Theorem algextdeglem8 33714

Description: Lemma for algextdeg 33715. The dimension of the univariate polynomial remainder ring (𝐻 “_s 𝑃) is the degree of the minimal polynomial. (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 2730	. . . 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 20699	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
7	4, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ DivRing)
8	7	drngringd 20646	. . . . . . . . 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 2729	. . . . . . . . . . 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 6861	. . . . . . . . . . 11 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹))
16	11, 12, 4, 13, 14, 15	minplym1p 33703	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
18		eqid 2729	. . . . . . . . . 10 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾)
19		eqid 2729	. . . . . . . . . 10 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾)
20	18, 19	mon1puc1p 26056	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
21	9, 17, 20	syl2anc 584	. . . . . . . 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 26064	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
25	9, 10, 21, 24	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
26		eqid 2729	. . . . . . . . . 10 ⊢ (deg1‘𝐾) = (deg1‘𝐾)
27	22, 23, 2, 18, 26	r1pdeglt 26065	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → ((deg1‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < ((deg1‘𝐾)‘(𝑀‘𝐴)))
28	9, 10, 21, 27	syl3anc 1373	. . . . . . . 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 6861	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	23, 31	eqtri 2752	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
34		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐸) = (0g‘𝐸)
35	12	fldcrngd 20651	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ CRing)
36		sdrgsubrg 20700	. . . . . . . . . . . . . . 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 33683	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
39	38, 14	sseldd 3947	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
40		eqid 2729	. . . . . . . . . . . 12 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)}
41		eqid 2729	. . . . . . . . . . . 12 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
42		eqid 2729	. . . . . . . . . . . 12 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
43	30, 32, 33, 12, 4, 39, 34, 40, 41, 42, 13	minplycl 33696	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
44	43, 2	eleqtrrdi 2839	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
45	5, 29, 23, 2, 44, 37	ressdeg1 33535	. . . . . . . . 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 5135	. . . . . . 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 20650	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ DivRing)
50	49	drngringd 20646	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Ring)
51		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
52		eqid 2729	. . . . . . . . . . . . 13 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾)
53		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾))
54		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸))
55	51, 5, 23, 2, 37, 52, 53, 54	ressply1bas2 22112	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))))
56		inss2 4201	. . . . . . . . . . . 12 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸))
57	55, 56	eqsstrdi 3991	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸)))
58	57, 44	sseldd 3947	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)))
59	11, 12, 4, 13, 14	irngnminplynz 33702	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
60	29, 51, 11, 54	deg1nn0cl 25993	. . . . . . . . . 10 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
61	50, 58, 59, 60	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
62	23, 26, 48, 61, 8, 2	ply1degleel 33561	. . . . . . . 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 713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
65	64	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
66		oveq1 7394	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝𝑅(𝑀‘𝐴)) = (𝑞𝑅(𝑀‘𝐴)))
67	66	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ 𝑞 = (𝑞𝑅(𝑀‘𝐴))))
68		eqcom 2736	. . . . . . . 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 33560	. . . . . . . 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 7402	. . . . . . . . . . 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 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ≤ (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1))
76	26, 23, 2	deg1cl 25988	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝑈 → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
77	71, 76	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
78	61	nn0zd 12555	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℤ)
79	45, 78	eqeltrrd 2829	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
80	79	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
