Theorem algextdeglem4 34402

Description: Lemma for algextdeg 34407. By lmhmqusker 34018, the surjective module homomorphism 𝐺 described in algextdeglem2 34400 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 21054	. . . . . . . 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 20838	. . . . . 6 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6		eqid 2760	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
7	6	subgss 19346	. . . . . 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 17425	. . . . 5 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
11	8, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
12	11	fveq2d 6894	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
13	12	fveq2d 6894	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14		eqid 2760	. . . . 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 34400	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29		eqid 2760	. . . . 5 ⊢ (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30		eqid 2760	. . . . 5 ⊢ (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
31	9	fveq2i 6893	. . . . . . . . . . 11 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	21, 31	eqtri 2783	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33	18	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐸 ∈ Field)
34	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐹 ∈ (SubDRing‘𝐸))
35		eqid 2760	. . . . . . . . . . . . 13 ⊢ (0g‘𝐸) = (0g‘𝐸)
36	18	fldcrngd 21004	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ CRing)
37	20, 9, 6, 35, 36, 4	irngssv 34370	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
38	37, 19	sseldd 3910	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
39	38	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐴 ∈ (Base‘𝐸))
40		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
41	6, 20, 32, 22, 33, 34, 39, 40	evls1fldgencl 34352	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
42	41	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	23	rnmptss 7128	. . . . . . . 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 21003	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ DivRing)
46	20, 32, 6, 22, 36, 4, 38, 23	evls1maprhm 22703	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐸))
47		rnrhmsubrg 20866	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐸) → ran 𝐺 ∈ (SubRing‘𝐸))
48	46, 47	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ (SubRing‘𝐸))
49	15	oveq1i 7434	. . . . . . . . . . 11 ⊢ (𝐿 ↾s ran 𝐺) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺)
50		ovex 7457	. . . . . . . . . . . 12 ⊢ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V
51		ressabs 17435	. . . . . . . . . . . 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 599	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
53	49, 52	eqtrid 2807	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
54		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐿) = (0g‘𝐿)
55	38	snssd 4743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
56	8, 55	unssd 4189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
57	6, 45, 56	fldgensdrg 33926	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58		issdrg 21054	. . . . . . . . . . . . . . . . . . 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 20863	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ ran 𝐺 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) → (𝐺 ∈ (𝑃 RingHom 𝐸) ↔ 𝐺 ∈ (𝑃 RingHom 𝐿)))
62	61	biimpa 482	. . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐿))
64		rhmghm 20723	. . . . . . . . . . . . . . . 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 19507	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 = (𝐽 ∘ 𝑁))
67	66	rneqd 5921	. . . . . . . . . . . . 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 7459	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 ~QG 𝑍) ∈ V)
71	3	simp3d 1162	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
72	32, 71	ply1lvec 34141	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LVec)
73	68, 69, 70, 72	qusbas 17726	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
75	54	ghmker 19465	. . . . . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝑃))
78	22, 74, 24, 77	qusrn 34010	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
80	20, 32, 6, 22, 36, 4, 38, 23, 79	evls1maplmhm 22704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
81	80	elexd 3473	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 ∈ V)
82	81	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
83	82	imaexd 7932	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → (𝐺 “ 𝑝) ∈ V)
84	83	uniexd 7763	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → ∪ (𝐺 “ 𝑝) ∈ V)
85	27, 84	dmmptd 6689	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐽 = (Base‘𝑄))
86	73, 78, 85	3eqtr4rd 2806	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐽 = ran 𝑁)
87		rncoeq 5964	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = ran 𝐽)
