Theorem algextdeglem4 34174

Description: Lemma for algextdeg 34179. By lmhmqusker 33790, the surjective module homomorphism 𝐺 described in algextdeglem2 34172 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 20941	. . . . . . . 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 20726	. . . . . 6 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
6		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
7	6	subgss 19237	. . . . . 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 17320	. . . . 5 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
11	8, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
12	11	fveq2d 6889	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘(Base‘𝐾)))
13	12	fveq2d 6889	. 2 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐿)‘𝐹)) = (dim‘((subringAlg ‘𝐿)‘(Base‘𝐾))))
14		eqid 2765	. . . . 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 34172	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
29		eqid 2765	. . . . 5 ⊢ (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})
30		eqid 2765	. . . . 5 ⊢ (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
31	9	fveq2i 6888	. . . . . . . . . . 11 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	21, 31	eqtri 2788	. . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (0g‘𝐸) = (0g‘𝐸)
36	18	fldcrngd 20892	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ CRing)
37	20, 9, 6, 35, 36, 4	irngssv 34142	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
38	37, 19	sseldd 3939	. . . . . . . . . . 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 34124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
42	41	ralrimiva 3159	. . . . . . . 8 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 ((𝑂‘𝑝)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
43	23	rnmptss 7122	. . . . . . . 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 20891	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ DivRing)
46	20, 32, 6, 22, 36, 4, 38, 23	evls1maprhm 22586	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑃 RingHom 𝐸))
47		rnrhmsubrg 20754	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑃 RingHom 𝐸) → ran 𝐺 ∈ (SubRing‘𝐸))
48	46, 47	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ (SubRing‘𝐸))
49	15	oveq1i 7429	. . . . . . . . . . 11 ⊢ (𝐿 ↾s ran 𝐺) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ↾s ran 𝐺)
50		ovex 7452	. . . . . . . . . . . 12 ⊢ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ V
51		ressabs 17330	. . . . . . . . . . . 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 2812	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐸 ↾s ran 𝐺))
54		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐿) = (0g‘𝐿)
55	38	snssd 4754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸))
56	8, 55	unssd 4145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸))
57	6, 45, 56	fldgensdrg 33699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸))
58		issdrg 20941	. . . . . . . . . . . . . . . . . . 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 20751	. . . . . . . . . . . . . . . . . 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 20612	. . . . . . . . . . . . . . . 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 19398	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 = (𝐽 ∘ 𝑁))
67	66	rneqd 5930	. . . . . . . . . . . . 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 7454	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 ~QG 𝑍) ∈ V)
71	3	simp3d 1162	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
72	32, 71	ply1lvec 33913	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ LVec)
73	68, 69, 70, 72	qusbas 17621	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 / (𝑃 ~QG 𝑍)) = (Base‘𝑄))
74		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 / (𝑃 ~QG 𝑍)) = (𝑈 / (𝑃 ~QG 𝑍))
75	54	ghmker 19356	. . . . . . . . . . . . . . . . . 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 2869	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ (NrmSGrp‘𝑃))
78	22, 74, 24, 77	qusrn 33782	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝑁 = (𝑈 / (𝑃 ~QG 𝑍)))
79		eqid 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
80	20, 32, 6, 22, 36, 4, 38, 23, 79	evls1maplmhm 22587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐸)‘𝐹)))
81	80	elexd 3480	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺 ∈ V)
82	81	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → 𝐺 ∈ V)
83	82	imaexd 7919	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → (𝐺 “ 𝑝) ∈ V)
84	83	uniexd 7750	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (Base‘𝑄)) → ∪ (𝐺 “ 𝑝) ∈ V)
85	27, 84	dmmptd 6684	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐽 = (Base‘𝑄))
86	73, 78, 85	3eqtr4rd 2811	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐽 = ran 𝑁)
87		rncoeq 5973	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 = ran 𝐽)
90	89	oveq2d 7435	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) = (𝐿 ↾s ran 𝐽))