81		degltlem1 25977	. . . . . . . . . 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 584	. . . . . . . . 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 20707	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
85	12, 4, 84	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
86	5, 85	eqeltrid 2832	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Field)
87		fldidom 20680	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
88	86, 87	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ IDomn)
89	88	idomdomd 20635	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Domn)
90	89	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → 𝐾 ∈ Domn)
91	8, 16, 20	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
92	91	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
93	23, 2, 18, 22, 26, 90, 71, 92	r1pid2 26067	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((𝑞𝑅(𝑀‘𝐴)) = 𝑞 ↔ ((deg1‘𝐾)‘𝑞) < ((deg1‘𝐾)‘(𝑀‘𝐴))))
94	83, 93	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (𝑞𝑅(𝑀‘𝐴)) = 𝑞)
95	69, 71, 94	rspcedvdw 3591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
96	95	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
97		algextdeglem.h	. . . . . 6 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
98	97	fompt 7090	. . . . 5 ⊢ (𝐻:𝑈–onto→𝑇 ↔ (∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ∧ ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴))))
99	65, 96, 98	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝐻:𝑈–onto→𝑇)
100	23	ply1ring 22132	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑃 ∈ Ring)
101	8, 100	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring)
102	1, 3, 99, 101	imasbas 17475	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝐻 “s 𝑃)))
103	71	ex 412	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ 𝑇 → 𝑞 ∈ 𝑈))
104	103	ssrdv 3952	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
105		eqid 2729	. . . . 5 ⊢ (𝑃 ↾s 𝑇) = (𝑃 ↾s 𝑇)
106	105, 2	ressbas2 17208	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
107	104, 106	syl 17	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
108		ssidd 3970	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑇)
109		eqid 2729	. . . . . . 7 ⊢ (𝐻 “s 𝑃) = (𝐻 “s 𝑃)
110		eqid 2729	. . . . . . 7 ⊢ (Base‘(𝐻 “s 𝑃)) = (Base‘(𝐻 “s 𝑃))
111	104	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ⊆ 𝑈)
112		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)
113	111, 112	sseldd 3947	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑈)
114		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇)
115	111, 114	sseldd 3947	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
116		foeq3 6770	. . . . . . . . . 10 ⊢ (𝑇 = (Base‘(𝐻 “s 𝑃)) → (𝐻:𝑈–onto→𝑇 ↔ 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃))))
117	102, 116	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐻:𝑈–onto→𝑇 ↔ 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃))))
118	99, 117	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
119	118	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
120	23, 2, 22, 18, 97, 8, 91	r1plmhm 33575	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
121	120	lmhmghmd 32978	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)))
122		ghmmhm 19158	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
123	121, 122	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
124	123	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
125		eqid 2729	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
126		eqid 2729	. . . . . . 7 ⊢ (+g‘(𝐻 “s 𝑃)) = (+g‘(𝐻 “s 𝑃))
127	109, 2, 110, 113, 115, 119, 124, 125, 126	mhmimasplusg 32979	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥(+g‘𝑃)𝑦)))
128		algextdeg.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))
129	12	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐸 ∈ Field)
130	4	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐹 ∈ (SubDRing‘𝐸))
131	14	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐴 ∈ (𝐸 IntgRing 𝐹))
132		algextdeglem.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))
133		algextdeglem.n	. . . . . . . . 9 ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))
134		algextdeglem.z	. . . . . . . . 9 ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)})
135		algextdeglem.q	. . . . . . . . 9 ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))
136		algextdeglem.j	. . . . . . . . 9 ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))
137	5, 128, 29, 13, 129, 130, 131, 30, 23, 2, 132, 133, 134, 135, 136, 22, 97, 48, 113	algextdeglem7 33713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥 ∈ 𝑇 ↔ (𝐻‘𝑥) = 𝑥))
138	112, 137	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘𝑥) = 𝑥)
139	5, 128, 29, 13, 129, 130, 131, 30, 23, 2, 132, 133, 134, 135, 136, 22, 97, 48, 115	algextdeglem7 33713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑦 ∈ 𝑇 ↔ (𝐻‘𝑦) = 𝑦))
140	114, 139	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘𝑦) = 𝑦)
141	138, 140	oveq12d 7405	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥(+g‘(𝐻 “s 𝑃))𝑦))
142	101	ringgrpd 20151	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Grp)
