Theorem algextdeglem4 34110

Description: Lemma for algextdeg 34115. By lmhmqusker 33726, the surjective module homomorphism 𝐺 described in algextdeglem2 34108 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 = (𝐸 ↾s 𝐹)
algextdeg.l	⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
algextdeg.d	⊢ 𝐷 = (deg1‘𝐸)
algextdeg.m	⊢ 𝑀 = (𝐸 minPoly 𝐹)
algextdeg.f	⊢ (𝜑 → 𝐸 ∈ Field)
algextdeg.e	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
algextdeg.a	⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
algextdeglem.o	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
algextdeglem.y	⊢ 𝑃 = (Poly1‘𝐾)
algextdeglem.u	⊢ 𝑈 = (Base‘𝑃)
algextdeglem.g	⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
algextdeglem.n	⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
algextdeglem.z	⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})
algextdeglem.q	⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
algextdeglem.j	⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))

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.

Step	Hyp	Ref	Expression
1		algextdeg.e	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
2		issdrg 20891	. . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
3	1, 2	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
4	3	simp2d 1161	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
5		subrgsubg 20676	. . . . . 6 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
7	6	subgss 19188	. . . . . 6 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
8	4, 5, 7	3syl 19	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
9		algextdeg.k	. . . . . 6 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
10	9, 6	ressbas2 17293	. . . . 5 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
11	8, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
12	11	fveq2d 6885	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
13	12	fveq2d 6885	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14		eqid 2763	. . . . 5 ⊢ (0g‘((subringAlg ‘𝐿)‘𝐹)) = (0g‘((subringAlg ‘𝐿)‘𝐹))
15		algextdeg.l	. . . . . 6 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
16		algextdeg.d	. . . . . 6 ⊢ 𝐷 = (deg1‘𝐸)
17		algextdeg.m	. . . . . 6 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
18		algextdeg.f	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Field)
19		algextdeg.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
20		algextdeglem.o	. . . . . 6 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
21		algextdeglem.y	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝐾)
22		algextdeglem.u	. . . . . 6 ⊢ 𝑈 = (Base‘𝑃)
23		algextdeglem.g	. . . . . 6 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
24		algextdeglem.n	. . . . . 6 ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
25		algextdeglem.z	. . . . . 6 ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})
26		algextdeglem.q	. . . . . 6 ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
27		algextdeglem.j	. . . . . 6 ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))
28	9, 15, 16, 17, 18, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27	algextdeglem2 34108	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29		eqid 2763	. . . . 5 ⊢ (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30		eqid 2763	. . . . 5 ⊢ (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
31	9	fveq2i 6884	. . . . . . . . . . 11 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	21, 31	eqtri 2786	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33	18	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐸 ∈ Field)
34	1	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
35		eqid 2763	. . . . . . . . . . . . 13 ⊢ (0g‘𝐸) = (0g‘𝐸)
36	18	fldcrngd 20842	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ CRing)
37	20, 9, 6, 35, 36, 4	irngssv 34078	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
38	37, 19	sseldd 3938	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
39	38	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐴 ∈ (Base‘𝐸))
40		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
41	6, 20, 32, 22, 33, 34, 39, 40	evls1fldgencl 34060	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
42	41	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	23	rnmptss 7118	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44	42, 43	syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
45	18	flddrngd 20841	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ DivRing)
46	20, 32, 6, 22, 36, 4, 38, 23	evls1maprhm 22536	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐸))
47		rnrhmsubrg 20704	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐸) → ran 𝐺 ∈ (SubRing‘𝐸))
48	46, 47	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ (SubRing‘𝐸))
49	15	oveq1i 7420	. . . . . . . . . . 11 ⊢ (𝐿 ↾s ran 𝐺) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺)
