Theorem algextdeglem8 34467

Description: Lemma for algextdeg 34468. 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 2764	. . . 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 21098	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
7	4, 6	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ DivRing)
8	7	drngringd 21039	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Ring)
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝐾 ∈ Ring)
10		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → 𝑝 ∈ 𝑈)
11		eqid 2763	. . . . . . . . . . 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 6919	. . . . . . . . . . 11 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹))
16	11, 12, 4, 13, 14, 15	minplym1p 34456	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
17	16	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
18		eqid 2763	. . . . . . . . . 10 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾)
19		eqid 2763	. . . . . . . . . 10 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾)
20	18, 19	mon1puc1p 26524	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
21	9, 17, 20	syl2anc 596	. . . . . . . 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 26532	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
25	9, 10, 21, 24	syl3anc 1399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
26		eqid 2763	. . . . . . . . . 10 ⊢ (deg1‘𝐾) = (deg1‘𝐾)
27	22, 23, 2, 18, 26	r1pdeglt 26533	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → ((deg1‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < ((deg1‘𝐾)‘(𝑀‘𝐴)))
28	9, 10, 21, 27	syl3anc 1399	. . . . . . . 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 6919	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	23, 31	eqtri 2786	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
34		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐸) = (0g‘𝐸)
35	12	fldcrngd 21046	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ CRing)
36		sdrgsubrg 21099	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
37	4, 36	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
38	30, 5, 33, 34, 35, 37	irngssv 34431	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
39	38, 14	sseldd 3914	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
40		eqid 2763	. . . . . . . . . . . 12 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)}
41		eqid 2763	. . . . . . . . . . . 12 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
42		eqid 2763	. . . . . . . . . . . 12 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
43	30, 32, 33, 12, 4, 39, 34, 40, 41, 42, 13	minplycl 34449	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
44	43, 2	eleqtrrdi 2875	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
45	5, 29, 23, 2, 44, 37	ressdeg1 34201	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) = ((deg1‘𝐾)‘(𝑀‘𝐴)))
46	45	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝐷‘(𝑀‘𝐴)) = ((deg1‘𝐾)‘(𝑀‘𝐴)))
47	28, 46	breqtrrd 5139	. . . . . . 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 21045	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ DivRing)
50	49	drngringd 21039	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Ring)
51		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
52		eqid 2763	. . . . . . . . . . . . 13 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾)
53		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾))
54		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸))
55	51, 5, 23, 2, 37, 52, 53, 54	ressply1bas2 22600	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))))
56		inss2 4137	. . . . . . . . . . . 12 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸))
57	55, 56	eqsstrdi 3957	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸)))
58	57, 44	sseldd 3914	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)))
59	11, 12, 4, 13, 14	irngnminplynz 34455	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
60	29, 51, 11, 54	deg1nn0cl 26461	. . . . . . . . . 10 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
61	50, 58, 59, 60	syl3anc 1399	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
