Theorem algextdeglem8 34406

Description: Lemma for algextdeg 34407. 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 2761	. . . 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 21056	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐾 ∈ DivRing)
7	4, 6	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ DivRing)
8	7	drngringd 20997	. . . . . . . . 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 2760	. . . . . . . . . . 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 6893	. . . . . . . . . . 11 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹))
16	11, 12, 4, 13, 14, 15	minplym1p 34395	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
17	16	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑀‘𝐴) ∈ (Monic1p‘𝐾))
18		eqid 2760	. . . . . . . . . 10 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾)
19		eqid 2760	. . . . . . . . . 10 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾)
20	18, 19	mon1puc1p 26478	. . . . . . . . 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 26486	. . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
25	9, 10, 21, 24	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑈) → (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑈)
26		eqid 2760	. . . . . . . . . 10 ⊢ (deg1‘𝐾) = (deg1‘𝐾)
27	22, 23, 2, 18, 26	r1pdeglt 26487	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑝 ∈ 𝑈 ∧ (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) → ((deg1‘𝐾)‘(𝑝𝑅(𝑀‘𝐴))) < ((deg1‘𝐾)‘(𝑀‘𝐴)))
28	9, 10, 21, 27	syl3anc 1398	. . . . . . . 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 6893	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹))
32	23, 31	eqtri 2783	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
33		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐸) = (Base‘𝐸)
34		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐸) = (0g‘𝐸)
35	12	fldcrngd 21004	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ CRing)
36		sdrgsubrg 21057	. . . . . . . . . . . . . . 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 34370	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸))
39	38, 14	sseldd 3910	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸))
40		eqid 2760	. . . . . . . . . . . 12 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)}
41		eqid 2760	. . . . . . . . . . . 12 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
42		eqid 2760	. . . . . . . . . . . 12 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹))
43	30, 32, 33, 12, 4, 39, 34, 40, 41, 42, 13	minplycl 34388	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
44	43, 2	eleqtrrdi 2871	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
45	5, 29, 23, 2, 44, 37	ressdeg1 34148	. . . . . . . . 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 5132	. . . . . . 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 21003	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ DivRing)
50	49	drngringd 20997	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Ring)
51		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸)
52		eqid 2760	. . . . . . . . . . . . 13 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾)
53		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾))
54		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸))
55	51, 5, 23, 2, 37, 52, 53, 54	ressply1bas2 22554	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))))
56		inss2 4133	. . . . . . . . . . . 12 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸))
57	55, 56	eqsstrdi 3953	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸)))
58	57, 44	sseldd 3910	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)))
59	11, 12, 4, 13, 14	irngnminplynz 34394	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸)))
60	29, 51, 11, 54	deg1nn0cl 26415	. . . . . . . . . 10 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
61	50, 58, 59, 60	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0)
62	23, 26, 48, 61, 8, 2	ply1degleel 34177	. . . . . . . 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 3154	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇)
