2777 . . . 4 (𝜑𝐾 = (𝐿s (Base‘𝐾)))
23015subsubrg 20615 . . . . . . 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 2840 . . . 4 (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234 brfldext 33675 . . . . 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 33682 . . 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 2785 1 (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  Vcvv 3478  cun 3961  wss 3963  {csn 4631   cuni 4912   class class class wbr 5148  cmpt 5231   I cid 5582  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  ccom 5693  cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  Basecbs 17245  s cress 17274  0gc0g 17486   /s cqus 17552  Mndcmnd 18760  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~QG cqg 19153   GrpHom cghm 19243  1rcur 20199  Ringcrg 20251  CRingccrg 20252  opprcoppr 20350  Irredcir 20373   RingHom crh 20486  NzRingcnzr 20529  SubRingcsubrg 20586  DivRingcdr 20746  Fieldcfield 20747  SubDRingcsdrg 20804   LMHom clmhm 21036   LMIso clmim 21037  LVecclvec 21119  subringAlg csra 21188  LIdealclidl 21234  RSpancrsp 21235  PIDcpid 21364  var1cv1 22193  Poly1cpl1 22194   evalSub1 ces1 22333  deg1cdg1 26108  idlGen1pcig1p 26184   fldGen cfldgen 33292  MaxIdealcmxidl 33467  dimcldim 33626  /FldExtcfldext 33666  [:]cextdg 33669   IntgRing cirng 33698   minPoly cminply 33707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-imas 17555  df-qus 17556  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-gim 19290  df-cntz 19348  df-oppg 19377  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-irred 20376  df-invr 20405  df-dvr 20418  df-rhm 20489  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-rlreg 20711  df-domn 20712  df-idom 20713  df-drng 20748  df-field 20749  df-sdrg 20805  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lmim 21040  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-2idl 21278  df-lpidl 21350  df-lpir 21351  df-pid 21365  df-cnfld 21383  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evls1 22335  df-evl1 22336  df-mdeg 26109  df-deg1 26110  df-mon1 26185  df-uc1p 26186  df-q1p 26187  df-r1p 26188  df-ig1p 26189  df-fldgen 33293  df-mxidl 33468  df-dim 33627  df-fldext 33670  df-extdg 33671  df-irng 33699  df-minply 33708
This theorem is referenced by:  algextdeg  33731
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