Theorem algextdeglem8 33084

Description: Lemma for algextdeg 33085. (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))
algextdeglem.r	⊢ 𝑅 = (rem1p‘𝐾)
algextdeglem.h	⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
algextdeglem.t	⊢ 𝑇 = (◡( deg1 ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))

Assertion

Ref	Expression
algextdeglem8	⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))

Distinct variable groups:   𝐴,𝑝   𝐸,𝑝   𝐹,𝑝,𝑥   𝐺,𝑝,𝑥   𝐻,𝑝,𝑥   𝐽,𝑝,𝑥   𝐾,𝑝   𝐿,𝑝,𝑥   𝑀,𝑝   𝑥,𝑁   𝑂,𝑝   𝑃,𝑝,𝑥   𝑄,𝑝,𝑥   𝑅,𝑝   𝑇,𝑝,𝑥   𝑈,𝑝,𝑥   𝑍,𝑝,𝑥   𝜑,𝑝,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑝)   𝑅(𝑥)   𝐸(𝑥)   𝐾(𝑥)   𝑀(𝑥)   𝑁(𝑝)   𝑂(𝑥)

Proof of Theorem algextdeglem8

Dummy variables 𝑞 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2732	. . . 4 ⊢ (𝜑 → (𝐻 “s 𝑃) = (𝐻 “s 𝑃))
2		algextdeglem.u	. . . . 5 ⊢ 𝑈 = (Base‘𝑃)
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑈 = (Base‘𝑃))
4		algextdeg.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5		algextdeg.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
6	5	sdrgdrng 20553	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
7	4, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ DivRing)
8	7	drngringd 20512	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐾 ∈ Ring)
10		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
11		eqid 2731	. . . . . . . . . . 11 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸))
12		algextdeg.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Field)
13		algextdeg.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
14		algextdeg.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))
15	5	fveq2i 6894	. . . . . . . . . . 11 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹))
16	11, 12, 4, 13, 14, 15	minplym1p 33076	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
18		eqid 2731	. . . . . . . . . 10 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾)
19		eqid 2731	. . . . . . . . . 10 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾)
20	18, 19	mon1puc1p 25917	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
21	9, 17, 20	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
22		algextdeglem.r	. . . . . . . . 9 ⊢ 𝑅 = (rem1p‘𝐾)
23		algextdeglem.y	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝐾)
24	22, 23, 2, 18	r1pcl 25924	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
25	9, 10, 21, 24	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
26		eqid 2731	. . . . . . . . . 10 ⊢ ( deg1 ‘𝐾) = ( deg1 ‘𝐾)
27	22, 23, 2, 18, 26	r1pdeglt 25925	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → (( deg1 ‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < (( deg1 ‘𝐾)‘(𝑀‘𝐴)))
28	9, 10, 21, 27	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (( deg1 ‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < (( deg1 ‘𝐾)‘(𝑀‘𝐴)))
29		algextdeg.d	. . . . . . . . . 10 ⊢ 𝐷 = ( deg1 ‘𝐸)
30		algextdeglem.o	. . . . . . . . . . . 12 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
31	5	fveq2i 6894	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	23, 31	eqtri 2759	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
34		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐸) = (0g‘𝐸)
35	12	fldcrngd 20517	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ CRing)
36		sdrgsubrg 20554	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
37	4, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
38	30, 5, 33, 34, 35, 37	irngssv 33056	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
39	38, 14	sseldd 3983	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
40		eqid 2731	. . . . . . . . . . . 12 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)}
41		eqid 2731	. . . . . . . . . . . 12 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
42		eqid 2731	. . . . . . . . . . . 12 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
43	30, 32, 33, 12, 4, 39, 34, 40, 41, 42, 13	minplycl 33071	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
