Theorem algextdeglem4 33710

Description: Lemma for algextdeg 33715. By lmhmqusker 33388, the surjective module homomorphism 𝐺 described in algextdeglem2 33708 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 20697	. . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
3	1, 2	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing))
4	3	simp2d 1143	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
5		subrgsubg 20486	. . . . . 6 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
7	6	subgss 19059	. . . . . 6 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
8	4, 5, 7	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
9		algextdeg.k	. . . . . 6 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
10	9, 6	ressbas2 17208	. . . . 5 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
11	8, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
12	11	fveq2d 6862	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
13	12	fveq2d 6862	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14		eqid 2729	. . . . 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 33708	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29		eqid 2729	. . . . 5 ⊢ (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30		eqid 2729	. . . . 5 ⊢ (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
31	9	fveq2i 6861	. . . . . . . . . . 11 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	21, 31	eqtri 2752	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐸 ∈ Field)
34	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
35		eqid 2729	. . . . . . . . . . . . 13 ⊢ (0g‘𝐸) = (0g‘𝐸)
36	18	fldcrngd 20651	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ CRing)
37	20, 9, 6, 35, 36, 4	irngssv 33683	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
38	37, 19	sseldd 3947	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐴 ∈ (Base‘𝐸))
40		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
41	6, 20, 32, 22, 33, 34, 39, 40	evls1fldgencl 33665	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
42	41	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	23	rnmptss 7095	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
45	18	flddrngd 20650	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ DivRing)
46	20, 32, 6, 22, 36, 4, 38, 23	evls1maprhm 22263	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐸))
47		rnrhmsubrg 20514	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐸) → ran 𝐺 ∈ (SubRing‘𝐸))
48	46, 47	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ (SubRing‘𝐸))
49	15	oveq1i 7397	. . . . . . . . . . 11 ⊢ (𝐿 ↾s ran 𝐺) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺)
50		ovex 7420	. . . . . . . . . . . 12 ⊢ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V
51		ressabs 17218	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
53	49, 52	eqtrid 2776	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
54		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐿) = (0g‘𝐿)
55	38	snssd 4773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
56	8, 55	unssd 4155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
57	6, 45, 56	fldgensdrg 33264	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58		issdrg 20697	. . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ DivRing))
60	59	simp2d 1143	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
61	15	resrhm2b 20511	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → (𝐺 ∈ (𝑃 RingHom 𝐸) ↔ 𝐺 ∈ (𝑃 RingHom 𝐿)))
62	61	biimpa 476	. . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐿))
64		rhmghm 20393	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐿) → 𝐺 ∈ (𝑃 GrpHom 𝐿))
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝐿))
66	54, 65, 25, 26, 27, 22, 24	ghmquskerco 19216	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 = (𝐽 ∘ 𝑁))
67	66	rneqd 5902	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 ~QG 𝑍) ∈ V)
71	3	simp3d 1144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
72	32, 71	ply1lvec 33528	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LVec)
73	68, 69, 70, 72	qusbas 17508	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
75	54	ghmker 19174	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ (𝑃 GrpHom 𝐿) → (◡𝐺 “ {(0g‘𝐿)}) ∈ (NrmSGrp‘𝑃))
76	65, 75	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) ∈ (NrmSGrp‘𝑃))
77	25, 76	eqeltrid 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝑃))
78	22, 74, 24, 77	qusrn 33380	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
80	20, 32, 6, 22, 36, 4, 38, 23, 79	evls1maplmhm 22264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
81	80	elexd 3471	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 ∈ V)
82	81	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
83	82	imaexd 7892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → (𝐺 “ 𝑝) ∈ V)
84	83	uniexd 7718	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → ∪ (𝐺 “ 𝑝) ∈ V)
85	27, 84	dmmptd 6663	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐽 = (Base‘𝑄))
86	73, 78, 85	3eqtr4rd 2775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐽 = ran 𝑁)