66		oveq1 7431	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝𝑅(𝑀‘𝐴)) = (𝑞𝑅(𝑀‘𝐴)))
67	66	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑞 = (𝑝𝑅(𝑀‘𝐴)) ↔ 𝑞 = (𝑞𝑅(𝑀‘𝐴))))
68		eqcom 2767	. . . . . . . 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 34176	. . . . . . . 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 7439	. . . . . . . . . . 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 5130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ≤ (((deg1‘𝐾)‘(𝑀‘𝐴)) − 1))
76	26, 23, 2	deg1cl 26410	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝑈 → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
77	71, 76	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘𝑞) ∈ (ℕ0 ∪ {-∞}))
78	61	nn0zd 12725	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℤ)
79	45, 78	eqeltrrd 2861	. . . . . . . . . . 11 ⊢ (𝜑 → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
80	79	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ((deg1‘𝐾)‘(𝑀‘𝐴)) ∈ ℤ)
81		degltlem1 26399	. . . . . . . . . 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 21064	. . . . . . . . . . . . . 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 2864	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Field)
87		fldidom 21038	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
88	86, 87	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ IDomn)
89	88	idomdomd 20986	. . . . . . . . . 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 26489	. . . . . . . 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 3579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑇) → ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
96	95	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴)))
97		algextdeglem.h	. . . . . 6 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))
98	97	fompt 7123	. . . . 5 ⊢ (𝐻:𝑈–onto→𝑇 ↔ (∀𝑝 ∈ 𝑈 (𝑝𝑅(𝑀‘𝐴)) ∈ 𝑇 ∧ ∀𝑞 ∈ 𝑇 ∃𝑝 ∈ 𝑈 𝑞 = (𝑝𝑅(𝑀‘𝐴))))
99	65, 96, 98	sylanbrc 595	. . . 4 ⊢ (𝜑 → 𝐻:𝑈–onto→𝑇)
100	23	ply1ring 22574	. . . . 5 ⊢ (𝐾 ∈ Ring → 𝑃 ∈ Ring)
101	8, 100	syl 18	. . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring)
102	1, 3, 99, 101	imasbas 17693	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝐻 “s 𝑃)))
103	71	ex 418	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ 𝑇 → 𝑞 ∈ 𝑈))
104	103	ssrdv 3915	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
105		eqid 2760	. . . . 5 ⊢ (𝑃 ↾s 𝑇) = (𝑃 ↾s 𝑇)
106	105, 2	ressbas2 17425	. . . 4 ⊢ (𝑇 ⊆ 𝑈 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
107	104, 106	syl 18	. . 3 ⊢ (𝜑 → 𝑇 = (Base‘(𝑃 ↾s 𝑇)))
108		ssidd 3932	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑇)
109		eqid 2760	. . . . . . 7 ⊢ (𝐻 “s 𝑃) = (𝐻 “s 𝑃)
110		eqid 2760	. . . . . . 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 3910	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑥 ∈ 𝑈)
114		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑇)
115	111, 114	sseldd 3910	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
116		foeq3 6799	. . . . . . . . . 10 ⊢ (𝑇 = (Base‘(𝐻 “s 𝑃)) → (𝐻:𝑈–onto→𝑇 ↔ 𝐻:𝑈–onto→(Base‘(𝐻 “s 𝑃))))
117	102, 116	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐻:𝑈–onto→𝑇 ↔ 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 34191	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (𝑃 LMHom (𝐻 “s 𝑃)))
121	120	lmhmghmd 33647	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom (𝐻 “s 𝑃)))
122		ghmmhm 19449	. . . . . . . . 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 2760	. . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃)
126		eqid 2760	. . . . . . 7 ⊢ (+g‘(𝐻 “s 𝑃)) = (+g‘(𝐻 “s 𝑃))
127	109, 2, 110, 113, 115, 119, 124, 125, 126	mhmimasplusg 33648	. . . . . 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 34405	. . . . . . . 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 34405	. . . . . . . 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 7442	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → ((𝐻‘𝑥)(+g‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝑥(+g‘(𝐻 “s 𝑃))𝑦))
142	101	ringgrpd 20478	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Grp)