143	23, 7	ply1lvec 33528	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ LVec)
144	23, 26, 48, 61, 8	ply1degltlss 33562	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (LSubSp‘𝑃))
145		eqid 2729	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
146	105, 145	lsslvec 21016	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LVec ∧ 𝑇 ∈ (LSubSp‘𝑃)) → (𝑃 ↾s 𝑇) ∈ LVec)
147	143, 144, 146	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ LVec)
148	147	lvecgrpd 21015	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ Grp)
149	2	issubg 19058	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubGrp‘𝑃) ↔ (𝑃 ∈ Grp ∧ 𝑇 ⊆ 𝑈 ∧ (𝑃 ↾s 𝑇) ∈ Grp))
150	142, 104, 148, 149	syl3anbrc 1344	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑃))
151	150	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ∈ (SubGrp‘𝑃))
152	125	subgcl 19068	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑃) ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
153	151, 112, 114, 152	syl3anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
154	142	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑃 ∈ Grp)
155	2, 125, 154, 113, 115	grpcld 18879	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑈)
156	5, 128, 29, 13, 129, 130, 131, 30, 23, 2, 132, 133, 134, 135, 136, 22, 97, 48, 155	algextdeglem7 33713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝑥(+g‘𝑃)𝑦) ∈ 𝑇 ↔ (𝐻‘(𝑥(+g‘𝑃)𝑦)) = (𝑥(+g‘𝑃)𝑦)))
157	153, 156	mpbid 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘(𝑥(+g‘𝑃)𝑦)) = (𝑥(+g‘𝑃)𝑦))
158	127, 141, 157	3eqtr3d 2772	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘𝑃)𝑦))
159		fvex 6871	. . . . . . . . 9 ⊢ (deg1‘𝐾) ∈ V
160		cnvexg 7900	. . . . . . . . 9 ⊢ ((deg1‘𝐾) ∈ V → ◡(deg1‘𝐾) ∈ V)
161		imaexg 7889	. . . . . . . . 9 ⊢ (◡(deg1‘𝐾) ∈ V → (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) ∈ V)
162	159, 160, 161	mp2b 10	. . . . . . . 8 ⊢ (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) ∈ V
163	48, 162	eqeltri 2824	. . . . . . 7 ⊢ 𝑇 ∈ V
164	105, 125	ressplusg 17254	. . . . . . 7 ⊢ (𝑇 ∈ V → (+g‘𝑃) = (+g‘(𝑃 ↾s 𝑇)))
165	163, 164	ax-mp 5	. . . . . 6 ⊢ (+g‘𝑃) = (+g‘(𝑃 ↾s 𝑇))
166	165	oveqi 7400	. . . . 5 ⊢ (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦)
167	158, 166	eqtrdi 2780	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦))
168	167	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦))
169		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑦 ∈ 𝑇)
170	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐸 ∈ Field)
171	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 ∈ (SubDRing‘𝐸))
172	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐴 ∈ (𝐸 IntgRing 𝐹))
173	104	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑇 ⊆ 𝑈)
174	173, 169	sseldd 3947	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑦 ∈ 𝑈)
175	5, 128, 29, 13, 170, 171, 172, 30, 23, 2, 132, 133, 134, 135, 136, 22, 97, 48, 174	algextdeglem7 33713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑦 ∈ 𝑇 ↔ (𝐻‘𝑦) = 𝑦))
176	169, 175	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘𝑦) = 𝑦)
177	176	oveq2d 7403	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦))
178		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ 𝐹)
179	33	sdrgss 20702	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
180	5, 33	ressbas2 17208	. . . . . . . . . 10 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
181	4, 179, 180	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
182	23	ply1sca 22137	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑃))
183	8, 182	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑃))
184	183	fveq2d 6862	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑃)))
185	181, 184	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘𝑃)))
186	185	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 = (Base‘(Scalar‘𝑃)))
187	178, 186	eleqtrd 2830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ (Base‘(Scalar‘𝑃)))
188	118	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
189	120	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
190		eqid 2729	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191		eqid 2729	. . . . . 6 ⊢ ( ·𝑠 ‘(𝐻 “s 𝑃)) = ( ·𝑠 ‘(𝐻 “s 𝑃))
192		eqid 2729	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
193	109, 2, 110, 187, 174, 188, 189, 190, 191, 192	lmhmimasvsca 32980	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
194	177, 193	eqtr3d 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
195	64, 97	fmptd 7086	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑈⟶𝑇)
196	195	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈⟶𝑇)
197		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
198	143	lveclmodd 21014	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
199	198	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑃 ∈ LMod)
200	2, 197, 190, 192, 199, 187, 174	lmodvscld 20785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑈)
201	196, 200	ffvelcdmd 7057	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) ∈ 𝑇)
202	194, 201	eqeltrd 2828	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) ∈ 𝑇)