50		ovex 7443	. . . . . . . . . . . 12 ⊢ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V
51		ressabs 17303	. . . . . . . . . . . 12 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
52	50, 44, 51	sylancr 598	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
53	49, 52	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
54		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐿) = (0g‘𝐿)
55	38	snssd 4752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
56	8, 55	unssd 4145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
57	6, 45, 56	fldgensdrg 33635	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58		issdrg 20891	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
59	57, 58	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
60	59	simp2d 1161	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
61	15	resrhm2b 20701	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → (𝐺 ∈ (𝑃 RingHom 𝐸) ↔ 𝐺 ∈ (𝑃 RingHom 𝐿)))
62	61	biimpa 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∧ 𝐺 ∈ (𝑃 RingHom 𝐸)) → 𝐺 ∈ (𝑃 RingHom 𝐿))
63	60, 44, 46, 62	syl21anc 850	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐿))
64		rhmghm 20562	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐿) → 𝐺 ∈ (𝑃 GrpHom 𝐿))
65	63, 64	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝐿))
66	54, 65, 25, 26, 27, 22, 24	ghmquskerco 19349	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 = (𝐽 ∘ 𝑁))
67	66	rneqd 5928	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐺 = ran (𝐽 ∘ 𝑁))
68	26	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍)))
69	22	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = (Base‘𝑃))
70		ovexd 7445	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 ~QG 𝑍) ∈ V)
71	3	simp3d 1162	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
72	32, 71	ply1lvec 33849	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LVec)
73	68, 69, 70, 72	qusbas 17594	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
75	54	ghmker 19307	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ (𝑃 GrpHom 𝐿) → (◡𝐺 “ {(0g‘𝐿)}) ∈ (NrmSGrp‘𝑃))
76	65, 75	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) ∈ (NrmSGrp‘𝑃))
77	25, 76	eqeltrid 2867	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝑃))
78	22, 74, 24, 77	qusrn 33718	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79		eqid 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
80	20, 32, 6, 22, 36, 4, 38, 23, 79	evls1maplmhm 22537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
81	80	elexd 3478	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 ∈ V)
82	81	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
83	82	imaexd 7909	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → (𝐺 “ 𝑝) ∈ V)
84	83	uniexd 7740	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → ∪ (𝐺 “ 𝑝) ∈ V)
85	27, 84	dmmptd 6680	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐽 = (Base‘𝑄))
86	73, 78, 85	3eqtr4rd 2809	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐽 = ran 𝑁)
87		rncoeq 5971	. . . . . . . . . . . . . 14 ⊢ (dom 𝐽 = ran 𝑁 → ran (𝐽 ∘ 𝑁) = ran 𝐽)
88	86, 87	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (𝐽 ∘ 𝑁) = ran 𝐽)
89	67, 88	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = ran 𝐽)
90	89	oveq2d 7426	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐿 ↾s ran 𝐽))
91		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s ran 𝐽) = (𝐿 ↾s ran 𝐽)
92	9	subrgcrng 20674	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
93	36, 4, 92	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
94	21	ply1crng 22358	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝑃 ∈ CRing)
95	93, 94	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
96	54, 63, 25, 26, 27, 95	rhmquskerlem 33733	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐿))
97	20, 32, 6, 22, 36, 4, 38, 23	evls1maprnss 22538	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ⊆ ran 𝐺)
98		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝐸) = (1r‘𝐸)
99	9, 98	subrg1 20681	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) = (1r‘𝐾))
100	4, 99	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐾))
101	98	subrg1cl 20679	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
102	4, 101	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
103	100, 102	eqeltrrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ∈ 𝐹)
104	97, 103	sseldd 3938	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐾) ∈ ran 𝐺)
105		drngnzr 20848	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
106	98, 35	nzrnz 20612	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ NzRing → (1r‘𝐸) ≠ (0g‘𝐸))
107	45, 105, 106	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ≠ (0g‘𝐸))
108	36	crnggrpd 20324	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	108	grpmndd 19008	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ Mnd)