143	23, 7	ply1lvec 34141	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ LVec)
144	23, 26, 48, 61, 8	ply1degltlss 34178	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (LSubSp‘𝑃))
145		eqid 2760	. . . . . . . . . . . . 13 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
146	105, 145	lsslvec 21393	. . . . . . . . . . . 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 21392	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ↾s 𝑇) ∈ Grp)
149	2	issubg 19345	. . . . . . . . . 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 19355	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑃) ∧ 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘𝑃)𝑦) ∈ 𝑇)
153	151, 112, 114, 152	syl3anc 1398	. . . . . . 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 19167	. . . . . . . 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 34405	. . . . . . 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 2803	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑇) ∧ 𝑦 ∈ 𝑇) → (𝑥(+g‘(𝐻 “s 𝑃))𝑦) = (𝑥(+g‘𝑃)𝑦))
159		fvex 6903	. . . . . . . . 9 ⊢ (deg1‘𝐾) ∈ V
160		cnvexg 7940	. . . . . . . . 9 ⊢ ((deg1‘𝐾) ∈ V → ◡(deg1‘𝐾) ∈ V)
161		imaexg 7929	. . . . . . . . 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 2856	. . . . . . 7 ⊢ 𝑇 ∈ V
164	105, 125	ressplusg 17471	. . . . . . 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 7437	. . . . 5 ⊢ (𝑥(+g‘𝑃)𝑦) = (𝑥(+g‘(𝑃 ↾s 𝑇))𝑦)
167	158, 166	eqtrdi 2811	. . . 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 3910	. . . . . . . 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 34405	. . . . . . 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 7440	. . . . 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 21059	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ (Base‘𝐸))
180	5, 33	ressbas2 17425	. . . . . . . . . 10 ⊢ (𝐹 ⊆ (Base‘𝐸) → 𝐹 = (Base‘𝐾))
181	4, 179, 180	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
182	23	ply1sca 22579	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Ring → 𝐾 = (Scalar‘𝑃))
183	8, 182	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑃))
184	183	fveq2d 6894	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑃)))
185	181, 184	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘𝑃)))
186	185	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐹 = (Base‘(Scalar‘𝑃)))
187	178, 186	eleqtrd 2862	. . . . . 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 2760	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
191		eqid 2760	. . . . . 6 ⊢ ( ·𝑠 ‘(𝐻 “s 𝑃)) = ( ·𝑠 ‘(𝐻 “s 𝑃))
192		eqid 2760	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
193	109, 2, 110, 187, 174, 188, 189, 190, 191, 192	lmhmimasvsca 33649	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))(𝐻‘𝑦)) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
194	177, 193	eqtr3d 2797	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)))
195	64, 97	fmptd 7119	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑈⟶𝑇)
196	195	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → 𝐻:𝑈⟶𝑇)
197		eqid 2760	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
198	143	lveclmodd 21391	. . . . . . 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 21163	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) ∈ 𝑈)
201	196, 200	ffvelcdmd 7090	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝐻‘(𝑥( ·𝑠 ‘𝑃)𝑦)) ∈ 𝑇)
202	194, 201	eqeltrd 2860	. . 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 21239	. . . . . 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 34405	. . . . 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 17524	. . . . . 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 7441	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
211	194, 207, 210	3eqtrd 2799	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝑇)) → (𝑥( ·𝑠 ‘(𝐻 “s 𝑃))𝑦) = (𝑥( ·𝑠 ‘(𝑃 ↾s 𝑇))𝑦))
212		eqid 2760	. . 3 ⊢ (Scalar‘(𝐻 “s 𝑃)) = (Scalar‘(𝐻 “s 𝑃))
213	105, 197	resssca 17523	. . . 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 17700	. . . . . 6 ⊢ (𝜑 → (Scalar‘𝑃) = (Scalar‘(𝐻 “s 𝑃)))
