Theorem algextdeglem4 33877

Description: Lemma for algextdeg 33882. By lmhmqusker 33498, the surjective module homomorphism 𝐺 described in algextdeglem2 33875 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 20721	. . . . . . . 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 20510	. . . . . 6 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
7	6	subgss 19057	. . . . . 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 17165	. . . . 5 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
11	8, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
12	11	fveq2d 6838	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
13	12	fveq2d 6838	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14		eqid 2736	. . . . 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 33875	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29		eqid 2736	. . . . 5 ⊢ (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30		eqid 2736	. . . . 5 ⊢ (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
31	9	fveq2i 6837	. . . . . . . . . . 11 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	21, 31	eqtri 2759	. . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (0g‘𝐸) = (0g‘𝐸)
36	18	fldcrngd 20675	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ CRing)
37	20, 9, 6, 35, 36, 4	irngssv 33845	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
38	37, 19	sseldd 3934	. . . . . . . . . . 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 33827	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
42	41	ralrimiva 3128	. . . . . . . 8 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	23	rnmptss 7068	. . . . . . . 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 20674	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ DivRing)
46	20, 32, 6, 22, 36, 4, 38, 23	evls1maprhm 22320	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐸))
47		rnrhmsubrg 20538	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐸) → ran 𝐺 ∈ (SubRing‘𝐸))
48	46, 47	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ (SubRing‘𝐸))
49	15	oveq1i 7368	. . . . . . . . . . 11 ⊢ (𝐿 ↾s ran 𝐺) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺)
50		ovex 7391	. . . . . . . . . . . 12 ⊢ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V
51		ressabs 17175	. . . . . . . . . . . 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 2783	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
54		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐿) = (0g‘𝐿)
55	38	snssd 4765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
56	8, 55	unssd 4144	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
57	6, 45, 56	fldgensdrg 33396	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58		issdrg 20721	. . . . . . . . . . . . . . . . . . 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 20535	. . . . . . . . . . . . . . . . . 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 20419	. . . . . . . . . . . . . . . 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 19213	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 = (𝐽 ∘ 𝑁))
67	66	rneqd 5887	. . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 ~QG 𝑍) ∈ V)
71	3	simp3d 1144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
72	32, 71	ply1lvec 33640	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LVec)
73	68, 69, 70, 72	qusbas 17466	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
75	54	ghmker 19171	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝑃))
78	22, 74, 24, 77	qusrn 33490	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
80	20, 32, 6, 22, 36, 4, 38, 23, 79	evls1maplmhm 22321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
81	80	elexd 3464	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 ∈ V)
82	81	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
83	82	imaexd 7858	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → (𝐺 “ 𝑝) ∈ V)
84	83	uniexd 7687	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → ∪ (𝐺 “ 𝑝) ∈ V)
85	27, 84	dmmptd 6637	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐽 = (Base‘𝑄))
86	73, 78, 85	3eqtr4rd 2782	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐽 = ran 𝑁)
87		rncoeq 5931	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = ran 𝐽)
90	89	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐿 ↾s ran 𝐽))
91		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s ran 𝐽) = (𝐿 ↾s ran 𝐽)
92	9	subrgcrng 20508	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
93	36, 4, 92	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
94	21	ply1crng 22139	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝑃 ∈ CRing)
95	93, 94	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
96	54, 63, 25, 26, 27, 95	rhmquskerlem 33506	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐿))
97	20, 32, 6, 22, 36, 4, 38, 23	evls1maprnss 22322	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ⊆ ran 𝐺)
98		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝐸) = (1r‘𝐸)
99	9, 98	subrg1 20515	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) = (1r‘𝐾))
100	4, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐾))
101	98	subrg1cl 20513	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
102	4, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
103	100, 102	eqeltrrd 2837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ∈ 𝐹)
104	97, 103	sseldd 3934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐾) ∈ ran 𝐺)
105		drngnzr 20681	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
106	98, 35	nzrnz 20448	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ NzRing → (1r‘𝐸) ≠ (0g‘𝐸))
107	45, 105, 106	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ≠ (0g‘𝐸))
108	36	crnggrpd 20182	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	108	grpmndd 18876	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ Mnd)