44	43, 2	eleqtrrdi 2843	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
45	5, 29, 23, 2, 44, 37	ressdeg1 32940	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) = (( deg1 ‘𝐾)‘(𝑀‘𝐴)))
46	45	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝐷‘(𝑀‘𝐴)) = (( deg1 ‘𝐾)‘(𝑀‘𝐴)))
47	28, 46	breqtrrd 5176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (( deg1 ‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < (𝐷‘(𝑀‘𝐴)))
48		algextdeglem.t	. . . . . . . . 9 ⊢ 𝑇 = (◡( deg1 ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))
49	12	flddrngd 20516	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ DivRing)
50	49	drngringd 20512	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Ring)
51		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
52		eqid 2731	. . . . . . . . . . . . 13 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾)
53		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾))
54		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸))
55	51, 5, 23, 2, 37, 52, 53, 54	ressply1bas2 21983	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))))
56		inss2 4229	. . . . . . . . . . . 12 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸))
57	55, 56	eqsstrdi 4036	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸)))
58	57, 44	sseldd 3983	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)))
59	11, 12, 4, 13, 14	irngnminplynz 33075	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
60	29, 51, 11, 54	deg1nn0cl 25855	. . . . . . . . . 10 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
61	50, 58, 59, 60	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
62	23, 26, 48, 61, 8, 2	ply1degleel 32956	. . . . . . . 8 ⊢ (𝜑 → ((𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ↔ ((𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈 ∧ (( deg1 ‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < (𝐷‘(𝑀‘𝐴)))))
63	62	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → ((𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ↔ ((𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈 ∧ (( deg1 ‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < (𝐷‘(𝑀‘𝐴)))))
64	25, 47, 63	mpbir2and 710	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
65	64	ralrimiva 3145	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
66		oveq1 7419	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝𝑅(𝑀‘𝐴)) = (𝑞𝑅(𝑀‘𝐴)))
67	66	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ 𝑞 = (𝑞𝑅(𝑀‘𝐴))))
68		eqcom 2738	. . . . . . . 8 ⊢ (𝑞 = (𝑞𝑅(𝑀‘𝐴)) ↔ (𝑞𝑅(𝑀‘𝐴)) = 𝑞)
69	67, 68	bitrdi 287	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ (𝑞𝑅(𝑀‘𝐴)) = 𝑞))
70	23, 26, 48, 61, 8, 2	ply1degltel 32955	. . . . . . . 8 ⊢ (𝜑 → (𝑞 ∈ 𝑇 ↔ (𝑞 ∈ 𝑈 ∧ (( deg1 ‘𝐾)‘𝑞) ≤ ((𝐷‘(𝑀‘𝐴)) − 1))))
71	70	simprbda 498	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → 𝑞 ∈ 𝑈)
72	70	simplbda 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (( deg1 ‘𝐾)‘𝑞) ≤ ((𝐷‘(𝑀‘𝐴)) − 1))
73	45	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷‘(𝑀‘𝐴)) − 1) = ((( deg1 ‘𝐾)‘(𝑀‘𝐴)) − 1))
74	73	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((𝐷‘(𝑀‘𝐴)) − 1) = ((( deg1 ‘𝐾)‘(𝑀‘𝐴)) − 1))
75	72, 74	breqtrd 5174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (( deg1 ‘𝐾)‘𝑞) ≤ ((( deg1 ‘𝐾)‘(𝑀‘𝐴)) − 1))
76	26, 23, 2	deg1cl 25850	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝑈 → (( deg1 ‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
77	71, 76	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (( deg1 ‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
78	61	nn0zd 12591	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℤ)
79	45, 78	eqeltrrd 2833	. . . . . . . . . . 11 ⊢ (𝜑 → (( deg1 ‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
80	79	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (( deg1 ‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
81		degltlem1 25839	. . . . . . . . . 10 ⊢ (((( deg1 ‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}) ∧ (( deg1 ‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ) → ((( deg1 ‘𝐾)‘𝑞) < (( deg1 ‘𝐾)‘(𝑀‘𝐴)) ↔ (( deg1 ‘𝐾)‘𝑞) ≤ ((( deg1 ‘𝐾)‘(𝑀‘𝐴)) − 1)))