62	23, 26, 48, 61, 8, 2	ply1degleel 34230	. . . . . . . 8 ⊢ (𝜑 → ((𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ↔ ((𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈 ∧ ((deg1‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < (𝐷‘(𝑀‘𝐴)))))
63	62	adantr 486	. . . . . . 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 726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
65	64	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
66		oveq1 7458	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝𝑅(𝑀‘𝐴)) = (𝑞𝑅(𝑀‘𝐴)))
67	66	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ 𝑞 = (𝑞𝑅(𝑀‘𝐴))))
68		eqcom 2770	. . . . . . . 8 ⊢ (𝑞 = (𝑞𝑅(𝑀‘𝐴)) ↔ (𝑞𝑅(𝑀‘𝐴)) = 𝑞)
69	67, 68	bitrdi 290	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ (𝑞𝑅(𝑀‘𝐴)) = 𝑞))
70	23, 26, 48, 61, 8, 2	ply1degltel 34229	. . . . . . . 8 ⊢ (𝜑 → (𝑞 ∈ 𝑇 ↔ (𝑞 ∈ 𝑈 ∧ ((deg1‘𝐾)‘𝑞) ≤ ((𝐷‘(𝑀‘𝐴)) − 1))))
71	70	simprbda 504	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → 𝑞 ∈ 𝑈)
72	70	simplbda 505	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ≤ ((𝐷‘(𝑀‘𝐴)) − 1))
73	45	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷‘(𝑀‘𝐴)) − 1) = (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1))
74	73	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((𝐷‘(𝑀‘𝐴)) − 1) = (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1))
75	72, 74	breqtrd 5137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ≤ (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1))
76	26, 23, 2	deg1cl 26456	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝑈 → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
77	71, 76	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
78	61	nn0zd 12758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℤ)
79	45, 78	eqeltrrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
80	79	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
81		degltlem1 26445	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (((deg1‘𝐾)‘𝑞) < ((deg1‘𝐾)‘(𝑀‘𝐴)) ↔ ((deg1‘𝐾)‘𝑞) ≤ (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1)))
83	75, 82	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) < ((deg1‘𝐾)‘(𝑀‘𝐴)))
84		fldsdrgfld 21106	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
85	12, 4, 84	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
86	5, 85	eqeltrid 2868	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Field)
87		fldidom 21080	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
88	86, 87	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ IDomn)
89	88	idomdomd 21028	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Domn)
90	89	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → 𝐾 ∈ Domn)
91	8, 16, 20	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
92	91	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾))
93	23, 2, 18, 22, 26, 90, 71, 92	r1pid2 26535	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((𝑞𝑅(𝑀‘𝐴)) = 𝑞 ↔ ((deg1‘𝐾)‘𝑞) < ((deg1‘𝐾)‘(𝑀‘𝐴))))
94	83, 93	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → (𝑞𝑅(𝑀‘𝐴)) = 𝑞)
95	69, 71, 94	rspcedvdw 3583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
96	95	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
97		algextdeglem.h	. . . . . 6 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
98	97	fompt 7150	. . . . 5 ⊢ (𝐻:𝑈⟶onto𝑇 ↔ (∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ∧ ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴))))
99	65, 96, 98	sylanbrc 595	. . . 4 ⊢ (𝜑 → 𝐻:𝑈⟶onto𝑇)
100	23	ply1ring 22620	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑃 ∈ Ring)
101	8, 100	syl 18	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring)
102	1, 3, 99, 101	imasbas 17734	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝐻 “s 𝑃)))
103	71	ex 418	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ 𝑇 → 𝑞 ∈ 𝑈))
104	103	ssrdv 3919	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
105		eqid 2763	. . . . 5 ⊢ (𝑃 ↾s 𝑇) = (𝑃 ↾s 𝑇)
106	105, 2	ressbas2 17460	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
107	104, 106	syl 18	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
108		ssidd 3936	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑇)