87		rncoeq 5943	. . . . . . . . . . . . . 14 ⊢ (dom 𝐽 = ran 𝑁 → ran (𝐽 ∘ 𝑁) = ran 𝐽)
88	86, 87	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (𝐽 ∘ 𝑁) = ran 𝐽)
89	67, 88	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = ran 𝐽)
90	89	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐿 ↾s ran 𝐽))
91		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s ran 𝐽) = (𝐿 ↾s ran 𝐽)
92	9	subrgcrng 20484	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
93	36, 4, 92	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
94	21	ply1crng 22083	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝑃 ∈ CRing)
95	93, 94	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
96	54, 63, 25, 26, 27, 95	rhmquskerlem 33396	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐿))
97	20, 32, 6, 22, 36, 4, 38, 23	evls1maprnss 22265	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ⊆ ran 𝐺)
98		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝐸) = (1r‘𝐸)
99	9, 98	subrg1 20491	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) = (1r‘𝐾))
100	4, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐾))
101	98	subrg1cl 20489	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
102	4, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
103	100, 102	eqeltrrd 2829	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ∈ 𝐹)
104	97, 103	sseldd 3947	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐾) ∈ ran 𝐺)
105		drngnzr 20657	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
106	98, 35	nzrnz 20424	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ NzRing → (1r‘𝐸) ≠ (0g‘𝐸))
107	45, 105, 106	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ≠ (0g‘𝐸))
108	36	crnggrpd 20156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	108	grpmndd 18878	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ Mnd)
110		sdrgsubrg 20700	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111		subrgsubg 20486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸))
112	57, 110, 111	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸))
113	35	subg0cl 19066	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
114	112, 113	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
115	6, 45, 56	fldgenssv 33265	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
116	15, 6, 35	ress0g 18689	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐿))
118	107, 100, 117	3netr3d 3001	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ≠ (0g‘𝐿))
119		nelsn 4630	. . . . . . . . . . . . . . 15 ⊢ ((1r‘𝐾) ≠ (0g‘𝐿) → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
120	118, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
121		nelne1 3022	. . . . . . . . . . . . . 14 ⊢ (((1r‘𝐾) ∈ ran 𝐺 ∧ ¬ (1r‘𝐾) ∈ {(0g‘𝐿)}) → ran 𝐺 ≠ {(0g‘𝐿)})
122	104, 120, 121	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐺 ≠ {(0g‘𝐿)})
123	89, 122	eqnetrrd 2993	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 ≠ {(0g‘𝐿)})
124		eqid 2729	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑃) = (oppr‘𝑃)
125	9	sdrgdrng 20699	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126		drngnzr 20657	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
127	1, 125, 126	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
128	21	ply1nz 26027	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129	127, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ NzRing)
130		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
131		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
132	9	fveq2i 6861	. . . . . . . . . . . . . . . 16 ⊢ (idlGen1p‘𝐾) = (idlGen1p‘(𝐸 ↾s 𝐹))
133	20, 32, 6, 18, 1, 38, 35, 130, 131, 132	ply1annig1p 33694	. . . . . . . . . . . . . . 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 6031	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)}))
136	25, 135	eqtr4id 2783	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)}))
137	22	mpteq1i 5198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
138	23, 137	eqtri 2752	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
139	20, 32, 6, 36, 4, 38, 35, 130, 138	ply1annidllem 33691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)}))
140	136, 139	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
141		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
142	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplyval 33695	. . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑃) = (0g‘𝑃)
147		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
148	147, 18, 1, 141, 19	irngnminplynz 33702	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
149		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
150	149, 9, 21, 22, 4, 147	ressply10g 33536	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
151	148, 150	neeqtrd 2994	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘𝑃))
152	20, 32, 6, 18, 1, 38, 141, 146, 151	minplyirred 33701	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154		fldsdrgfld 20707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
155	18, 1, 154	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
156	9, 155	eqeltrid 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Field)