110		sdrgsubrg 20724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111		subrgsubg 20510	. . . . . . . . . . . . . . . . . . 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 19064	. . . . . . . . . . . . . . . . . 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 33397	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
116	15, 6, 35	ress0g 18687	. . . . . . . . . . . . . . . . 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 3008	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ≠ (0g‘𝐿))
119		nelsn 4623	. . . . . . . . . . . . . . 15 ⊢ ((1r‘𝐾) ≠ (0g‘𝐿) → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
120	118, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
121		nelne1 3029	. . . . . . . . . . . . . 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 3000	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 ≠ {(0g‘𝐿)})
124		eqid 2736	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑃) = (oppr‘𝑃)
125	9	sdrgdrng 20723	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126		drngnzr 20681	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
127	1, 125, 126	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
128	21	ply1nz 26083	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129	127, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ NzRing)
130		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
131		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
132	9	fveq2i 6837	. . . . . . . . . . . . . . . 16 ⊢ (idlGen1p‘𝐾) = (idlGen1p‘(𝐸 ↾s 𝐹))
133	20, 32, 6, 18, 1, 38, 35, 130, 131, 132	ply1annig1p 33861	. . . . . . . . . . . . . . 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 4592	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {(0g‘𝐸)} = {(0g‘𝐿)})
135	134	imaeq2d 6019	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)}))
136	25, 135	eqtr4id 2790	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)}))
137	22	mpteq1i 5189	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
138	23, 137	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
139	20, 32, 6, 36, 4, 38, 35, 130, 138	ply1annidllem 33858	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)}))
140	136, 139	eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
141		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
142	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplyval 33862	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) = ((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}))
143	142	sneqd 4592	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} = {((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})
144	143	fveq2d 6838	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑃) = (0g‘𝑃)
147		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
148	147, 18, 1, 141, 19	irngnminplynz 33869	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
149		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
150	149, 9, 21, 22, 4, 147	ressply10g 33648	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
151	148, 150	neeqtrd 3001	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘𝑃))
152	20, 32, 6, 18, 1, 38, 141, 146, 151	minplyirred 33868	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154		fldsdrgfld 20731	. . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Field)
157	21	ply1pid 26144	. . . . . . . . . . . . . . . . 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 33863	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160	159, 22	eleqtrrdi 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
161	95	crngringd 20181	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
162	160	snssd 4765	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
164	131, 22, 163	rspcl 21190	. . . . . . . . . . . . . . . . 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 33553	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘𝑃))
169		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170	169, 124	crngmxidl 33550	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
171	95, 170	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
172	168, 171	eleqtrd 2838	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘(oppr‘𝑃)))
173	124, 26, 129, 168, 172	qsdrngi 33576	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ DivRing)
174	91, 54, 96, 123, 173	rndrhmcl 33378	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐽) ∈ DivRing)
175	90, 174	eqeltrd 2836	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) ∈ DivRing)
176	53, 175	eqeltrrd 2837	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾s ran 𝐺) ∈ DivRing)
177		issdrg 20721	. . . . . . . . 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 6834	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (var1‘𝐾) → (𝑂‘𝑝) = (𝑂‘(var1‘𝐾)))
180	179	fveq1d 6836	. . . . . . . . . . . . 13 ⊢ (𝑝 = (var1‘𝐾) → ((𝑂‘𝑝)‘𝐴) = ((𝑂‘(var1‘𝐾))‘𝐴))
181	180	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑝 = (var1‘𝐾) → (𝐴 = ((𝑂‘𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴)))
182	9, 71	eqeltrid 2840	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ DivRing)
183	182	drngringd 20670	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
184		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
185	184, 21, 22	vr1cl 22158	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (var1‘𝐾) ∈ 𝑈)
186	183, 185	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (var1‘𝐾) ∈ 𝑈)
187	20, 184, 9, 6, 36, 4	evls1var 22282	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(var1‘𝐾)) = ( I ↾ (Base‘𝐸)))
188	187	fveq1d 6836	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(var1‘𝐾))‘𝐴) = (( I ↾ (Base‘𝐸))‘𝐴))
189		fvresi 7119	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴))
192	181, 186, 191	rspcedvdw 3579	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑝 ∈ 𝑈 𝐴 = ((𝑂‘𝑝)‘𝐴))
193	23, 192, 19	elrnmptd 5912	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ran 𝐺)
194	193	snssd 4765	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ ran 𝐺)