110		sdrgsubrg 20894	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111		subrgsubg 20676	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸))
112	57, 110, 111	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸))
113	35	subg0cl 19195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
114	112, 113	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
115	6, 45, 56	fldgenssv 33636	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
116	15, 6, 35	ress0g 18815	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) → (0g‘𝐸) = (0g‘𝐿))
117	109, 114, 115, 116	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐿))
118	107, 100, 117	3netr3d 3034	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ≠ (0g‘𝐿))
119		nelsn 4632	. . . . . . . . . . . . . . 15 ⊢ ((1r‘𝐾) ≠ (0g‘𝐿) → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
120	118, 119	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
121		nelne1 3055	. . . . . . . . . . . . . 14 ⊢ (((1r‘𝐾) ∈ ran 𝐺 ∧ ¬ (1r‘𝐾) ∈ {(0g‘𝐿)}) → ran 𝐺 ≠ {(0g‘𝐿)})
122	104, 120, 121	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐺 ≠ {(0g‘𝐿)})
123	89, 122	eqnetrrd 3026	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 ≠ {(0g‘𝐿)})
124		eqid 2763	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑃) = (oppr‘𝑃)
125	9	sdrgdrng 20893	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126		drngnzr 20848	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
127	1, 125, 126	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
128	21	ply1nz 26279	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129	127, 128	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ NzRing)
130		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
131		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
132	9	fveq2i 6884	. . . . . . . . . . . . . . . 16 ⊢ (idlGen1p‘𝐾) = (idlGen1p‘(𝐸 ↾s 𝐹))
133	20, 32, 6, 18, 1, 38, 35, 130, 131, 132	ply1annig1p 34094	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = ((RSpan‘𝑃)‘{((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})}))
134	117	sneqd 4601	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {(0g‘𝐸)} = {(0g‘𝐿)})
135	134	imaeq2d 6062	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)}))
136	25, 135	eqtr4id 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)}))
137	22	mpteq1i 5202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
138	23, 137	eqtri 2786	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
139	20, 32, 6, 36, 4, 38, 35, 130, 138	ply1annidllem 34091	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)}))
140	136, 139	eqtr4d 2801	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
141		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
142	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplyval 34095	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) = ((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}))
143	142	sneqd 4601	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} = {((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})
144	143	fveq2d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})}))
145	133, 140, 144	3eqtr4d 2808	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑃) = (0g‘𝑃)
147		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
148	147, 18, 1, 141, 19	irngnminplynz 34102	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
149		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
150	149, 9, 21, 22, 4, 147	ressply10g 33857	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
151	148, 150	neeqtrd 3027	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘𝑃))
152	20, 32, 6, 18, 1, 38, 141, 146, 151	minplyirred 34101	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154		fldsdrgfld 20901	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
155	18, 1, 154	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
156	9, 155	eqeltrid 2867	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Field)
157	21	ply1pid 26340	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Field → 𝑃 ∈ PID)
158	156, 157	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ PID)
159	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplycl 34096	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160	159, 22	eleqtrrdi 2874	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
161	95	crngringd 20323	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
162	160	snssd 4752	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
164	131, 22, 163	rspcl 21364	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Ring ∧ {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈) → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
165	161, 162, 164	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
166	22, 131, 146, 153, 158, 160, 151, 165	mxidlirred 33755	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (MaxIdeal‘𝑃) ↔ ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃)))
167	152, 166	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (MaxIdeal‘𝑃))