203	144	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑇 ∈ (LSubSp‘𝑃))
204	197, 190, 192, 145	lssvscl 20861	. . . . . 6 ⊢ (((𝑃 ∈ LMod ∧ 𝑇 ∈ (LSubSp‘𝑃)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇)
205	199, 203, 187, 169, 204	syl22anc 838	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇)
206	5, 128, 29, 13, 170, 171, 172, 30, 23, 2, 132, 133, 134, 135, 136, 22, 97, 48, 200	algextdeglem7 33713	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → ((𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇 ↔ (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) = (𝑥( ·𝑠 ‘𝑃)𝑦)))
207	205, 206	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) = (𝑥( ·𝑠 ‘𝑃)𝑦))
208	105, 190	ressvsca 17307	. . . . . 6 ⊢ (𝑇 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(𝑃 ↾s 𝑇)))
209	163, 208	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(𝑃 ↾s 𝑇)))
210	209	oveqd 7404	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
211	194, 207, 210	3eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
212		eqid 2729	. . 3 ⊢ (Scalar‘(𝐻 “s 𝑃)) = (Scalar‘(𝐻 “s 𝑃))
213	105, 197	resssca 17306	. . . 4 ⊢ (𝑇 ∈ V → (Scalar‘𝑃) = (Scalar‘(𝑃 ↾s 𝑇)))
214	163, 213	ax-mp 5	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘(𝑃 ↾s 𝑇))
215	1, 3, 99, 101, 197	imassca 17482	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑃) = (Scalar‘(𝐻 “s 𝑃)))
216	183, 215	eqtrd 2764	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘(𝐻 “s 𝑃)))
217	216	fveq2d 6862	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(𝐻 “s 𝑃))))
218	181, 217	eqtrd 2764	. . 3 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘(𝐻 “s 𝑃))))
219	215	fveq2d 6862	. . . . . 6 ⊢ (𝜑 → (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘(𝐻 “s 𝑃))))
220	219	oveqd 7404	. . . . 5 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘𝑃))𝑦) = (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦))
221	220	eqcomd 2735	. . . 4 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦) = (𝑥(+g‘(Scalar‘𝑃))𝑦))
222	221	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦) = (𝑥(+g‘(Scalar‘𝑃))𝑦))
223		lmhmlvec2 33615	. . . 4 ⊢ ((𝑃 ∈ LVec ∧ 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃))) → (𝐻 “s 𝑃) ∈ LVec)
224	143, 120, 223	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐻 “s 𝑃) ∈ LVec)
225	102, 107, 108, 168, 202, 211, 212, 214, 218, 185, 222, 224, 147	dimpropd 33604	. 2 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (dim‘(𝑃 ↾s 𝑇)))
226	23, 26, 48, 61, 7, 105	ply1degltdim 33619	. 2 ⊢ (𝜑 → (dim‘(𝑃 ↾s 𝑇)) = (𝐷‘(𝑀‘𝐴)))
227	225, 226	eqtrd 2764	1 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188  ^◡ccnv 5637  dom cdm 5638   “ cima 5641  ⟶wf 6507  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387  [cec 8669  1c1 11069  -∞cmnf 11206   < clt 11208   ≤ cle 11209   − cmin 11405  ℕ₀cn0 12442  ℤcz 12529  [,)cico 13308  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   “_s cimas 17467   /_s cqus 17468   MndHom cmhm 18708  Grpcgrp 18865  SubGrpcsubg 19052   ~_QG cqg 19054   GrpHom cghm 19144  Ringcrg 20142  SubRingcsubrg 20478  Domncdomn 20601  IDomncidom 20602  DivRingcdr 20638  Fieldcfield 20639  SubDRingcsdrg 20695  LModclmod 20766  LSubSpclss 20837   LMHom clmhm 20926  LVecclvec 21009  RSpancrsp 21117  PwSer₁cps1 22059  Poly₁cpl1 22061   evalSub₁ ces1 22200  deg₁cdg1 25959  Monic_1pcmn1 26031  Unic_1pcuc1p 26032  rem_1pcr1p 26034  idlGen_1pcig1p 26035   fldGen cfldgen 33260  dimcldim 33594   IntgRing cirng 33678   minPoly cminply 33689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594  ax-ac2 10416  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-rpss 7699  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-inf 9394  df-oi 9463  df-r1 9717  df-rank 9718  df-dju 9854  df-card 9892  df-acn 9895  df-ac 10069  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-xnn0 12516  df-z 12530  df-dec 12650  df-uz 12794  df-ico 13312  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ocomp 17241  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-imas 17471  df-mre 17547  df-mrc 17548  df-mri 17549  df-acs 17550  df-proset 18255  df-drs 18256  df-poset 18274  df-ipo 18487  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-srg 20096  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-invr 20297  df-rhm 20381  df-nzr 20422  df-subrng 20455  df-subrg 20479  df-rlreg 20603  df-domn 20604  df-idom 20605  df-drng 20640  df-field 20641  df-sdrg 20696  df-lmod 20768  df-lss 20838  df-lsp 20878  df-lmhm 20929  df-lbs 20982  df-lvec 21010  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-rsp 21119  df-cnfld 21265  df-dsmm 21641  df-frlm 21656  df-uvc 21692  df-lindf 21715  df-linds 21716  df-assa 21762  df-asp 21763  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-evls 21981  df-evl 21982  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-evls1 22202  df-evl1 22203  df-mdeg 25960  df-deg1 25961  df-mon1 26036  df-uc1p 26037  df-q1p 26038  df-r1p 26039  df-ig1p 26040  df-dim 33595  df-irng 33679  df-minply 33690

This theorem is referenced by:  algextdeg  33715