Theorem algextdeglem8 33868

Description: Lemma for algextdeg 33869. 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 2737	. . . 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 20767	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
7	4, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ DivRing)
8	7	drngringd 20714	. . . . . . . . 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 2736	. . . . . . . . . . 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 6843	. . . . . . . . . . 11 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹))
16	11, 12, 4, 13, 14, 15	minplym1p 33857	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
18		eqid 2736	. . . . . . . . . 10 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾)
19		eqid 2736	. . . . . . . . . 10 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾)
20	18, 19	mon1puc1p 26116	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
21	9, 17, 20	syl2anc 585	. . . . . . . 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 26124	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
25	9, 10, 21, 24	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
26		eqid 2736	. . . . . . . . . 10 ⊢ (deg1‘𝐾) = (deg1‘𝐾)
27	22, 23, 2, 18, 26	r1pdeglt 26125	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → ((deg1‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < ((deg1‘𝐾)‘(𝑀‘𝐴)))
28	9, 10, 21, 27	syl3anc 1374	. . . . . . . 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 6843	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	23, 31	eqtri 2759	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
34		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐸) = (0g‘𝐸)
35	12	fldcrngd 20719	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ CRing)
36		sdrgsubrg 20768	. . . . . . . . . . . . . . 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 33832	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
39	38, 14	sseldd 3922	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
40		eqid 2736	. . . . . . . . . . . 12 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)}
41		eqid 2736	. . . . . . . . . . . 12 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
42		eqid 2736	. . . . . . . . . . . 12 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
43	30, 32, 33, 12, 4, 39, 34, 40, 41, 42, 13	minplycl 33850	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
44	43, 2	eleqtrrdi 2847	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
45	5, 29, 23, 2, 44, 37	ressdeg1 33626	. . . . . . . . 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 5113	. . . . . . 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 20718	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ DivRing)
50	49	drngringd 20714	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Ring)
51		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
52		eqid 2736	. . . . . . . . . . . . 13 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾)
53		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾))
54		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸))
55	51, 5, 23, 2, 37, 52, 53, 54	ressply1bas2 22191	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))))
56		inss2 4178	. . . . . . . . . . . 12 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸))
57	55, 56	eqsstrdi 3966	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸)))
58	57, 44	sseldd 3922	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)))
59	11, 12, 4, 13, 14	irngnminplynz 33856	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
60	29, 51, 11, 54	deg1nn0cl 26053	. . . . . . . . . 10 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
61	50, 58, 59, 60	syl3anc 1374	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
62	23, 26, 48, 61, 8, 2	ply1degleel 33655	. . . . . . . 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 714	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
65	64	ralrimiva 3129	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
66		oveq1 7374	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝𝑅(𝑀‘𝐴)) = (𝑞𝑅(𝑀‘𝐴)))
67	66	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ 𝑞 = (𝑞𝑅(𝑀‘𝐴))))
68		eqcom 2743	. . . . . . . 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 33654	. . . . . . . 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 7382	. . . . . . . . . . 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 5111	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ≤ (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1))
76	26, 23, 2	deg1cl 26048	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝑈 → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
77	71, 76	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
78	61	nn0zd 12549	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℤ)
79	45, 78	eqeltrrd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
80	79	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
81		degltlem1 26037	. . . . . . . . . 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 585	. . . . . . . . 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 20775	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
85	12, 4, 84	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
86	5, 85	eqeltrid 2840	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Field)