168	145, 167	eqeltrd 2863	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘𝑃))
169		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170	169, 124	crngmxidl 33752	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
171	95, 170	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
172	168, 171	eleqtrd 2865	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘(oppr‘𝑃)))
173	124, 26, 129, 168, 172	qsdrngi 33777	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ DivRing)
174	91, 54, 96, 123, 173	rndrhmcl 33617	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐽) ∈ DivRing)
175	90, 174	eqeltrd 2863	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) ∈ DivRing)
176	53, 175	eqeltrrd 2864	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾s ran 𝐺) ∈ DivRing)
177		issdrg 20891	. . . . . . . . 9 ⊢ (ran 𝐺 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ ran 𝐺 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s ran 𝐺) ∈ DivRing))
178	45, 48, 176, 177	syl3anbrc 1362	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ∈ (SubDRing‘𝐸))
179		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (var1‘𝐾) → (𝑂‘𝑝) = (𝑂‘(var1‘𝐾)))
180	179	fveq1d 6883	. . . . . . . . . . . . 13 ⊢ (𝑝 = (var1‘𝐾) → ((𝑂‘𝑝)‘𝐴) = ((𝑂‘(var1‘𝐾))‘𝐴))
181	180	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑝 = (var1‘𝐾) → (𝐴 = ((𝑂‘𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴)))
182	9, 71	eqeltrid 2867	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ DivRing)
183	182	drngringd 20835	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
184		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
185	184, 21, 22	vr1cl 22377	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (var1‘𝐾) ∈ 𝑈)
186	183, 185	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (var1‘𝐾) ∈ 𝑈)
187	20, 184, 9, 6, 36, 4	evls1var 22498	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(var1‘𝐾)) = ( I ↾ (Base‘𝐸)))
188	187	fveq1d 6883	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(var1‘𝐾))‘𝐴) = (( I ↾ (Base‘𝐸))‘𝐴))
189		fvresi 7171	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Base‘𝐸) → (( I ↾ (Base‘𝐸))‘𝐴) = 𝐴)
190	38, 189	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (( I ↾ (Base‘𝐸))‘𝐴) = 𝐴)
191	188, 190	eqtr2d 2799	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴))
192	181, 186, 191	rspcedvdw 3584	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑝 ∈ 𝑈 𝐴 = ((𝑂‘𝑝)‘𝐴))
193	23, 192, 19	elrnmptd 5953	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ran 𝐺)
194	193	snssd 4752	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ ran 𝐺)
195	97, 194	unssd 4145	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
196	6, 45, 178, 195	fldgenssp 33639	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
197	44, 196	eqssd 3954	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = (𝐸 fldGen (𝐹 ∪ {𝐴})))
198	15, 6	ressbas2 17293	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿))
199	115, 198	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿))
200		eqidd 2764	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
201	6, 45, 56	fldgenssid 33634	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202	201	unssad 4146	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203	202, 199	sseqtrd 3973	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
204	200, 203	srabase 21298	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205	197, 199, 204	3eqtrd 2802	. . . . 5 ⊢ (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206		imaeq2 6058	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝐺 “ 𝑞) = (𝐺 “ 𝑝))
207	206	unieqd 4885	. . . . . 6 ⊢ (𝑞 = 𝑝 → ∪ (𝐺 “ 𝑞) = ∪ (𝐺 “ 𝑝))
208	207	cbvmptv 5215	. . . . 5 ⊢ (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)) = (𝑝 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑝))
209	14, 28, 29, 30, 205, 208	lmhmqusker 33726	. . . 4 ⊢ (𝜑 → (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)) ∈ ((𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) LMIso ((subringAlg ‘𝐿)‘𝐹)))
210		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐿))
211	200, 210, 203	sralmod0 21309	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212	211	sneqd 4601	. . . . . . . . . . . 12 ⊢ (𝜑 → {(0g‘𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213	212	imaeq2d 6062	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
214	25, 213	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
215	214	oveq2d 7426	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ~QG 𝑍) = (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
216	215	oveq2d 7426	. . . . . . . 8 ⊢ (𝜑 → (𝑃 /s (𝑃 ~QG 𝑍)) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
217	26, 216	eqtrid 2810	. . . . . . 7 ⊢ (𝜑 → 𝑄 = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218	217	fveq2d 6885	. . . . . 6 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219	218	mpteq1d 5201	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝)) = (𝑝 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑝)))
