r1plmhm - Mathbox for Thierry Arnoux

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Theorem r1plmhm 33816

Description: The univariate polynomial remainder function 𝐹 is a module homomorphism. Its image (𝐹 “_s 𝑃) is sometimes called the "ring of remainders". (Contributed by Thierry Arnoux, 2-Apr-2025.)

**Hypotheses**
Ref	Expression
r1plmhm.1	⊢ 𝑃 = (Poly₁‘𝑅)
r1plmhm.2	⊢ 𝑈 = (Base‘𝑃)
r1plmhm.4	⊢ 𝐸 = (rem_1p‘𝑅)
r1plmhm.5	⊢ 𝑁 = (Unic_1p‘𝑅)
r1plmhm.6	⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))
r1plmhm.9	⊢ (𝜑 → 𝑅 ∈ Ring)
r1plmhm.10	⊢ (𝜑 → 𝑀 ∈ 𝑁)

**Assertion**
Ref	Expression
r1plmhm	⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃)))

Distinct variable groups: 𝑓,𝐸 𝑓,𝑀 𝑃,𝑓 𝑈,𝑓 𝜑,𝑓 Allowed substitution hints: 𝑅(𝑓) 𝐹(𝑓) 𝑁(𝑓)

Proof of Theorem r1plmhm Dummy variables 𝑝 𝑎 𝑏 𝑘 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1plmhm.2	. . 3 ⊢ 𝑈 = (Base‘𝑃)
2		r1plmhm.9	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑅 ∈ Ring)
4		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑓 ∈ 𝑈)
5		r1plmhm.10	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑁)
6	5	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑀 ∈ 𝑁)
7		r1plmhm.4	. . . . . 6 ⊢ 𝐸 = (rem_1p‘𝑅)
8		r1plmhm.1	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
9		r1plmhm.5	. . . . . 6 ⊢ 𝑁 = (Unic_1p‘𝑅)
10	7, 8, 1, 9	r1pcl 26277	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝑈 ∧ 𝑀 ∈ 𝑁) → (𝑓𝐸𝑀) ∈ 𝑈)
11	3, 4, 6, 10	syl3anc 1394	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → (𝑓𝐸𝑀) ∈ 𝑈)
12		r1plmhm.6	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))
13	11, 12	fmptd 7099	. . 3 ⊢ (𝜑 → 𝐹:𝑈⟶𝑈)
14		eqid 2765	. . 3 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
15		anass 473	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ↔ (𝜑 ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)))
16	2	ad6antr 748	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑅 ∈ Ring)
17		simp-6r 799	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑎 ∈ 𝑈)
18	5	ad6antr 748	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑀 ∈ 𝑁)
19		simplr 780	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝐹‘𝑝))
20		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝑓𝐸𝑀) = (𝑎𝐸𝑀))
21		ovexd 7435	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑎𝐸𝑀) ∈ V)
22	12, 20, 17, 21	fvmptd3 7003	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
23		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑝 → (𝑓𝐸𝑀) = (𝑝𝐸𝑀))
24		simp-4r 795	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑝 ∈ 𝑈)
25		ovexd 7435	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑝𝐸𝑀) ∈ V)
26	12, 23, 24, 25	fvmptd3 7003	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑝) = (𝑝𝐸𝑀))
27	19, 22, 26	3eqtr3d 2808	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑎𝐸𝑀) = (𝑝𝐸𝑀))
28		simp-5r 797	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑏 ∈ 𝑈)
29	8, 1, 9, 7, 16, 17, 18, 27, 14, 24, 28	r1padd1 33815	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
30		oveq1 7407	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑎(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀))
31	8	ply1ring 22367	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
32	2, 31	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Ring)
33	32	ringgrpd 20315	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ Grp)
34	33	ad6antr 748	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑃 ∈ Grp)
35	1, 14, 34, 17, 28	grpcld 19004	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑎(+_g‘𝑃)𝑏) ∈ 𝑈)
36		ovexd 7435	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀) ∈ V)
37	12, 30, 35, 36	fvmptd3 7003	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀))
38		oveq1 7407	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑝(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
39	1, 14, 34, 24, 28	grpcld 19004	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑝(+_g‘𝑃)𝑏) ∈ 𝑈)
40		ovexd 7435	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀) ∈ V)
41	12, 38, 39, 40	fvmptd3 7003	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑝(+_g‘𝑃)𝑏)) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