90	89	oveq2d 7440	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐿 ↾s ran 𝐽))
91		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s ran 𝐽) = (𝐿 ↾s ran 𝐽)
92	9	subrgcrng 20836	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
93	36, 4, 92	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
94	21	ply1crng 22525	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝑃 ∈ CRing)
95	93, 94	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
96	54, 63, 25, 26, 27, 95	rhmquskerlem 34025	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐿))
97	20, 32, 6, 22, 36, 4, 38, 23	evls1maprnss 22705	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ⊆ ran 𝐺)
98		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝐸) = (1r‘𝐸)
99	9, 98	subrg1 20843	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) = (1r‘𝐾))
100	4, 99	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐾))
101	98	subrg1cl 20841	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
102	4, 101	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
103	100, 102	eqeltrrd 2861	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ∈ 𝐹)
104	97, 103	sseldd 3910	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐾) ∈ ran 𝐺)
105		drngnzr 21011	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
106	98, 35	nzrnz 20774	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ NzRing → (1r‘𝐸) ≠ (0g‘𝐸))
107	45, 105, 106	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ≠ (0g‘𝐸))
108	36	crnggrpd 20483	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	108	grpmndd 19166	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ Mnd)
110		sdrgsubrg 21057	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111		subrgsubg 20838	. . . . . . . . . . . . . . . . . . 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 19353	. . . . . . . . . . . . . . . . . 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 33927	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
116	15, 6, 35	ress0g 18963	. . . . . . . . . . . . . . . . 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 3031	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ≠ (0g‘𝐿))
119		nelsn 4623	. . . . . . . . . . . . . . 15 ⊢ ((1r‘𝐾) ≠ (0g‘𝐿) → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
120	118, 119	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
121		nelne1 3052	. . . . . . . . . . . . . 14 ⊢ (((1r‘𝐾) ∈ ran 𝐺 ∧ ¬ (1r‘𝐾) ∈ {(0g‘𝐿)}) → ran 𝐺 ≠ {(0g‘𝐿)})
122	104, 120, 121	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐺 ≠ {(0g‘𝐿)})
123	89, 122	eqnetrrd 3023	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 ≠ {(0g‘𝐿)})
124		eqid 2760	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑃) = (oppr‘𝑃)
125	9	sdrgdrng 21056	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126		drngnzr 21011	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
127	1, 125, 126	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
128	21	ply1nz 26449	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129	127, 128	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ NzRing)
130		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
131		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
132	9	fveq2i 6893	. . . . . . . . . . . . . . . 16 ⊢ (idlGen1p‘𝐾) = (idlGen1p‘(𝐸 ↾s 𝐹))
133	20, 32, 6, 18, 1, 38, 35, 130, 131, 132	ply1annig1p 34386	. . . . . . . . . . . . . . 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 6054	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)}))
136	25, 135	eqtr4id 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)}))
137	22	mpteq1i 5195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
138	23, 137	eqtri 2783	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
139	20, 32, 6, 36, 4, 38, 35, 130, 138	ply1annidllem 34383	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)}))
140	136, 139	eqtr4d 2798	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
141		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
142	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplyval 34387	. . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑃) = (0g‘𝑃)
147		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
148	147, 18, 1, 141, 19	irngnminplynz 34394	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
149		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
150	149, 9, 21, 22, 4, 147	ressply10g 34149	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
151	148, 150	neeqtrd 3024	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘𝑃))
152	20, 32, 6, 18, 1, 38, 141, 146, 151	minplyirred 34393	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154		fldsdrgfld 21064	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
155	18, 1, 154	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
156	9, 155	eqeltrid 2864	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Field)