82	77, 80, 81	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((( deg1 ‘𝐾)‘𝑞) < (( deg1 ‘𝐾)‘(𝑀‘𝐴)) ↔ (( deg1 ‘𝐾)‘𝑞) ≤ ((( deg1 ‘𝐾)‘(𝑀‘𝐴)) − 1)))
83	75, 82	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (( deg1 ‘𝐾)‘𝑞) < (( deg1 ‘𝐾)‘(𝑀‘𝐴)))
84		fldsdrgfld 20561	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
85	12, 4, 84	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
86	5, 85	eqeltrid 2836	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ Field)
87		fldidom 21127	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ IDomn)
89	88	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → 𝐾 ∈ IDomn)
90	8, 16, 20	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
91	90	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
92	23, 2, 18, 22, 89, 26, 71, 91	r1pid2 32969	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((𝑞𝑅(𝑀‘𝐴)) = 𝑞 ↔ (( deg1 ‘𝐾)‘𝑞) < (( deg1 ‘𝐾)‘(𝑀‘𝐴))))
93	83, 92	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (𝑞𝑅(𝑀‘𝐴)) = 𝑞)
94	69, 71, 93	rspcedvdw 3615	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
95	94	ralrimiva 3145	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
96		algextdeglem.h	. . . . . 6 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
97	96	fompt 7119	. . . . 5 ⊢ (𝐻:𝑈–onto→𝑇 ↔ (∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ∧ ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴))))
98	65, 95, 97	sylanbrc 582	. . . 4 ⊢ (𝜑 → 𝐻:𝑈–onto→𝑇)
99	23	ply1ring 22003	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑃 ∈ Ring)
100	8, 99	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring)
101	1, 3, 98, 100	imasbas 17465	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝐻 “s 𝑃)))
102	71	ex 412	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ 𝑇 → 𝑞 ∈ 𝑈))
103	102	ssrdv 3988	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
104		eqid 2731	. . . . 5 ⊢ (𝑃 ↾s 𝑇) = (𝑃 ↾s 𝑇)
105	104, 2	ressbas2 17189	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
106	103, 105	syl 17	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
107		ssidd 4005	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑇)
108		eqid 2731	. . . . . . 7 ⊢ (𝐻 “s 𝑃) = (𝐻 “s 𝑃)
109		eqid 2731	. . . . . . 7 ⊢ (Base‘(𝐻 “s 𝑃)) = (Base‘(𝐻 “s 𝑃))
110	103	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ⊆ 𝑈)
111		simplr 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)
112	110, 111	sseldd 3983	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑈)
113		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇)
114	110, 113	sseldd 3983	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
115		foeq3 6803	. . . . . . . . . 10 ⊢ (𝑇 = (Base‘(𝐻 “s 𝑃)) → (𝐻:𝑈–onto→𝑇 ↔ 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃))))
116	101, 115	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐻:𝑈–onto→𝑇 ↔ 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃))))
117	98, 116	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
118	117	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
119	23, 2, 22, 18, 96, 8, 90	r1plmhm 32970	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
120	119	lmhmghmd 32480	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)))
121		ghmmhm 19144	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
122	120, 121	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
123	122	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
124		eqid 2731	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
125		eqid 2731	. . . . . . 7 ⊢ (+g‘(𝐻 “s 𝑃)) = (+g‘(𝐻 “s 𝑃))
126	108, 2, 109, 112, 114, 118, 123, 124, 125	mhmimasplusg 32481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥(+g‘𝑃)𝑦)))
127		algextdeg.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
128	12	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐸 ∈ Field)
129	4	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐹 ∈ (SubDRing‘𝐸))