216	183, 215	eqtrd 2795	. . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘(𝐻 “s 𝑃)))
217	216	fveq2d 6894	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(𝐻 “s 𝑃))))
218	181, 217	eqtrd 2795	. . 3 ⊢ (𝜑 → 𝐹 = (Base‘(Scalar‘(𝐻 “s 𝑃))))
219	215	fveq2d 6894	. . . . . 6 ⊢ (𝜑 → (+g‘(Scalar‘𝑃)) = (+g‘(Scalar‘(𝐻 “s 𝑃))))
220	219	oveqd 7441	. . . . 5 ⊢ (𝜑 → (𝑥(+g‘(Scalar‘𝑃))𝑦) = (𝑥(+g‘(Scalar‘(𝐻 “s 𝑃)))𝑦))
221	220	eqcomd 2766	. . . 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 34301	. . . 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 34291	. 2 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (dim‘(𝑃 ↾s 𝑇)))
226	23, 26, 48, 61, 7, 105	ply1degltdim 34305	. 2 ⊢ (𝜑 → (dim‘(𝑃 ↾s 𝑇)) = (𝐷‘(𝑀‘𝐴)))
227	225, 226	eqtrd 2795	1 ⊢ (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  {csn 4584  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7424  [cec 8723  1c1 11208  -∞cmnf 11348   < clt 11350   ≤ cle 11351   − cmin 11548  ℕ₀cn0 12613  ℤcz 12700  [,)cico 13485  Basecbs 17396   ↾_s cress 17417  +_gcplusg 17437  Scalarcsca 17440   ·_𝑠 cvsca 17441  0_gc0g 17619   “_s cimas 17685   /_s cqus 17686   MndHom cmhm 18985  Grpcgrp 19153  SubGrpcsubg 19339   ~_QG cqg 19341   GrpHom cghm 19436  Ringcrg 20468  SubRingcsubrg 20830  Domncdomn 20953  IDomncidom 20954  DivRingcdr 20989  Fieldcfield 20990  SubDRingcsdrg 21052  LModclmod 21144  LSubSpclss 21215   LMHom clmhm 21303  LVecclvec 21386  RSpancrsp 21494  PwSer₁cps1 22502  Poly₁cpl1 22504   evalSub₁ ces1 22640  deg₁cdg1 26381  Monic_1pcmn1 26453  Unic_1pcuc1p 26454  rem_1pcr1p 26456  idlGen_1pcig1p 26457   fldGen cfldgen 33922  dimcldim 34281   IntgRing cirng 34365   minPoly cminply 34381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-ac2 10548  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285  ax-addf 11286

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  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 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-ofr 7698  df-rpss 7743  df-om 7882  df-1st 8005  df-2nd 8006  df-supp 8185  df-tpos 8251  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-oadd 8488  df-er 8725  df-map 8857  df-pm 8858  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-fsupp 9362  df-sup 9441  df-inf 9442  df-oi 9511  df-dju 9989  df-card 10027  df-acn 10030  df-ac 10202  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-xnn0 12687  df-z 12701  df-dec 12822  df-uz 12973  df-ico 13489  df-fz 13647  df-fzo 13796  df-seq 14152  df-hash 14482  df-struct 17334  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-ress 17418  df-plusg 17450  df-mulr 17451  df-starv 17452  df-sca 17453  df-vsca 17454  df-ip 17455  df-tset 17456  df-ple 17457  df-ocomp 17458  df-ds 17459  df-unif 17460  df-hom 17461  df-cco 17462  df-0g 17621  df-gsum 17622  df-prds 17627  df-pws 17629  df-imas 17689  df-mre 17765  df-mrc 17766  df-mri 17767  df-acs 17768  df-proset 18477  df-drs 18478  df-poset 18496  df-ipo 18711  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-mhm 18987  df-submnd 18988  df-grp 19156  df-minusg 19157  df-sbg 19158  df-mulg 19287  df-subg 19342  df-ghm 19437  df-cntz 19540  df-cmn 20005  df-abl 20006  df-mgp 20370  df-rng 20384  df-ur 20417  df-srg 20422  df-ring 20470  df-cring 20471  df-oppr 20576  df-dvdsr 20596  df-unit 20597  df-invr 20627  df-rhm 20711  df-nzr 20772  df-subrng 20807  df-subrg 20831  df-rlreg 20955  df-domn 20956  df-idom 20957  df-drng 20991  df-field 20992  df-sdrg 21053  df-lmod 21146  df-lss 21216  df-lsp 21256  df-lmhm 21306  df-lbs 21359  df-lvec 21387  df-sra 21457  df-rgmod 21458  df-lidl 21495  df-rsp 21496  df-cnfld 21688  df-dsmm 22047  df-frlm 22062  df-uvc 22098  df-lindf 22121  df-linds 22122  df-assa 22170  df-asp 22171  df-ascl 22172  df-psr 22226  df-mvr 22227  df-mpl 22228  df-opsr 22230  df-evls 22392  df-evl 22393  df-psr1 22507  df-vr1 22508  df-ply1 22509  df-coe1 22510  df-evls1 22642  df-evl1 22643  df-mdeg 26382  df-deg1 26383  df-mon1 26458  df-uc1p 26459  df-q1p 26460  df-r1p 26461  df-ig1p 26462  df-dim 34282  df-irng 34366  df-minply 34382

This theorem is used by:  algextdeg  34407