109		eqid 2763	. . . . . . 7 ⊢ (𝐻 “s 𝑃) = (𝐻 “s 𝑃)
110		eqid 2763	. . . . . . 7 ⊢ (Base‘(𝐻 “s 𝑃)) = (Base‘(𝐻 “s 𝑃))
111	104	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ⊆ 𝑈)
112		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑇)
113	111, 112	sseldd 3914	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑈)
114		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇)
115	111, 114	sseldd 3914	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
116		foeq3 6824	. . . . . . . . . 10 ⊢ (𝑇 = (Base‘(𝐻 “s 𝑃)) → (𝐻:𝑈⟶onto𝑇 ↔ 𝐻:𝑈⟶onto(Base‘(𝐻 “s 𝑃))))
117	102, 116	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐻:𝑈⟶onto𝑇 ↔ 𝐻:𝑈⟶onto(Base‘(𝐻 “s 𝑃))))
118	99, 117	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → 𝐻:𝑈⟶onto(Base‘(𝐻 “s 𝑃)))
119	118	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻:𝑈⟶onto(Base‘(𝐻 “s 𝑃)))
120	23, 2, 22, 18, 97, 8, 91	r1plmhm 34246	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
121	120	lmhmghmd 33696	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)))
122		ghmmhm 19490	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
123	121, 122	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
124	123	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐻 ∈ (𝑃 MndHom (𝐻 “s 𝑃)))
125		eqid 2763	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
126		eqid 2763	. . . . . . 7 ⊢ (+g‘(𝐻 “s 𝑃)) = (+g‘(𝐻 “s 𝑃))
127	109, 2, 110, 113, 115, 119, 124, 125, 126	mhmimasplusg 33697	. . . . . 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 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐸 ∈ Field)
130	4	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝐹 ∈ (SubDRing‘𝐸))
131	14	ad2antrr 739	. . . . . . . . 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 34466	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥 ∈ 𝑇 ↔ (𝐻‘𝑥) = 𝑥))
138	112, 137	mpbid 235	. . . . . . 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 34466	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑦 ∈ 𝑇 ↔ (𝐻‘𝑦) = 𝑦))
140	114, 139	mpbid 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘𝑦) = 𝑦)
141	138, 140	oveq12d 7469	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥(+g‘(𝐻 “s 𝑃))𝑦))
142	101	ringgrpd 20520	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Grp)
143	23, 7	ply1lvec 34194	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ LVec)
144	23, 26, 48, 61, 8	ply1degltlss 34231	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (LSubSp‘𝑃))
145		eqid 2763	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
146	105, 145	lsslvec 21435	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LVec ∧ 𝑇 ∈ (LSubSp‘𝑃)) → (𝑃 ↾s 𝑇) ∈ LVec)
147	143, 144, 146	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ LVec)
148	147	lvecgrpd 21434	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ Grp)
149	2	issubg 19386	. . . . . . . . . 10 ⊢ (𝑇 ∈ (SubGrp‘𝑃) ↔ (𝑃 ∈ Grp ∧ 𝑇 ⊆ 𝑈 ∧ (𝑃 ↾s 𝑇) ∈ Grp))
150	142, 104, 148, 149	syl3anbrc 1362	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑃))
151	150	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑇 ∈ (SubGrp‘𝑃))
152	125	subgcl 19396	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑃) ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
153	151, 112, 114, 152	syl3anc 1399	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
154	142	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑃 ∈ Grp)
155	2, 125, 154, 113, 115	grpcld 19208	. . . . . . . 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 34466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝑥(+g‘𝑃)𝑦) ∈ 𝑇 ↔ (𝐻‘(𝑥(+g‘𝑃)𝑦)) = (𝑥(+g‘𝑃)𝑦)))
157	153, 156	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝐻‘(𝑥(+g‘𝑃)𝑦)) = (𝑥(+g‘𝑃)𝑦))
158	127, 141, 157	3eqtr3d 2806	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘𝑃)𝑦))
159		fvex 6929	. . . . . . . . 9 ⊢ (deg1‘𝐾) ∈ V
160		cnvexg 7970	. . . . . . . . 9 ⊢ ((deg1‘𝐾) ∈ V → ◡(deg1‘𝐾) ∈ V)
161		imaexg 7959	. . . . . . . . 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 2860	. . . . . . 7 ⊢ 𝑇 ∈ V
164	105, 125	ressplusg 17508	. . . . . . 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 7464	. . . . 5 ⊢ (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦)