91		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝐿 ↾s ran 𝐽) = (𝐿 ↾s ran 𝐽)
92	9	subrgcrng 20724	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝐾 ∈ CRing)
93	36, 4, 92	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ CRing)
94	21	ply1crng 22408	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → 𝑃 ∈ CRing)
95	93, 94	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ CRing)
96	54, 63, 25, 26, 27, 95	rhmquskerlem 33797	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐿))
97	20, 32, 6, 22, 36, 4, 38, 23	evls1maprnss 22588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ⊆ ran 𝐺)
98		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝐸) = (1r‘𝐸)
99	9, 98	subrg1 20731	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) = (1r‘𝐾))
100	4, 99	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐾))
101	98	subrg1cl 20729	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
102	4, 101	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
103	100, 102	eqeltrrd 2866	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ∈ 𝐹)
104	97, 103	sseldd 3939	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝐾) ∈ ran 𝐺)
105		drngnzr 20898	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
106	98, 35	nzrnz 20662	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ∈ NzRing → (1r‘𝐸) ≠ (0g‘𝐸))
107	45, 105, 106	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1r‘𝐸) ≠ (0g‘𝐸))
108	36	crnggrpd 20373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ Grp)
109	108	grpmndd 19057	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸 ∈ Mnd)
110		sdrgsubrg 20944	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸))
111		subrgsubg 20726	. . . . . . . . . . . . . . . . . . 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 19244	. . . . . . . . . . . . . . . . . 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 33700	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸))
116	15, 6, 35	ress0g 18855	. . . . . . . . . . . . . . . . 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 3036	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1r‘𝐾) ≠ (0g‘𝐿))
119		nelsn 4634	. . . . . . . . . . . . . . 15 ⊢ ((1r‘𝐾) ≠ (0g‘𝐿) → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
120	118, 119	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (1r‘𝐾) ∈ {(0g‘𝐿)})
121		nelne1 3057	. . . . . . . . . . . . . 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 3028	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 ≠ {(0g‘𝐿)})
124		eqid 2765	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑃) = (oppr‘𝑃)
125	9	sdrgdrng 20943	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
126		drngnzr 20898	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
127	1, 125, 126	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ NzRing)
128	21	ply1nz 26330	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ NzRing → 𝑃 ∈ NzRing)
129	127, 128	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ NzRing)
130		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}
131		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
132	9	fveq2i 6888	. . . . . . . . . . . . . . . 16 ⊢ (idlGen1p‘𝐾) = (idlGen1p‘(𝐸 ↾s 𝐹))
133	20, 32, 6, 18, 1, 38, 35, 130, 131, 132	ply1annig1p 34158	. . . . . . . . . . . . . . 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 4603	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {(0g‘𝐸)} = {(0g‘𝐿)})
135	134	imaeq2d 6064	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)}))
136	25, 135	eqtr4id 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)}))
137	22	mpteq1i 5204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
138	23, 137	eqtri 2788	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))
139	20, 32, 6, 36, 4, 38, 35, 130, 138	ply1annidllem 34155	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)}))
140	136, 139	eqtr4d 2803	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})
141		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
142	20, 32, 6, 18, 1, 38, 35, 130, 131, 132, 141	minplyval 34159	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) = ((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}))
143	142	sneqd 4603	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} = {((idlGen1p‘𝐾)‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})
144	143	fveq2d 6889	. . . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}))
146		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑃) = (0g‘𝑃)
147		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
148	147, 18, 1, 141, 19	irngnminplynz 34166	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
149		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
150	149, 9, 21, 22, 4, 147	ressply10g 33921	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0g‘(Poly1‘𝐸)) = (0g‘𝑃))
151	148, 150	neeqtrd 3029	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ≠ (0g‘𝑃))
152	20, 32, 6, 18, 1, 38, 141, 146, 151	minplyirred 34165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Irred‘𝑃))