130	14	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐴 ∈ (𝐸 IntgRing 𝐹))
131		algextdeglem.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
132		algextdeglem.n	. . . . . . . . 9 ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
133		algextdeglem.z	. . . . . . . . 9 ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})
134		algextdeglem.q	. . . . . . . . 9 ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
135		algextdeglem.j	. . . . . . . . 9 ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))
136	5, 127, 29, 13, 128, 129, 130, 30, 23, 2, 131, 132, 133, 134, 135, 22, 96, 48, 112	algextdeglem7 33083	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥 ∈ 𝑇 ↔ (𝐻‘𝑥) = 𝑥))
137	111, 136	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘𝑥) = 𝑥)
138	5, 127, 29, 13, 128, 129, 130, 30, 23, 2, 131, 132, 133, 134, 135, 22, 96, 48, 114	algextdeglem7 33083	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑦 ∈ 𝑇 ↔ (𝐻‘𝑦) = 𝑦))
139	113, 138	mpbid 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘𝑦) = 𝑦)
140	137, 139	oveq12d 7430	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥(+g‘(𝐻 “s 𝑃))𝑦))
141	100	ringgrpd 20140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Grp)
142	23, 7	ply1lvec 32927	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ LVec)
143	23, 26, 48, 61, 8	ply1degltlss 32957	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (LSubSp‘𝑃))
144		eqid 2731	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
145	104, 144	lsslvec 20868	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LVec ∧ 𝑇 ∈ (LSubSp‘𝑃)) → (𝑃 ↾s 𝑇) ∈ LVec)
146	142, 143, 145	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ LVec)
147	146	lvecgrpd 20867	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ Grp)
148	2	issubg 19046	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubGrp‘𝑃) ↔ (𝑃 ∈ Grp ∧ 𝑇 ⊆ 𝑈 ∧ (𝑃 ↾s 𝑇) ∈ Grp))
149	141, 103, 147, 148	syl3anbrc 1342	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑃))
150	149	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ∈ (SubGrp‘𝑃))
151	124	subgcl 19056	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑃) ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
152	150, 111, 113, 151	syl3anc 1370	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
153	141	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑃 ∈ Grp)
154	2, 124, 153, 112, 114	grpcld 18872	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑈)
155	5, 127, 29, 13, 128, 129, 130, 30, 23, 2, 131, 132, 133, 134, 135, 22, 96, 48, 154	algextdeglem7 33083	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝑥(+g‘𝑃)𝑦) ∈ 𝑇 ↔ (𝐻‘(𝑥(+g‘𝑃)𝑦)) = (𝑥(+g‘𝑃)𝑦)))
156	152, 155	mpbid 231	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘(𝑥(+g‘𝑃)𝑦)) = (𝑥(+g‘𝑃)𝑦))
157	126, 140, 156	3eqtr3d 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘𝑃)𝑦))
158		fvex 6904	. . . . . . . . 9 ⊢ ( deg1 ‘𝐾) ∈ V
159		cnvexg 7919	. . . . . . . . 9 ⊢ (( deg1 ‘𝐾) ∈ V → ◡( deg1 ‘𝐾) ∈ V)
160		imaexg 7910	. . . . . . . . 9 ⊢ (◡( deg1 ‘𝐾) ∈ V → (◡( deg1 ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) ∈ V)
161	158, 159, 160	mp2b 10	. . . . . . . 8 ⊢ (◡( deg1 ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) ∈ V
162	48, 161	eqeltri 2828	. . . . . . 7 ⊢ 𝑇 ∈ V
163	104, 124	ressplusg 17242	. . . . . . 7 ⊢ (𝑇 ∈ V → (+g‘𝑃) = (+g‘(𝑃 ↾s 𝑇)))
164	162, 163	ax-mp 5	. . . . . 6 ⊢ (+g‘𝑃) = (+g‘(𝑃 ↾s 𝑇))
165	164	oveqi 7425	. . . . 5 ⊢ (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦)
166	157, 165	eqtrdi 2787	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦))
167	166	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦))
168		simprr 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑦 ∈ 𝑇)
169	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐸 ∈ Field)
170	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 ∈ (SubDRing‘𝐸))
171	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐴 ∈ (𝐸 IntgRing 𝐹))
172	103	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑇 ⊆ 𝑈)