167	158, 166	eqtrdi 2814	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦))
168	167	anasss 472	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦))
169		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑦 ∈ 𝑇)
170	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐸 ∈ Field)
171	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 ∈ (SubDRing‘𝐸))
172	14	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐴 ∈ (𝐸 IntgRing 𝐹))
173	104	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑇 ⊆ 𝑈)
174	173, 169	sseldd 3914	. . . . . . . 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 34466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑦 ∈ 𝑇 ↔ (𝐻‘𝑦) = 𝑦))
176	169, 175	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘𝑦) = 𝑦)
177	176	oveq2d 7467	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦))
178		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ 𝐹)
179	33	sdrgss 21101	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
180	5, 33	ressbas2 17460	. . . . . . . . . 10 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
181	4, 179, 180	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
182	23	ply1sca 22625	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑃))
183	8, 182	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑃))
184	183	fveq2d 6920	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑃)))
185	181, 184	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘𝑃)))
186	185	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 = (Base‘(Scalar‘𝑃)))
187	178, 186	eleqtrd 2866	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑥 ∈ (Base‘(Scalar‘𝑃)))
188	118	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈⟶onto(Base‘(𝐻 “s 𝑃)))
189	120	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
190		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘(𝐻 “s 𝑃)) = ( ·𝑠 ‘(𝐻 “s 𝑃))
192		eqid 2763	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
193	109, 2, 110, 187, 174, 188, 189, 190, 191, 192	lmhmimasvsca 33698	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
194	177, 193	eqtr3d 2800	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
195	64, 97	fmptd 7146	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑈⟶𝑇)
196	195	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈⟶𝑇)
197		eqid 2763	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
198	143	lveclmodd 21433	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
199	198	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑃 ∈ LMod)
200	2, 197, 190, 192, 199, 187, 174	lmodvscld 21205	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑈)
201	196, 200	ffvelcdmd 7117	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) ∈ 𝑇)
202	194, 201	eqeltrd 2864	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) ∈ 𝑇)
203	144	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝑇 ∈ (LSubSp‘𝑃))
204	197, 190, 192, 145	lssvscl 21281	. . . . . 6 ⊢ (((𝑃 ∈ LMod ∧ 𝑇 ∈ (LSubSp‘𝑃)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇)
205	199, 203, 187, 169, 204	syl22anc 852	. . . . 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 34466	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → ((𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑇 ↔ (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) = (𝑥( ·𝑠 ‘𝑃)𝑦)))
207	205, 206	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) = (𝑥( ·𝑠 ‘𝑃)𝑦))
208	105, 190	ressvsca 17561	. . . . . 6 ⊢ (𝑇 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(𝑃 ↾s 𝑇)))
209	163, 208	mp1i 14	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(𝑃 ↾s 𝑇)))
210	209	oveqd 7468	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
211	194, 207, 210	3eqtrd 2802	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
212		eqid 2763	. . 3 ⊢ (Scalar‘(𝐻 “s 𝑃)) = (Scalar‘(𝐻 “s 𝑃))
213	105, 197	resssca 17560	. . . 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 17741	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑃) = (Scalar‘(𝐻 “s 𝑃)))
216	183, 215	eqtrd 2798	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘(𝐻 “s 𝑃)))
217	216	fveq2d 6920	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(𝐻 “s 𝑃))))