42	29, 37, 41	3eqtr4d 2810	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑝(+_g‘𝑃)𝑏)))
43	32	ringabld 20357	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Abel)
44	43	ad6antr 748	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑃 ∈ Abel)
45	1, 14	ablcom 19860	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Abel ∧ 𝑝 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) → (𝑝(+_g‘𝑃)𝑏) = (𝑏(+_g‘𝑃)𝑝))
46	44, 24, 28, 45	syl3anc 1394	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑝(+_g‘𝑃)𝑏) = (𝑏(+_g‘𝑃)𝑝))
47	46	fveq2d 6875	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑝(+_g‘𝑃)𝑏)) = (𝐹‘(𝑏(+_g‘𝑃)𝑝)))
48	42, 47	eqtrd 2800	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑏(+_g‘𝑃)𝑝)))
49		simpr 489	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝐹‘𝑞))
50		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (𝑓𝐸𝑀) = (𝑏𝐸𝑀))
51		ovexd 7435	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑏𝐸𝑀) ∈ V)
52	12, 50, 28, 51	fvmptd3 7003	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
53		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑞 → (𝑓𝐸𝑀) = (𝑞𝐸𝑀))
54		simpllr 787	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑞 ∈ 𝑈)
55		ovexd 7435	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑞𝐸𝑀) ∈ V)
56	12, 53, 54, 55	fvmptd3 7003	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑞) = (𝑞𝐸𝑀))
57	49, 52, 56	3eqtr3d 2808	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑏𝐸𝑀) = (𝑞𝐸𝑀))
58	8, 1, 9, 7, 16, 28, 18, 57, 14, 54, 24	r1padd1 33815	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
59		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀))
60	1, 14, 34, 28, 24	grpcld 19004	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑏(+_g‘𝑃)𝑝) ∈ 𝑈)
61		ovexd 7435	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀) ∈ V)
62	12, 59, 60, 61	fvmptd3 7003	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑏(+_g‘𝑃)𝑝)) = ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀))
63		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑓 = (𝑞(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
64	1, 14, 34, 54, 24	grpcld 19004	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑞(+_g‘𝑃)𝑝) ∈ 𝑈)
65		ovexd 7435	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀) ∈ V)
66	12, 63, 64, 65	fvmptd3 7003	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑞(+_g‘𝑃)𝑝)) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
67	58, 62, 66	3eqtr4d 2810	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑏(+_g‘𝑃)𝑝)) = (𝐹‘(𝑞(+_g‘𝑃)𝑝)))
68	1, 14	ablcom 19860	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Abel ∧ 𝑞 ∈ 𝑈 ∧ 𝑝 ∈ 𝑈) → (𝑞(+_g‘𝑃)𝑝) = (𝑝(+_g‘𝑃)𝑞))
69	44, 54, 24, 68	syl3anc 1394	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑞(+_g‘𝑃)𝑝) = (𝑝(+_g‘𝑃)𝑞))
70	69	fveq2d 6875	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑞(+_g‘𝑃)𝑝)) = (𝐹‘(𝑝(+_g‘𝑃)𝑞)))
71	48, 67, 70	3eqtrd 2804	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑝(+_g‘𝑃)𝑞)))
72	71	expl 462	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑝(+_g‘𝑃)𝑞))))
73	72	anasss 471	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ (𝑝 ∈ 𝑈 ∧ 𝑞 ∈ 𝑈)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑝(+_g‘𝑃)𝑞))))
74	15, 73	sylanbr 593	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) ∧ (𝑝 ∈ 𝑈 ∧ 𝑞 ∈ 𝑈)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑝(+_g‘𝑃)𝑞))))
75	74	3impa 1125	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) ∧ (𝑝 ∈ 𝑈 ∧ 𝑞 ∈ 𝑈)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+_g‘𝑃)𝑏)) = (𝐹‘(𝑝(+_g‘𝑃)𝑞))))
76		eqid 2765	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
77		eqid 2765	. . 3 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
78		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝐹‘𝑏))
79		simpr2 1212	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑈)
80		ovexd 7435	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) ∈ V)
81	12, 20, 79, 80	fvmptd3 7003	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
82		simpr3 1213	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
83		ovexd 7435	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑏𝐸𝑀) ∈ V)