157	21	ply1pid 26510	. . . . . . . . . . . . . . . . 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 34388	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160	159, 22	eleqtrrdi 2871	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
161	95	crngringd 20482	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
162	160	snssd 4743	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
164	131, 22, 163	rspcl 21527	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Ring ∧ {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈) → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
165	161, 162, 164	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) ∈ (LIdeal‘𝑃))
166	22, 131, 146, 153, 158, 160, 151, 165	mxidlirred 34047	. . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘𝑃))
169		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170	169, 124	crngmxidl 34044	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
171	95, 170	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
172	168, 171	eleqtrd 2862	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘(oppr‘𝑃)))
173	124, 26, 129, 168, 172	qsdrngi 34069	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ DivRing)
174	91, 54, 96, 123, 173	rndrhmcl 33908	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐽) ∈ DivRing)
175	90, 174	eqeltrd 2860	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) ∈ DivRing)
176	53, 175	eqeltrrd 2861	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾s ran 𝐺) ∈ DivRing)
177		issdrg 21054	. . . . . . . . 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 6890	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (var1‘𝐾) → (𝑂‘𝑝) = (𝑂‘(var1‘𝐾)))
180	179	fveq1d 6892	. . . . . . . . . . . . 13 ⊢ (𝑝 = (var1‘𝐾) → ((𝑂‘𝑝)‘𝐴) = ((𝑂‘(var1‘𝐾))‘𝐴))
181	180	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑝 = (var1‘𝐾) → (𝐴 = ((𝑂‘𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴)))
182	9, 71	eqeltrid 2864	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ DivRing)
183	182	drngringd 20997	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
184		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
185	184, 21, 22	vr1cl 22544	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (var1‘𝐾) ∈ 𝑈)
186	183, 185	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (var1‘𝐾) ∈ 𝑈)
187	20, 184, 9, 6, 36, 4	evls1var 22665	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(var1‘𝐾)) = ( I ↾ (Base‘𝐸)))
188	187	fveq1d 6892	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(var1‘𝐾))‘𝐴) = (( I ↾ (Base‘𝐸))‘𝐴))
189		fvresi 7182	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 5946	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ran 𝐺)
194	193	snssd 4743	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ ran 𝐺)
195	97, 194	unssd 4189	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
196	6, 45, 178, 195	fldgenssp 33930	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
197	44, 196	eqssd 3926	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = (𝐸 fldGen (𝐹 ∪ {𝐴})))
198	15, 6	ressbas2 17425	. . . . . . 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 2761	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
201	6, 45, 56	fldgenssid 33925	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202	201	unssad 4190	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203	202, 199	sseqtrd 3945	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
204	200, 203	srabase 21461	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205	197, 199, 204	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206		imaeq2 6050	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝐺 “ 𝑞) = (𝐺 “ 𝑝))
207	206	unieqd 4879	. . . . . 6 ⊢ (𝑞 = 𝑝 → ∪ (𝐺 “ 𝑞) = ∪ (𝐺 “ 𝑝))
208	207	cbvmptv 5208	. . . . 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 34018	. . . 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 2761	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐿))
211	200, 210, 203	sralmod0 21472	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212	211	sneqd 4592	. . . . . . . . . . . 12 ⊢ (𝜑 → {(0g‘𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213	212	imaeq2d 6054	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
214	25, 213	eqtrid 2807	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
215	214	oveq2d 7440	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ~QG 𝑍) = (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
216	215	oveq2d 7440	. . . . . . . 8 ⊢ (𝜑 → (𝑃 /s (𝑃 ~QG 𝑍)) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
217	26, 216	eqtrid 2807	. . . . . . 7 ⊢ (𝜑 → 𝑄 = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218	217	fveq2d 6894	. . . . . 6 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219	218	mpteq1d 5194	. . . . 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 2820	. . . 4 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)))
221	217	oveq1d 7439	. . . 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 2875	. . 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 34401	. . 3 ⊢ (𝜑 → 𝑄 ∈ LVec)