220	219, 27, 208	3eqtr4g 2823	. . . 4 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)))
221	217	oveq1d 7425	. . . 4 ⊢ (𝜑 → (𝑄 LMIso ((subringAlg ‘𝐿)‘𝐹)) = ((𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) LMIso ((subringAlg ‘𝐿)‘𝐹)))
222	209, 220, 221	3eltr4d 2878	. . 3 ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso ((subringAlg ‘𝐿)‘𝐹)))
223	9, 15, 16, 17, 18, 1, 19, 20, 21, 22, 23, 24, 25, 26, 27	algextdeglem3 34109	. . 3 ⊢ (𝜑 → 𝑄 ∈ LVec)
224	222, 223	lmimdim 33994	. 2 ⊢ (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
225	6, 18, 56	fldgenfld 33641	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
226	15, 225	eqeltrid 2867	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Field)
227	9, 15, 16, 17, 18, 1, 19	algextdeglem1 34107	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
228	11	oveq2d 7426	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
229	227, 228	eqtr3d 2800	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
230	15	subsubrg 20697	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐹 ∈ (SubRing‘𝐿) ↔ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))))
231	230	biimpar 482	. . . . . 6 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))) → 𝐹 ∈ (SubRing‘𝐿))
232	60, 4, 202, 231	syl12anc 849	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
233	11, 232	eqeltrrd 2864	. . . 4 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234		brfldext 34035	. . . . 5 ⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
235	234	biimpar 482	. . . 4 ⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
236	226, 156, 229, 233, 235	syl22anc 851	. . 3 ⊢ (𝜑 → 𝐿/FldExt𝐾)
237		extdgval 34043	. . 3 ⊢ (𝐿/FldExt𝐾 → (𝐿[:]𝐾) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
238	236, 237	syl 18	. 2 ⊢ (𝜑 → (𝐿[:]𝐾) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
239	13, 224, 238	3eqtr4d 2808	1 ⊢ (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  {csn 4589  ∪ cuni 4872   class class class wbr 5109   ↦ cmpt 5192   I cid 5555  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410  [cec 8688   / cqs 8689  Basecbs 17264   ↾_s cress 17285  0_gc0g 17487   /_s cqus 17554  Mndcmnd 18787  SubGrpcsubg 19181  NrmSGrpcnsg 19182   ~_QG cqg 19183   GrpHom cghm 19278  1_rcur 20258  Ringcrg 20310  CRingccrg 20311  opp_rcoppr 20414  Irredcir 20434   RingHom crh 20547  NzRingcnzr 20609  SubRingcsubrg 20668  DivRingcdr 20827  Fieldcfield 20828  SubDRingcsdrg 20889   LMHom clmhm 21140   LMIso clmim 21141  LVecclvec 21223  subringAlg csra 21292  LIdealclidl 21330  RSpancrsp 21331  PIDcpid 21504  var₁cv1 22336  Poly₁cpl1 22337   evalSub₁ ces1 22473  deg₁cdg1 26211  idlGen_1pcig1p 26287   fldGen cfldgen 33631  MaxIdealcmxidl 33742  dimcldim 33989  /_FldExtcfldext 34028  [:]cextdg 34030   IntgRing cirng 34073   minPoly cminply 34089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606  ax-ac2 10442  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-rpss 7720  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-inf 9399  df-oi 9468  df-r1 9732  df-rank 9733  df-dju 9883  df-card 9921  df-acn 9924  df-ac 10096  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034  df-hash 14363  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ocomp 17326  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-0g 17489  df-gsum 17490  df-prds 17495  df-pws 17497  df-imas 17557  df-qus 17558  df-mre 17633  df-mrc 17634  df-mri 17635  df-acs 17636  df-proset 18345  df-drs 18346  df-poset 18364  df-ipo 18579  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-nsg 19185  df-eqg 19186  df-ghm 19279  df-gim 19324  df-cntz 19382  df-oppg 19411  df-lsm 19701  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-srg 20264  df-ring 20312  df-cring 20313  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-irred 20437  df-invr 20466  df-dvr 20479  df-rhm 20550  df-nzr 20610  df-subrng 20645  df-subrg 20669  df-rlreg 20793  df-domn 20794  df-idom 20795  df-drng 20829  df-field 20830  df-sdrg 20890  df-lmod 20983  df-lss 21053  df-lsp 21093  df-lmhm 21143  df-lmim 21144  df-lbs 21196  df-lvec 21224  df-sra 21294  df-rgmod 21295  df-lidl 21332  df-rsp 21333  df-2idl 21389  df-lpidl 21490  df-lpir 21491  df-pid 21505  df-cnfld 21523  df-dsmm 21882  df-frlm 21897  df-uvc 21933  df-lindf 21956  df-linds 21957  df-assa 22003  df-asp 22004  df-ascl 22005  df-psr 22059  df-mvr 22060  df-mpl 22061  df-opsr 22063  df-evls 22225  df-evl 22226  df-psr1 22340  df-vr1 22341  df-ply1 22342  df-coe1 22343  df-evls1 22475  df-evl1 22476  df-mdeg 26212  df-deg1 26213  df-mon1 26288  df-uc1p 26289  df-q1p 26290  df-r1p 26291  df-ig1p 26292  df-fldgen 33632  df-mxidl 33743  df-dim 33990  df-fldext 34031  df-extdg 34032  df-irng 34074  df-minply 34090

This theorem is referenced by:  algextdeg  34115