84	12, 50, 82, 83	fvmptd3 7003	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
85	78, 81, 84	3eqtr3d 2808	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) = (𝑏𝐸𝑀))
86	85	oveq2d 7416	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)(𝑎𝐸𝑀)) = (𝑘( ·_𝑠 ‘𝑃)(𝑏𝐸𝑀)))
87	2	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑅 ∈ Ring)
88	5	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑀 ∈ 𝑁)
89		eqid 2765	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
90		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
91		simpr1 1211	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
92	8	ply1sca 22372	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
93	2, 92	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
94	93	fveq2d 6875	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
95	94	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
96	91, 95	eleqtrrd 2868	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑘 ∈ (Base‘𝑅))
97	8, 1, 9, 7, 87, 79, 88, 89, 90, 96	r1pvsca 33812	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀) = (𝑘( ·_𝑠 ‘𝑃)(𝑎𝐸𝑀)))
98	8, 1, 9, 7, 87, 82, 88, 89, 90, 96	r1pvsca 33812	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀) = (𝑘( ·_𝑠 ‘𝑃)(𝑏𝐸𝑀)))
99	86, 97, 98	3eqtr4d 2810	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
100		oveq1 7407	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑎) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀))
101	8	ply1lmod 22371	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
102	87, 101	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑃 ∈ LMod)
103	1, 76, 89, 77, 102, 91, 79	lmodvscld 20969	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑎) ∈ 𝑈)
104		ovexd 7435	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀) ∈ V)
105	12, 100, 103, 104	fvmptd3 7003	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑎)) = ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀))
106		oveq1 7407	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
107	1, 76, 89, 77, 102, 91, 82	lmodvscld 20969	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑏) ∈ 𝑈)
108		ovexd 7435	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀) ∈ V)
109	12, 106, 107, 108	fvmptd3 7003	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑏)) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
110	99, 105, 109	3eqtr4d 2810	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑎)) = (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑏)))
111	110	an32s 664	. . . 4 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) → (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑎)) = (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑏)))
112	111	ex 417	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑎)) = (𝐹‘(𝑘( ·_𝑠 ‘𝑃)𝑏))))
113	2, 101	syl 18	. . 3 ⊢ (𝜑 → 𝑃 ∈ LMod)
114	1, 13, 14, 75, 76, 77, 112, 113, 89	imaslmhm 33592	. 2 ⊢ (𝜑 → ((𝐹 “_s 𝑃) ∈ LMod ∧ 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃))))
115	114	simprd 500	1 ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ↦ cmpt 5186 ‘cfv 6525 (class class class)co 7400 Basecbs 17259 +_gcplusg 17300 Scalarcsca 17303 ·_𝑠 cvsca 17304 “_s cimas 17548 Grpcgrp 18990 Abelcabl 19842 Ringcrg 20306 LModclmod 20950 LMHom clmhm 21109 Poly₁cpl1 22297 Unic_1pcuc1p 26245 rem_1pcr1p 26247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-fzo 13674 df-seq 14029 df-hash 14358 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-0g 17484 df-gsum 17485 df-prds 17490 df-pws 17492 df-imas 17552 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-ghm 19275 df-cntz 19378 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-cring 20309 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-subrng 20622 df-subrg 20646 df-rlreg 20770 df-lmod 20952 df-lss 21022 df-lmhm 21112 df-cnfld 21483 df-psr 22019 df-mvr 22020 df-mpl 22021 df-opsr 22023 df-psr1 22300 df-vr1 22301 df-ply1 22302 df-coe1 22303 df-mdeg 26173 df-deg1 26174 df-uc1p 26250 df-q1p 26251 df-r1p 26252

This theorem is referenced by: r1pquslmic 33817 algextdeglem8 34031

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