153		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)}) = ((RSpan‘𝑃)‘{((𝐸 minPoly 𝐹)‘𝐴)})
154		fldsdrgfld 20951	. . . . . . . . . . . . . . . . . . 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 2869	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Field)
157	21	ply1pid 26391	. . . . . . . . . . . . . . . . 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 34160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ (Base‘𝑃))
160	159, 22	eleqtrrdi 2876	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐸 minPoly 𝐹)‘𝐴) ∈ 𝑈)
161	95	crngringd 20372	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ Ring)
162	160	snssd 4754	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((𝐸 minPoly 𝐹)‘𝐴)} ⊆ 𝑈)
163		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
164	131, 22, 163	rspcl 21414	. . . . . . . . . . . . . . . . 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 33819	. . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘𝑃))
169		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (MaxIdeal‘𝑃) = (MaxIdeal‘𝑃)
170	169, 124	crngmxidl 33816	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ CRing → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
171	95, 170	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (MaxIdeal‘𝑃) = (MaxIdeal‘(oppr‘𝑃)))
172	168, 171	eleqtrd 2867	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ (MaxIdeal‘(oppr‘𝑃)))
173	124, 26, 129, 168, 172	qsdrngi 33841	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ DivRing)
174	91, 54, 96, 123, 173	rndrhmcl 33681	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 ↾s ran 𝐽) ∈ DivRing)
175	90, 174	eqeltrd 2865	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾s ran 𝐺) ∈ DivRing)
176	53, 175	eqeltrrd 2866	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾s ran 𝐺) ∈ DivRing)
177		issdrg 20941	. . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (var1‘𝐾) → (𝑂‘𝑝) = (𝑂‘(var1‘𝐾)))
180	179	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝑝 = (var1‘𝐾) → ((𝑂‘𝑝)‘𝐴) = ((𝑂‘(var1‘𝐾))‘𝐴))
181	180	eqeq2d 2776	. . . . . . . . . . . 12 ⊢ (𝑝 = (var1‘𝐾) → (𝐴 = ((𝑂‘𝑝)‘𝐴) ↔ 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴)))
182	9, 71	eqeltrid 2869	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ DivRing)
183	182	drngringd 20885	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
184		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
185	184, 21, 22	vr1cl 22427	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (var1‘𝐾) ∈ 𝑈)
186	183, 185	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (var1‘𝐾) ∈ 𝑈)
187	20, 184, 9, 6, 36, 4	evls1var 22548	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑂‘(var1‘𝐾)) = ( I ↾ (Base‘𝐸)))
188	187	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑂‘(var1‘𝐾))‘𝐴) = (( I ↾ (Base‘𝐸))‘𝐴))
189		fvresi 7177	. . . . . . . . . . . . . 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 2801	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = ((𝑂‘(var1‘𝐾))‘𝐴))
192	181, 186, 191	rspcedvdw 3586	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑝 ∈ 𝑈 𝐴 = ((𝑂‘𝑝)‘𝐴))
193	23, 192, 19	elrnmptd 5955	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ran 𝐺)
194	193	snssd 4754	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ ran 𝐺)
195	97, 194	unssd 4145	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ ran 𝐺)
196	6, 45, 178, 195	fldgenssp 33703	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ ran 𝐺)
197	44, 196	eqssd 3955	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = (𝐸 fldGen (𝐹 ∪ {𝐴})))
198	15, 6	ressbas2 17320	. . . . . . 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 2766	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹))
201	6, 45, 56	fldgenssid 33698	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
202	201	unssad 4146	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴})))
203	202, 199	sseqtrd 3974	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
204	200, 203	srabase 21348	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐹)))
205	197, 199, 204	3eqtrd 2804	. . . . 5 ⊢ (𝜑 → ran 𝐺 = (Base‘((subringAlg ‘𝐿)‘𝐹)))
206		imaeq2 6060	. . . . . . 7 ⊢ (𝑞 = 𝑝 → (𝐺 “ 𝑞) = (𝐺 “ 𝑝))
207	206	unieqd 4887	. . . . . 6 ⊢ (𝑞 = 𝑝 → ∪ (𝐺 “ 𝑞) = ∪ (𝐺 “ 𝑝))
208	207	cbvmptv 5217	. . . . 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 33790	. . . 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 2766	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐿))
211	200, 210, 203	sralmod0 21359	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹)))
212	211	sneqd 4603	. . . . . . . . . . . 12 ⊢ (𝜑 → {(0g‘𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))})
213	212	imaeq2d 6064	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
214	25, 213	eqtrid 2812	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))
215	214	oveq2d 7435	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ~QG 𝑍) = (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))