87		fldidom 20748	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
88	86, 87	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ IDomn)
89	88	idomdomd 20703	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Domn)
90	89	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → 𝐾 ∈ Domn)
91	8, 16, 20	syl2anc 585	. . . . . . . . . 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 26127	. . . . . . . 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 3567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
96	95	ralrimiva 3129	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
97		algextdeglem.h	. . . . . 6 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
98	97	fompt 7070	. . . . 5 ⊢ (𝐻:𝑈–onto→𝑇 ↔ (∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ∧ ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴))))
99	65, 96, 98	sylanbrc 584	. . . 4 ⊢ (𝜑 → 𝐻:𝑈–onto→𝑇)
100	23	ply1ring 22211	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑃 ∈ Ring)
101	8, 100	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring)
102	1, 3, 99, 101	imasbas 17476	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝐻 “s 𝑃)))
103	71	ex 412	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ 𝑇 → 𝑞 ∈ 𝑈))
104	103	ssrdv 3927	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
105		eqid 2736	. . . . 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 3945	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑇)
109		eqid 2736	. . . . . . 7 ⊢ (𝐻 “s 𝑃) = (𝐻 “s 𝑃)
110		eqid 2736	. . . . . . 7 ⊢ (Base‘(𝐻 “s 𝑃)) = (Base‘(𝐻 “s 𝑃))
111	104	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ⊆ 𝑈)
112		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)
113	111, 112	sseldd 3922	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑈)
114		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇)
115	111, 114	sseldd 3922	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
116		foeq3 6750	. . . . . . . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃)))
120	23, 2, 22, 18, 97, 8, 91	r1plmhm 33670	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
121	120	lmhmghmd 33097	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)))
122		ghmmhm 19201	. . . . . . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
125		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
126		eqid 2736	. . . . . . 7 ⊢ (+g‘(𝐻 “s 𝑃)) = (+g‘(𝐻 “s 𝑃))
127	109, 2, 110, 113, 115, 119, 124, 125, 126	mhmimasplusg 33098	. . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐸 ∈ Field)
130	4	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐹 ∈ (SubDRing‘𝐸))
131	14	ad2antrr 727	. . . . . . . . 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 33867	. . . . . . . 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 33867	. . . . . . . 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 7385	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥(+g‘(𝐻 “s 𝑃))𝑦))
142	101	ringgrpd 20223	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Grp)
143	23, 7	ply1lvec 33619	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ LVec)
144	23, 26, 48, 61, 8	ply1degltlss 33656	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (LSubSp‘𝑃))
145		eqid 2736	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
146	105, 145	lsslvec 21104	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LVec ∧ 𝑇 ∈ (LSubSp‘𝑃)) → (𝑃 ↾s 𝑇) ∈ LVec)
147	143, 144, 146	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ LVec)
148	147	lvecgrpd 21103	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ Grp)
149	2	issubg 19102	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubGrp‘𝑃) ↔ (𝑃 ∈ Grp ∧ 𝑇 ⊆ 𝑈 ∧ (𝑃 ↾s 𝑇) ∈ Grp))
150	142, 104, 148, 149	syl3anbrc 1345	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑃))
151	150	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ∈ (SubGrp‘𝑃))
152	125	subgcl 19112	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑃) ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
153	151, 112, 114, 152	syl3anc 1374	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
154	142	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑃 ∈ Grp)
155	2, 125, 154, 113, 115	grpcld 18923	. . . . . . . 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 33867	. . . . . . 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 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘𝑃)𝑦))
159		fvex 6853	. . . . . . . . 9 ⊢ (deg1‘𝐾) ∈ V
160		cnvexg 7875	. . . . . . . . 9 ⊢ ((deg1‘𝐾) ∈ V → ◡(deg1‘𝐾) ∈ V)
161		imaexg 7864	. . . . . . . . 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 2832	. . . . . . 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 7380	. . . . 5 ⊢ (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦)
167	158, 166	eqtrdi 2787	. . . 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 773	. . . . . . 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 3922	. . . . . . . 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 33867	. . . . . . 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 7383	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦))
178		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ 𝐹)
179	33	sdrgss 20770	. . . . . . . . . 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 22216	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑃))
183	8, 182	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑃))