173	172, 168	sseldd 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑦 ∈ 𝑈)
174	5, 127, 29, 13, 169, 170, 171, 30, 23, 2, 131, 132, 133, 134, 135, 22, 96, 48, 173	algextdeglem7 33083	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑦 ∈ 𝑇 ↔ (𝐻‘𝑦) = 𝑦))
175	168, 174	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘𝑦) = 𝑦)
176	175	oveq2d 7428	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦))
177		simprl 768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ 𝐹)
178	33	sdrgss 20556	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
179	5, 33	ressbas2 17189	. . . . . . . . . 10 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
180	4, 178, 179	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
181	23	ply1sca 22008	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑃))
182	8, 181	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑃))
183	182	fveq2d 6895	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑃)))
184	180, 183	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘𝑃)))
185	184	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 = (Base‘(Scalar‘𝑃)))
186	177, 185	eleqtrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ (Base‘(Scalar‘𝑃)))
187	117	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
188	119	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
189		eqid 2731	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
190		eqid 2731	. . . . . 6 ⊢ ( ·𝑠 ‘(𝐻 “s 𝑃)) = ( ·𝑠 ‘(𝐻 “s 𝑃))
191		eqid 2731	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
192	108, 2, 109, 186, 173, 187, 188, 189, 190, 191	lmhmimasvsca 32482	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
193	176, 192	eqtr3d 2773	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
194	64, 96	fmptd 7115	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑈⟶𝑇)
195	194	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈⟶𝑇)
196		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
197	142	lveclmodd 20866	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
198	197	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑃 ∈ LMod)
199	2, 196, 189, 191, 198, 186, 173	lmodvscld 20637	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑈)
200	195, 199	ffvelcdmd 7087	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) ∈ 𝑇)
201	193, 200	eqeltrd 2832	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) ∈ 𝑇)
202	143	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑇 ∈ (LSubSp‘𝑃))
203	196, 189, 191, 144	lssvscl 20714	. . . . . 6 ⊢ (((𝑃 ∈ LMod ∧ 𝑇 ∈ (LSubSp‘𝑃)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇)
204	198, 202, 186, 168, 203	syl22anc 836	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇)
205	5, 127, 29, 13, 169, 170, 171, 30, 23, 2, 131, 132, 133, 134, 135, 22, 96, 48, 199	algextdeglem7 33083	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → ((𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇 ↔ (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) = (𝑥( ·𝑠 ‘𝑃)𝑦)))
206	204, 205	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) = (𝑥( ·𝑠 ‘𝑃)𝑦))
207	104, 189	ressvsca 17296	. . . . . 6 ⊢ (𝑇 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(𝑃 ↾s 𝑇)))
208	162, 207	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(𝑃 ↾s 𝑇)))
209	208	oveqd 7429	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
210	193, 206, 209	3eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
211		eqid 2731	. . 3 ⊢ (Scalar‘(𝐻 “s 𝑃)) = (Scalar‘(𝐻 “s 𝑃))
212	104, 196	resssca 17295	. . . 4 ⊢ (𝑇 ∈ V → (Scalar‘𝑃) = (Scalar‘(𝑃 ↾s 𝑇)))
213	162, 212	ax-mp 5	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘(𝑃 ↾s 𝑇))
214	1, 3, 98, 100, 196	imassca 17472	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑃) = (Scalar‘(𝐻 “s 𝑃)))
215	182, 214	eqtrd 2771	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘(𝐻 “s 𝑃)))
216	215	fveq2d 6895	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(𝐻 “s 𝑃))))
217	180, 216	eqtrd 2771	. . 3 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘(𝐻 “s 𝑃))))