216	215	oveq2d 7435	. . . . . . . 8 ⊢ (𝜑 → (𝑃 /s (𝑃 ~QG 𝑍)) = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
217	26, 216	eqtrid 2812	. . . . . . 7 ⊢ (𝜑 → 𝑄 = (𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}))))
218	217	fveq2d 6889	. . . . . 6 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))))
219	218	mpteq1d 5203	. . . . 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 2825	. . . 4 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘(𝑃 /s (𝑃 ~QG (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})))) ↦ ∪ (𝐺 “ 𝑞)))
221	217	oveq1d 7434	. . . 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 2880	. . 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 34173	. . 3 ⊢ (𝜑 → 𝑄 ∈ LVec)
224	222, 223	lmimdim 34058	. 2 ⊢ (𝜑 → (dim‘𝑄) = (dim‘((subringAlg ‘𝐿)‘𝐹)))
225	6, 18, 56	fldgenfld 33705	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) ∈ Field)
226	15, 225	eqeltrid 2869	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Field)
227	9, 15, 16, 17, 18, 1, 19	algextdeglem1 34171	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
228	11	oveq2d 7435	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
229	227, 228	eqtr3d 2802	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
230	15	subsubrg 20747	. . . . . . 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 2866	. . . 4 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
234		brfldext 34099	. . . . 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 34107	. . 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 2810	1 ⊢ (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  {crab 3418  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  {csn 4591  ∪ cuni 4874   class class class wbr 5111   ↦ cmpt 5194   I cid 5557  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7419  [cec 8698   / cqs 8699  Basecbs 17291   ↾_s cress 17312  0_gc0g 17514   /_s cqus 17581  Mndcmnd 18824  SubGrpcsubg 19230  NrmSGrpcnsg 19231   ~_QG cqg 19232   GrpHom cghm 19327  1_rcur 20307  Ringcrg 20359  CRingccrg 20360  opp_rcoppr 20464  Irredcir 20484   RingHom crh 20597  NzRingcnzr 20659  SubRingcsubrg 20718  DivRingcdr 20877  Fieldcfield 20878  SubDRingcsdrg 20939   LMHom clmhm 21190   LMIso clmim 21191  LVecclvec 21273  subringAlg csra 21342  LIdealclidl 21380  RSpancrsp 21381  PIDcpid 21554  var₁cv1 22386  Poly₁cpl1 22387   evalSub₁ ces1 22523  deg₁cdg1 26262  idlGen_1pcig1p 26338   fldGen cfldgen 33695  MaxIdealcmxidl 33806  dimcldim 34053  /_FldExtcfldext 34092  [:]cextdg 34094   IntgRing cirng 34137   minPoly cminply 34153

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-reg 9561  ax-inf2 9617  ax-ac2 10462  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193  ax-addf 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-ofr 7685  df-rpss 7730  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-sup 9409  df-inf 9410  df-oi 9479  df-r1 9743  df-rank 9744  df-scott 9865  df-dju 9903  df-card 9941  df-acn 9944  df-ac 10116  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-xnn0 12593  df-z 12607  df-dec 12728  df-uz 12879  df-fz 13552  df-fzo 13700  df-seq 14056  df-hash 14385  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-starv 17347  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ocomp 17353  df-ds 17354  df-unif 17355  df-hom 17356  df-cco 17357  df-0g 17516  df-gsum 17517  df-prds 17522  df-pws 17524  df-imas 17584  df-qus 17585  df-mre 17660  df-mrc 17661  df-mri 17662  df-acs 17663  df-proset 18372  df-drs 18373  df-poset 18391  df-ipo 18606  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-mhm 18878  df-submnd 18879  df-grp 19047  df-minusg 19048  df-sbg 19049  df-mulg 19178  df-subg 19233  df-nsg 19234  df-eqg 19235  df-ghm 19328  df-gim 19373  df-cntz 19431  df-oppg 19460  df-lsm 19750  df-cmn 19896  df-abl 19897  df-mgp 20261  df-rng 20275  df-ur 20308  df-srg 20313  df-ring 20361  df-cring 20362  df-oppr 20465  df-dvdsr 20485  df-unit 20486  df-irred 20487  df-invr 20516  df-dvr 20529  df-rhm 20600  df-nzr 20660  df-subrng 20695  df-subrg 20719  df-rlreg 20843  df-domn 20844  df-idom 20845  df-drng 20879  df-field 20880  df-sdrg 20940  df-lmod 21033  df-lss 21103  df-lsp 21143  df-lmhm 21193  df-lmim 21194  df-lbs 21246  df-lvec 21274  df-sra 21344  df-rgmod 21345  df-lidl 21382  df-rsp 21383  df-2idl 21439  df-lpidl 21540  df-lpir 21541  df-pid 21555  df-cnfld 21573  df-dsmm 21932  df-frlm 21947  df-uvc 21983  df-lindf 22006  df-linds 22007  df-assa 22053  df-asp 22054  df-ascl 22055  df-psr 22109  df-mvr 22110  df-mpl 22111  df-opsr 22113  df-evls 22275  df-evl 22276  df-psr1 22390  df-vr1 22391  df-ply1 22392  df-coe1 22393  df-evls1 22525  df-evl1 22526  df-mdeg 26263  df-deg1 26264  df-mon1 26339  df-uc1p 26340  df-q1p 26341  df-r1p 26342  df-ig1p 26343  df-fldgen 33696  df-mxidl 33807  df-dim 34054  df-fldext 34095  df-extdg 34096  df-irng 34138  df-minply 34154

This theorem is used by:  algextdeg  34179