157	21	ply1pid 26088	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Field → 𝑃 ∈ PID)
158	156, 157	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ PID)
159	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplycl 33696	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160	159, 22	eleqtrrdi 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
161	95	crngringd 20155	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
162	160	snssd 4773	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
164	131, 22, 163	rspcl 21145	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Ring ∧ {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈) → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
165	161, 162, 164	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
166	22, 131, 146, 153, 158, 160, 151, 165	mxidlirred 33443	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (MaxIdeal‘𝑃) ↔ ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃)))
167	152, 166	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (MaxIdeal‘𝑃))
168	145, 167	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘𝑃))
169		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170	169, 124	crngmxidl 33440	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
171	95, 170	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
172	168, 171	eleqtrd 2830	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘(oppr‘𝑃)))
173	124, 26, 129, 168, 172	qsdrngi 33466	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ DivRing)
174	91, 54, 96, 123, 173	rndrhmcl 33246	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐽) ∈ DivRing)
175	90, 174	eqeltrd 2828	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) ∈ DivRing)
176	53, 175	eqeltrrd 2829	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾s ran 𝐺) ∈ DivRing)
177		issdrg 20697	. . . . . . . . 9 ⊢ (ran 𝐺 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ ran 𝐺 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s ran 𝐺) ∈ DivRing))
178	45, 48, 176, 177	syl3anbrc 1344	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ∈ (SubDRing‘𝐸))
179		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (var1‘𝐾) → (𝑂‘𝑝) = (𝑂‘(var1‘𝐾)))
180	179	fveq1d 6860	. . . . . . . . . . . . 13 ⊢ (𝑝 = (var1‘𝐾) → ((𝑂‘𝑝)‘𝐴) = ((𝑂‘(var1‘𝐾))‘𝐴))
181	180	eqeq2d 2740	. . . . . . . . . . . 12 ⊢ (𝑝 = (var1‘𝐾) → (𝐴 = ((𝑂‘𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴)))
182	9, 71	eqeltrid 2832	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ DivRing)
183	182	drngringd 20646	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
184		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
185	184, 21, 22	vr1cl 22102	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (var1‘𝐾) ∈ 𝑈)
186	183, 185	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (var1‘𝐾) ∈ 𝑈)
187	20, 184, 9, 6, 36, 4	evls1var 22225	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(var1‘𝐾)) = ( I ↾ (Base‘𝐸)))
188	187	fveq1d 6860	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(var1‘𝐾))‘𝐴) = (( I ↾ (Base‘𝐸))‘𝐴))
189		fvresi 7147	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (Base‘𝐸) → (( I ↾ (Base‘𝐸))‘𝐴) = 𝐴)
190	38, 189	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (( I ↾ (Base‘𝐸))‘𝐴) = 𝐴)
191	188, 190	eqtr2d 2765	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴))
192	181, 186, 191	rspcedvdw 3591	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑝 ∈ 𝑈 𝐴 = ((𝑂‘𝑝)‘𝐴))
193	23, 192, 19	elrnmptd 5927	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ran 𝐺)
194	193	snssd 4773	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ ran 𝐺)
195	97, 194	unssd 4155	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
196	6, 45, 178, 195	fldgenssp 33268	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
197	44, 196	eqssd 3964	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = (𝐸 fldGen (𝐹 ∪ {𝐴})))
198	15, 6	ressbas2 17208	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿))
199	115, 198	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿))
200		eqidd 2730	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
201	6, 45, 56	fldgenssid 33263	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202	201	unssad 4156	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203	202, 199	sseqtrd 3983	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
204	200, 203	srabase 21084	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205	197, 199, 204	3eqtrd 2768	. . . . 5 ⊢ (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206		imaeq2 6027	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝐺 “ 𝑞) = (𝐺 “ 𝑝))
207	206	unieqd 4884	. . . . . 6 ⊢ (𝑞 = 𝑝 → ∪ (𝐺 “ 𝑞) = ∪ (𝐺 “ 𝑝))
208	207	cbvmptv 5211	. . . . 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 33388	. . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐿))
211	200, 210, 203	sralmod0 21095	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212	211	sneqd 4601	. . . . . . . . . . . 12 ⊢ (𝜑 → {(0g‘𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213	212	imaeq2d 6031	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