224	222, 223	lmimdim 34286	. 2 ⊢ (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
225	6, 18, 56	fldgenfld 33932	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
226	15, 225	eqeltrid 2864	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Field)
227	9, 15, 16, 17, 18, 1, 19	algextdeglem1 34399	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
228	11	oveq2d 7440	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
229	227, 228	eqtr3d 2797	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
230	15	subsubrg 20859	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐹 ∈ (SubRing‘𝐿) ↔ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))))
231	230	biimpar 483	. . . . . 6 ⊢ (((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) ∧ (𝐹 ∈ (SubRing‘𝐸) ∧ 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))) → 𝐹 ∈ (SubRing‘𝐿))
232	60, 4, 202, 231	syl12anc 850	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
233	11, 232	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234		brfldext 34327	. . . . 5 ⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
235	234	biimpar 483	. . . 4 ⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
236	226, 156, 229, 233, 235	syl22anc 852	. . 3 ⊢ (𝜑 → 𝐿/FldExt𝐾)
237		extdgval 34335	. . 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 2805	1 ⊢ (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∪ cun 4148  {csn 4584  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185   I cid 5545  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  ‘cfv 6543  (class class class)co 7424  [cec 8722   / cqs 8723  Basecbs 17396   ↾_s cress 17417  0_gc0g 17619   /_s cqus 17686  Mndcmnd 18932  SubGrpcsubg 19339  NrmSGrpcnsg 19340   ~_QG cqg 19341   GrpHom cghm 19436  1_rcur 20416  Ringcrg 20468  CRingccrg 20469  opp_rcoppr 20575  Irredcir 20595   RingHom crh 20708  NzRingcnzr 20771  SubRingcsubrg 20830  DivRingcdr 20989  Fieldcfield 20990  SubDRingcsdrg 21052   LMHom clmhm 21303   LMIso clmim 21304  LVecclvec 21386  subringAlg csra 21455  LIdealclidl 21493  RSpancrsp 21494  PIDcpid 21669  var₁cv1 22503  Poly₁cpl1 22504   evalSub₁ ces1 22640  deg₁cdg1 26381  idlGen_1pcig1p 26457   fldGen cfldgen 33922  MaxIdealcmxidl 34034  dimcldim 34281  /_FldExtcfldext 34320  [:]cextdg 34322   IntgRing cirng 34365   minPoly cminply 34381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-ac2 10548  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285  ax-addf 11286

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-ofr 7698  df-rpss 7743  df-om 7882  df-1st 8005  df-2nd 8006  df-supp 8184  df-tpos 8250  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-er 8724  df-ec 8726  df-qs 8730  df-map 8856  df-pm 8857  df-ixp 8933  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-fsupp 9361  df-sup 9441  df-inf 9442  df-oi 9511  df-dju 9989  df-card 10027  df-acn 10030  df-ac 10202  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-xnn0 12687  df-z 12701  df-dec 12822  df-uz 12973  df-fz 13647  df-fzo 13796  df-seq 14152  df-hash 14482  df-struct 17334  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-ress 17418  df-plusg 17450  df-mulr 17451  df-starv 17452  df-sca 17453  df-vsca 17454  df-ip 17455  df-tset 17456  df-ple 17457  df-ocomp 17458  df-ds 17459  df-unif 17460  df-hom 17461  df-cco 17462  df-0g 17621  df-gsum 17622  df-prds 17627  df-pws 17629  df-imas 17689  df-qus 17690  df-mre 17765  df-mrc 17766  df-mri 17767  df-acs 17768  df-proset 18477  df-drs 18478  df-poset 18496  df-ipo 18711  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-mhm 18987  df-submnd 18988  df-grp 19156  df-minusg 19157  df-sbg 19158  df-mulg 19287  df-subg 19342  df-nsg 19343  df-eqg 19344  df-ghm 19437  df-gim 19482  df-cntz 19540  df-oppg 19569  df-lsm 19859  df-cmn 20005  df-abl 20006  df-mgp 20370  df-rng 20384  df-ur 20417  df-srg 20422  df-ring 20470  df-cring 20471  df-oppr 20576  df-dvdsr 20596  df-unit 20597  df-irred 20598  df-invr 20627  df-dvr 20640  df-rhm 20711  df-nzr 20772  df-subrng 20807  df-subrg 20831  df-rlreg 20955  df-domn 20956  df-idom 20957  df-drng 20991  df-field 20992  df-sdrg 21053  df-lmod 21146  df-lss 21216  df-lsp 21256  df-lmhm 21306  df-lmim 21307  df-lbs 21359  df-lvec 21387  df-sra 21457  df-rgmod 21458  df-lidl 21495  df-rsp 21496  df-2idl 21552  df-lpidl 21655  df-lpir 21656  df-pid 21670  df-cnfld 21688  df-dsmm 22047  df-frlm 22062  df-uvc 22098  df-lindf 22121  df-linds 22122  df-assa 22170  df-asp 22171  df-ascl 22172  df-psr 22226  df-mvr 22227  df-mpl 22228  df-opsr 22230  df-evls 22392  df-evl 22393  df-psr1 22507  df-vr1 22508  df-ply1 22509  df-coe1 22510  df-evls1 22642  df-evl1 22643  df-mdeg 26382  df-deg1 26383  df-mon1 26458  df-uc1p 26459  df-q1p 26460  df-r1p 26461  df-ig1p 26462  df-fldgen 33923  df-mxidl 34035  df-dim 34282  df-fldext 34323  df-extdg 34324  df-irng 34366  df-minply 34382

This theorem is used by:  algextdeg  34407