218	214	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘(𝐻 “s 𝑃))))
219	218	oveqd 7429	. . . . 5 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘𝑃))𝑦) = (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦))
220	219	eqcomd 2737	. . . 4 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦) = (𝑥(+g‘(Scalar‘𝑃))𝑦))
221	220	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦) = (𝑥(+g‘(Scalar‘𝑃))𝑦))
222		lmhmlvec2 33007	. . . 4 ⊢ ((𝑃 ∈ LVec ∧ 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃))) → (𝐻 “s 𝑃) ∈ LVec)
223	142, 119, 222	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐻 “s 𝑃) ∈ LVec)
224	101, 106, 107, 167, 201, 210, 211, 213, 217, 184, 221, 223, 146	dimpropd 32996	. 2 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (dim‘(𝑃 ↾s 𝑇)))
225	23, 26, 48, 61, 7, 104	ply1degltdim 33011	. 2 ⊢ (𝜑 → (dim‘(𝑃 ↾s 𝑇)) = (𝐷‘(𝑀‘𝐴)))
226	224, 225	eqtrd 2771	1 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3431  Vcvv 3473   ∪ cun 3946   ∩ cin 3947   ⊆ wss 3948  {csn 4628  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5675  dom cdm 5676   “ cima 5679  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7412  [cec 8707  1c1 11117  -∞cmnf 11253   < clt 11255   ≤ cle 11256   − cmin 11451  ℕ₀cn0 12479  ℤcz 12565  [,)cico 13333  Basecbs 17151   ↾_s cress 17180  +_gcplusg 17204  Scalarcsca 17207   ·_𝑠 cvsca 17208  0_gc0g 17392   “_s cimas 17457   /_s cqus 17458   MndHom cmhm 18706  Grpcgrp 18858  SubGrpcsubg 19040   ~_QG cqg 19042   GrpHom cghm 19131  Ringcrg 20131  SubRingcsubrg 20461  DivRingcdr 20504  Fieldcfield 20505  SubDRingcsdrg 20549  LModclmod 20618  LSubSpclss 20690   LMHom clmhm 20778  LVecclvec 20861  RSpancrsp 20933  IDomncidom 21101  PwSer₁cps1 21931  Poly₁cpl1 21933   evalSub₁ ces1 22065   deg₁ cdg1 25818  Monic_1pcmn1 25892  Unic_1pcuc1p 25893  rem_1pcr1p 25895  idlGen_1pcig1p 25896   fldGen cfldgen 32685  dimcldim 32986   IntgRing cirng 33051   minPoly cminply 33060

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-reg 9593  ax-inf2 9642  ax-ac2 10464  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195  ax-mulf 11196

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-rpss 7717  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oadd 8476  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-sup 9443  df-inf 9444  df-oi 9511  df-r1 9765  df-rank 9766  df-dju 9902  df-card 9940  df-acn 9943  df-ac 10117  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-xnn0 12552  df-z 12566  df-dec 12685  df-uz 12830  df-ico 13337  df-fz 13492  df-fzo 13635  df-seq 13974  df-hash 14298  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ocomp 17225  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-imas 17461  df-mre 17537  df-mrc 17538  df-mri 17539  df-acs 17540  df-proset 18255  df-drs 18256  df-poset 18273  df-ipo 18488  df-mgm 18568  df-sgrp 18647  df-mnd 18663  df-mhm 18708  df-submnd 18709  df-grp 18861  df-minusg 18862  df-sbg 18863  df-mulg 18991  df-subg 19043  df-ghm 19132  df-cntz 19226  df-cmn 19695  df-abl 19696  df-mgp 20033  df-rng 20051  df-ur 20080  df-srg 20085  df-ring 20133  df-cring 20134  df-oppr 20229  df-dvdsr 20252  df-unit 20253  df-invr 20283  df-rhm 20367  df-nzr 20408  df-subrng 20438  df-subrg 20463  df-drng 20506  df-field 20507  df-sdrg 20550  df-lmod 20620  df-lss 20691  df-lsp 20731  df-lmhm 20781  df-lbs 20834  df-lvec 20862  df-sra 20934  df-rgmod 20935  df-lidl 20936  df-rsp 20937  df-rlreg 21103  df-domn 21104  df-idom 21105  df-cnfld 21149  df-dsmm 21510  df-frlm 21525  df-uvc 21561  df-lindf 21584  df-linds 21585  df-assa 21631  df-asp 21632  df-ascl 21633  df-psr 21685  df-mvr 21686  df-mpl 21687  df-opsr 21689  df-evls 21859  df-evl 21860  df-psr1 21936  df-vr1 21937  df-ply1 21938  df-coe1 21939  df-evls1 22067  df-evl1 22068  df-mdeg 25819  df-deg1 25820  df-mon1 25897  df-uc1p 25898  df-q1p 25899  df-r1p 25900  df-ig1p 25901  df-dim 32987  df-irng 33052  df-minply 33061

This theorem is referenced by:  algextdeg  33085