218	181, 217	eqtrd 2798	. . 3 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘(𝐻 “s 𝑃))))
219	215	fveq2d 6920	. . . . . 6 ⊢ (𝜑 → (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘(𝐻 “s 𝑃))))
220	219	oveqd 7468	. . . . 5 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘𝑃))𝑦) = (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦))
221	220	eqcomd 2769	. . . 4 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦) = (𝑥(+g‘(Scalar‘𝑃))𝑦))
222	221	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦) = (𝑥(+g‘(Scalar‘𝑃))𝑦))
223		lmhmlvec2 34362	. . . 4 ⊢ ((𝑃 ∈ LVec ∧ 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃))) → (𝐻 “s 𝑃) ∈ LVec)
224	143, 120, 223	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐻 “s 𝑃) ∈ LVec)
225	102, 107, 108, 168, 202, 211, 212, 214, 218, 185, 222, 224, 147	dimpropd 34352	. 2 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (dim‘(𝑃 ↾s 𝑇)))
226	23, 26, 48, 61, 7, 105	ply1degltdim 34366	. 2 ⊢ (𝜑 → (dim‘(𝑃 ↾s 𝑇)) = (𝐷‘(𝑀‘𝐴)))
227	225, 226	eqtrd 2798	1 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  {csn 4588  ∪ cuni 4872   class class class wbr 5108   ↦ cmpt 5192  ^◡ccnv 5657  dom cdm 5658   “ cima 5661  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  [cec 8753  1c1 11240  -∞cmnf 11380   < clt 11382   ≤ cle 11383   − cmin 11580  ℕ₀cn0 12646  ℤcz 12733  [,)cico 13518  Basecbs 17431   ↾_s cress 17452  +_gcplusg 17472  Scalarcsca 17475   ·_𝑠 cvsca 17476  0_gc0g 17660   “_s cimas 17726   /_s cqus 17727   MndHom cmhm 19026  Grpcgrp 19194  SubGrpcsubg 19380   ~_QG cqg 19382   GrpHom cghm 19477  Ringcrg 20510  SubRingcsubrg 20872  Domncdomn 20995  IDomncidom 20996  DivRingcdr 21031  Fieldcfield 21032  SubDRingcsdrg 21094  LModclmod 21186  LSubSpclss 21257   LMHom clmhm 21345  LVecclvec 21428  RSpancrsp 21536  PwSer₁cps1 22548  Poly₁cpl1 22550   evalSub₁ ces1 22686  deg₁cdg1 26427  Monic_1pcmn1 26499  Unic_1pcuc1p 26500  rem_1pcr1p 26502  idlGen_1pcig1p 26503   fldGen cfldgen 33976  dimcldim 34342   IntgRing cirng 34426   minPoly cminply 34442

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-ac2 10580  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-addf 11318

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-ofr 7726  df-rpss 7771  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-er 8755  df-map 8887  df-pm 8888  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-sup 9471  df-inf 9472  df-oi 9541  df-dju 10019  df-card 10057  df-acn 10060  df-ac 10234  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-xnn0 12720  df-z 12734  df-dec 12855  df-uz 13006  df-ico 13522  df-fz 13680  df-fzo 13829  df-seq 14185  df-hash 14516  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-mulr 17487  df-starv 17488  df-sca 17489  df-vsca 17490  df-ip 17491  df-tset 17492  df-ple 17493  df-ocomp 17494  df-ds 17495  df-unif 17496  df-hom 17497  df-cco 17498  df-euf 17499  df-0g 17662  df-gsum 17663  df-prds 17668  df-pws 17670  df-imas 17730  df-mre 17806  df-mrc 17807  df-mri 17808  df-acs 17809  df-proset 18518  df-drs 18519  df-poset 18537  df-ipo 18752  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-cntz 19582  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-srg 20464  df-ring 20512  df-cring 20513  df-oppr 20618  df-dvdsr 20638  df-unit 20639  df-invr 20669  df-rhm 20753  df-nzr 20814  df-subrng 20849  df-subrg 20873  df-rlreg 20997  df-domn 20998  df-idom 20999  df-drng 21033  df-field 21034  df-sdrg 21095  df-lmod 21188  df-lss 21258  df-lsp 21298  df-lmhm 21348  df-lbs 21401  df-lvec 21429  df-sra 21499  df-rgmod 21500  df-lidl 21537  df-rsp 21538  df-cnfld 21730  df-dsmm 22093  df-frlm 22108  df-uvc 22144  df-lindf 22167  df-linds 22168  df-assa 22216  df-asp 22217  df-ascl 22218  df-psr 22272  df-mvr 22273  df-mpl 22274  df-opsr 22276  df-evls 22438  df-evl 22439  df-psr1 22553  df-vr1 22554  df-ply1 22555  df-coe1 22556  df-evls1 22688  df-evl1 22689  df-mdeg 26428  df-deg1 26429  df-mon1 26504  df-uc1p 26505  df-q1p 26506  df-r1p 26507  df-ig1p 26508  df-dim 34343  df-irng 34427  df-minply 34443

This theorem is used by:  algextdeg  34468