195	97, 194	unssd 4144	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
196	6, 45, 178, 195	fldgenssp 33400	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
197	44, 196	eqssd 3951	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = (𝐸 fldGen (𝐹 ∪ {𝐴})))
198	15, 6	ressbas2 17165	. . . . . . 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 2737	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
201	6, 45, 56	fldgenssid 33395	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202	201	unssad 4145	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203	202, 199	sseqtrd 3970	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
204	200, 203	srabase 21129	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205	197, 199, 204	3eqtrd 2775	. . . . 5 ⊢ (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206		imaeq2 6015	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝐺 “ 𝑞) = (𝐺 “ 𝑝))
207	206	unieqd 4876	. . . . . 6 ⊢ (𝑞 = 𝑝 → ∪ (𝐺 “ 𝑞) = ∪ (𝐺 “ 𝑝))
208	207	cbvmptv 5202	. . . . 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 33498	. . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐿))
211	200, 210, 203	sralmod0 21140	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212	211	sneqd 4592	. . . . . . . . . . . 12 ⊢ (𝜑 → {(0g‘𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213	212	imaeq2d 6019	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
214	25, 213	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
215	214	oveq2d 7374	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ~QG 𝑍) = (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
216	215	oveq2d 7374	. . . . . . . 8 ⊢ (𝜑 → (𝑃 /s (𝑃 ~QG 𝑍)) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
217	26, 216	eqtrid 2783	. . . . . . 7 ⊢ (𝜑 → 𝑄 = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218	217	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219	218	mpteq1d 5188	. . . . 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 2796	. . . 4 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)))
221	217	oveq1d 7373	. . . 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 2851	. . 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 33876	. . 3 ⊢ (𝜑 → 𝑄 ∈ LVec)
224	222, 223	lmimdim 33760	. 2 ⊢ (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
225	6, 18, 56	fldgenfld 33402	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
226	15, 225	eqeltrid 2840	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Field)
227	9, 15, 16, 17, 18, 1, 19	algextdeglem1 33874	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
228	11	oveq2d 7374	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
229	227, 228	eqtr3d 2773	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
230	15	subsubrg 20531	. . . . . . 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 2837	. . . 4 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234		brfldext 33802	. . . . 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 33810	. . 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 2781	1 ⊢ (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  {csn 4580  ∪ cuni 4863   class class class wbr 5098   ↦ cmpt 5179   I cid 5518  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7358  [cec 8633   / cqs 8634  Basecbs 17136   ↾_s cress 17157  0_gc0g 17359   /_s cqus 17426  Mndcmnd 18659  SubGrpcsubg 19050  NrmSGrpcnsg 19051   ~_QG cqg 19052   GrpHom cghm 19141  1_rcur 20116  Ringcrg 20168  CRingccrg 20169  opp_rcoppr 20272  Irredcir 20292   RingHom crh 20405  NzRingcnzr 20445  SubRingcsubrg 20502  DivRingcdr 20662  Fieldcfield 20663  SubDRingcsdrg 20719   LMHom clmhm 20971   LMIso clmim 20972  LVecclvec 21054  subringAlg csra 21123  LIdealclidl 21161  RSpancrsp 21162  PIDcpid 21291  var₁cv1 22116  Poly₁cpl1 22117   evalSub₁ ces1 22257  deg₁cdg1 26015  idlGen_1pcig1p 26091   fldGen cfldgen 33392  MaxIdealcmxidl 33540  dimcldim 33755  /_FldExtcfldext 33795  [:]cextdg 33797   IntgRing cirng 33840   minPoly cminply 33856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550  ax-ac2 10373  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-rpss 7668  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-ec 8637  df-qs 8641  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-inf 9346  df-oi 9415  df-r1 9676  df-rank 9677  df-dju 9813  df-card 9851  df-acn 9854  df-ac 10026  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ocomp 17198  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-imas 17429  df-qus 17430  df-mre 17505  df-mrc 17506  df-mri 17507  df-acs 17508  df-proset 18217  df-drs 18218  df-poset 18236  df-ipo 18451  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-nsg 19054  df-eqg 19055  df-ghm 19142  df-gim 19188  df-cntz 19246  df-oppg 19275  df-lsm 19565  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-srg 20122  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-irred 20295  df-invr 20324  df-dvr 20337  df-rhm 20408  df-nzr 20446  df-subrng 20479  df-subrg 20503  df-rlreg 20627  df-domn 20628  df-idom 20629  df-drng 20664  df-field 20665  df-sdrg 20720  df-lmod 20813  df-lss 20883  df-lsp 20923  df-lmhm 20974  df-lmim 20975  df-lbs 21027  df-lvec 21055  df-sra 21125  df-rgmod 21126  df-lidl 21163  df-rsp 21164  df-2idl 21205  df-lpidl 21277  df-lpir 21278  df-pid 21292  df-cnfld 21310  df-dsmm 21687  df-frlm 21702  df-uvc 21738  df-lindf 21761  df-linds 21762  df-assa 21808  df-asp 21809  df-ascl 21810  df-psr 21865  df-mvr 21866  df-mpl 21867  df-opsr 21869  df-evls 22029  df-evl 22030  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-evls1 22259  df-evl1 22260  df-mdeg 26016  df-deg1 26017  df-mon1 26092  df-uc1p 26093  df-q1p 26094  df-r1p 26095  df-ig1p 26096  df-fldgen 33393  df-mxidl 33541  df-dim 33756  df-fldext 33798  df-extdg 33799  df-irng 33841  df-minply 33857

This theorem is referenced by:  algextdeg  33882