214	25, 213	eqtrid 2776	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
215	214	oveq2d 7403	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ~QG 𝑍) = (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
216	215	oveq2d 7403	. . . . . . . 8 ⊢ (𝜑 → (𝑃 /s (𝑃 ~QG 𝑍)) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
217	26, 216	eqtrid 2776	. . . . . . 7 ⊢ (𝜑 → 𝑄 = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218	217	fveq2d 6862	. . . . . 6 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219	218	mpteq1d 5197	. . . . 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 2789	. . . 4 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)))
221	217	oveq1d 7402	. . . 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 2843	. . 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 33709	. . 3 ⊢ (𝜑 → 𝑄 ∈ LVec)
224	222, 223	lmimdim 33599	. 2 ⊢ (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
225	6, 18, 56	fldgenfld 33270	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
226	15, 225	eqeltrid 2832	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Field)
227	9, 15, 16, 17, 18, 1, 19	algextdeglem1 33707	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
228	11	oveq2d 7403	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
229	227, 228	eqtr3d 2766	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
230	15	subsubrg 20507	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐹 ∈ (SubRing‘𝐿) ↔ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))))
231	230	biimpar 477	. . . . . 6 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))) → 𝐹 ∈ (SubRing‘𝐿))
232	60, 4, 202, 231	syl12anc 836	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
233	11, 232	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234		brfldext 33641	. . . . 5 ⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
235	234	biimpar 477	. . . 4 ⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
236	226, 156, 229, 233, 235	syl22anc 838	. . 3 ⊢ (𝜑 → 𝐿/FldExt𝐾)
237		extdgval 33649	. . 3 ⊢ (𝐿/FldExt𝐾 → (𝐿[:]𝐾) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
238	236, 237	syl 17	. 2 ⊢ (𝜑 → (𝐿[:]𝐾) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
239	13, 224, 238	3eqtr4d 2774	1 ⊢ (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3405  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  {csn 4589  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188   I cid 5532  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  ‘cfv 6511  (class class class)co 7387  [cec 8669   / cqs 8670  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402   /_s cqus 17468  Mndcmnd 18661  SubGrpcsubg 19052  NrmSGrpcnsg 19053   ~_QG cqg 19054   GrpHom cghm 19144  1_rcur 20090  Ringcrg 20142  CRingccrg 20143  opp_rcoppr 20245  Irredcir 20265   RingHom crh 20378  NzRingcnzr 20421  SubRingcsubrg 20478  DivRingcdr 20638  Fieldcfield 20639  SubDRingcsdrg 20695   LMHom clmhm 20926   LMIso clmim 20927  LVecclvec 21009  subringAlg csra 21078  LIdealclidl 21116  RSpancrsp 21117  PIDcpid 21246  var₁cv1 22060  Poly₁cpl1 22061   evalSub₁ ces1 22200  deg₁cdg1 25959  idlGen_1pcig1p 26035   fldGen cfldgen 33260  MaxIdealcmxidl 33430  dimcldim 33594  /_FldExtcfldext 33634  [:]cextdg 33636   IntgRing cirng 33678   minPoly cminply 33689

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594  ax-ac2 10416  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-rpss 7699  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-ec 8673  df-qs 8677  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-inf 9394  df-oi 9463  df-r1 9717  df-rank 9718  df-dju 9854  df-card 9892  df-acn 9895  df-ac 10069  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-xnn0 12516  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ocomp 17241  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-imas 17471  df-qus 17472  df-mre 17547  df-mrc 17548  df-mri 17549  df-acs 17550  df-proset 18255  df-drs 18256  df-poset 18274  df-ipo 18487  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-nsg 19056  df-eqg 19057  df-ghm 19145  df-gim 19191  df-cntz 19249  df-oppg 19278  df-lsm 19566  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-irred 20268  df-invr 20297  df-dvr 20310  df-rhm 20381  df-nzr 20422  df-subrng 20455  df-subrg 20479  df-rlreg 20603  df-domn 20604  df-idom 20605  df-drng 20640  df-field 20641  df-sdrg 20696  df-lmod 20768  df-lss 20838  df-lsp 20878  df-lmhm 20929  df-lmim 20930  df-lbs 20982  df-lvec 21010  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-rsp 21119  df-2idl 21160  df-lpidl 21232  df-lpir 21233  df-pid 21247  df-cnfld 21265  df-dsmm 21641  df-frlm 21656  df-uvc 21692  df-lindf 21715  df-linds 21716  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-evl 21982  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-evls1 22202  df-evl1 22203  df-mdeg 25960  df-deg1 25961  df-mon1 26036  df-uc1p 26037  df-q1p 26038  df-r1p 26039  df-ig1p 26040  df-fldgen 33261  df-mxidl 33431  df-dim 33595  df-fldext 33637  df-extdg 33638  df-irng 33679  df-minply 33690

This theorem is referenced by:  algextdeg  33715