184	183	fveq2d 6844	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑃)))
185	181, 184	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘𝑃)))
186	185	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 = (Base‘(Scalar‘𝑃)))
187	178, 186	eleqtrd 2838	. . . . . 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 2736	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191		eqid 2736	. . . . . 6 ⊢ ( ·𝑠 ‘(𝐻 “s 𝑃)) = ( ·𝑠 ‘(𝐻 “s 𝑃))
192		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
193	109, 2, 110, 187, 174, 188, 189, 190, 191, 192	lmhmimasvsca 33099	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
194	177, 193	eqtr3d 2773	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
195	64, 97	fmptd 7066	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑈⟶𝑇)
196	195	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈⟶𝑇)
197		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
198	143	lveclmodd 21102	. . . . . . 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 20874	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑈)
201	196, 200	ffvelcdmd 7037	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) ∈ 𝑇)
202	194, 201	eqeltrd 2836	. . 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 20950	. . . . . 6 ⊢ (((𝑃 ∈ LMod ∧ 𝑇 ∈ (LSubSp‘𝑃)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇)
205	199, 203, 187, 169, 204	syl22anc 839	. . . . 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 33867	. . . . 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 7384	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
211	194, 207, 210	3eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
212		eqid 2736	. . 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 17483	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑃) = (Scalar‘(𝐻 “s 𝑃)))
216	183, 215	eqtrd 2771	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘(𝐻 “s 𝑃)))
217	216	fveq2d 6844	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(𝐻 “s 𝑃))))
218	181, 217	eqtrd 2771	. . 3 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘(𝐻 “s 𝑃))))
219	215	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘(𝐻 “s 𝑃))))
220	219	oveqd 7384	. . . . 5 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘𝑃))𝑦) = (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦))
221	220	eqcomd 2742	. . . 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 33763	. . . 4 ⊢ ((𝑃 ∈ LVec ∧ 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃))) → (𝐻 “s 𝑃) ∈ LVec)
224	143, 120, 223	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐻 “s 𝑃) ∈ LVec)
225	102, 107, 108, 168, 202, 211, 212, 214, 218, 185, 222, 224, 147	dimpropd 33753	. 2 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (dim‘(𝑃 ↾s 𝑇)))
226	23, 26, 48, 61, 7, 105	ply1degltdim 33767	. 2 ⊢ (𝜑 → (dim‘(𝑃 ↾s 𝑇)) = (𝐷‘(𝑀‘𝐴)))
227	225, 226	eqtrd 2771	1 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  {csn 4567  ∪ cuni 4850   class class class wbr 5085   ↦ cmpt 5166  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  ⟶wf 6494  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367  [cec 8641  1c1 11039  -∞cmnf 11177   < clt 11179   ≤ cle 11180   − cmin 11377  ℕ₀cn0 12437  ℤcz 12524  [,)cico 13300  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   “_s cimas 17468   /_s cqus 17469   MndHom cmhm 18749  Grpcgrp 18909  SubGrpcsubg 19096   ~_QG cqg 19098   GrpHom cghm 19187  Ringcrg 20214  SubRingcsubrg 20546  Domncdomn 20669  IDomncidom 20670  DivRingcdr 20706  Fieldcfield 20707  SubDRingcsdrg 20763  LModclmod 20855  LSubSpclss 20926   LMHom clmhm 21014  LVecclvec 21097  RSpancrsp 21205  PwSer₁cps1 22138  Poly₁cpl1 22140   evalSub₁ ces1 22278  deg₁cdg1 26019  Monic_1pcmn1 26091  Unic_1pcuc1p 26092  rem_1pcr1p 26094  idlGen_1pcig1p 26095   fldGen cfldgen 33371  dimcldim 33743   IntgRing cirng 33827   minPoly cminply 33843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-rpss 7677  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-inf 9356  df-oi 9425  df-r1 9688  df-rank 9689  df-dju 9825  df-card 9863  df-acn 9866  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-ico 13304  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  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 17472  df-mre 17548  df-mrc 17549  df-mri 17550  df-acs 17551  df-proset 18260  df-drs 18261  df-poset 18279  df-ipo 18494  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-srg 20168  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-rhm 20452  df-nzr 20490  df-subrng 20523  df-subrg 20547  df-rlreg 20671  df-domn 20672  df-idom 20673  df-drng 20708  df-field 20709  df-sdrg 20764  df-lmod 20857  df-lss 20927  df-lsp 20967  df-lmhm 21017  df-lbs 21070  df-lvec 21098  df-sra 21168  df-rgmod 21169  df-lidl 21206  df-rsp 21207  df-cnfld 21353  df-dsmm 21712  df-frlm 21727  df-uvc 21763  df-lindf 21786  df-linds 21787  df-assa 21833  df-asp 21834  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-opsr 21893  df-evls 22052  df-evl 22053  df-psr1 22143  df-vr1 22144  df-ply1 22145  df-coe1 22146  df-evls1 22280  df-evl1 22281  df-mdeg 26020  df-deg1 26021  df-mon1 26096  df-uc1p 26097  df-q1p 26098  df-r1p 26099  df-ig1p 26100  df-dim 33744  df-irng 33828  df-minply 33844

This theorem is referenced by:  algextdeg  33869