r1plmhm - Mathbox for Thierry Arnoux

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Theorem r1plmhm 33702

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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑅 ∈ Ring)
4		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑓 ∈ 𝑈)
5		r1plmhm.10	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑁)
6	5	adantr 480	. . . . 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 26132	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝑈 ∧ 𝑀 ∈ 𝑁) → (𝑓𝐸𝑀) ∈ 𝑈)
11	3, 4, 6, 10	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → (𝑓𝐸𝑀) ∈ 𝑈)
12		r1plmhm.6	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))
13	11, 12	fmptd 7068	. . 3 ⊢ (𝜑 → 𝐹:𝑈⟶𝑈)
14		eqid 2737	. . 3 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
15		anass 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ↔ (𝜑 ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)))
16	2	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑅 ∈ Ring)
17		simp-6r 788	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑎 ∈ 𝑈)
18	5	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑀 ∈ 𝑁)
19		simplr 769	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝐹‘𝑝))
20		oveq1 7375	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝑓𝐸𝑀) = (𝑎𝐸𝑀))
21		ovexd 7403	. . . . . . . . . . . . 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 6973	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
23		oveq1 7375	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑝 → (𝑓𝐸𝑀) = (𝑝𝐸𝑀))
24		simp-4r 784	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑝 ∈ 𝑈)
25		ovexd 7403	. . . . . . . . . . . . 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 6973	. . . . . . . . . . . 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 2780	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑎𝐸𝑀) = (𝑝𝐸𝑀))
28		simp-5r 786	. . . . . . . . . . 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 33700	. . . . . . . . . 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 7375	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑎(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀))
31	8	ply1ring 22200	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
32	2, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Ring)
33	32	ringgrpd 20189	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ Grp)
34	33	ad6antr 737	. . . . . . . . . . . 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 18889	. . . . . . . . . . 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 7403	. . . . . . . . . . 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 6973	. . . . . . . . . 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 7375	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑝(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
39	1, 14, 34, 24, 28	grpcld 18889	. . . . . . . . . . 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 7403	. . . . . . . . . . 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 6973	. . . . . . . . . 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 2782	. . . . . . . . 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 20230	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Abel)
44	43	ad6antr 737	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑃 ∈ Abel)
45	1, 14	ablcom 19740	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Abel ∧ 𝑝 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) → (𝑝(+_g‘𝑃)𝑏) = (𝑏(+_g‘𝑃)𝑝))
46	44, 24, 28, 45	syl3anc 1374	. . . . . . . . . 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 6846	. . . . . . . . 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 2772	. . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝐹‘𝑞))
50		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (𝑓𝐸𝑀) = (𝑏𝐸𝑀))
51		ovexd 7403	. . . . . . . . . . . 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 6973	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
53		oveq1 7375	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑞 → (𝑓𝐸𝑀) = (𝑞𝐸𝑀))
54		simpllr 776	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑞 ∈ 𝑈)
55		ovexd 7403	. . . . . . . . . . . 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 6973	. . . . . . . . . . 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 2780	. . . . . . . . . 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 33700	. . . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀))
60	1, 14, 34, 28, 24	grpcld 18889	. . . . . . . . . 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 7403	. . . . . . . . . 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 6973	. . . . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑓 = (𝑞(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
64	1, 14, 34, 54, 24	grpcld 18889	. . . . . . . . . 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 7403	. . . . . . . . . 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 6973	. . . . . . . . 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 2782	. . . . . . . 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 19740	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Abel ∧ 𝑞 ∈ 𝑈 ∧ 𝑝 ∈ 𝑈) → (𝑞(+_g‘𝑃)𝑝) = (𝑝(+_g‘𝑃)𝑞))
69	44, 54, 24, 68	syl3anc 1374	. . . . . . . . 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 6846	. . . . . . . 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 2776	. . . . . . 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 457	. . . . . 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 466	. . . . 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 583	. . . 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 1110	. . 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 2737	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
77		eqid 2737	. . 3 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
78		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝐹‘𝑏))
79		simpr2 1197	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑈)
80		ovexd 7403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) ∈ V)
81	12, 20, 79, 80	fvmptd3 6973	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
82		simpr3 1198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
83		ovexd 7403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑏𝐸𝑀) ∈ V)
84	12, 50, 82, 83	fvmptd3 6973	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
85	78, 81, 84	3eqtr3d 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) = (𝑏𝐸𝑀))
86	85	oveq2d 7384	. . . . . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑅 ∈ Ring)
88	5	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑀 ∈ 𝑁)
89		eqid 2737	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
90		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
91		simpr1 1196	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
92	8	ply1sca 22205	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
93	2, 92	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
94	93	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
95	94	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
96	91, 95	eleqtrrd 2840	. . . . . . . 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 33697	. . . . . . 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 33697	. . . . . . 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 2782	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑎) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀))
101	8	ply1lmod 22204	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
102	87, 101	syl 17	. . . . . . . 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 20842	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑎) ∈ 𝑈)
104		ovexd 7403	. . . . . . 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 6973	. . . . . 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 7375	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
107	1, 76, 89, 77, 102, 91, 82	lmodvscld 20842	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑏) ∈ 𝑈)
108		ovexd 7403	. . . . . . 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 6973	. . . . . 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 2782	. . . . 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 653	. . . 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 412	. . 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 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ LMod)
114	1, 13, 14, 75, 76, 77, 112, 113, 89	imaslmhm 33449	. 2 ⊢ (𝜑 → ((𝐹 “_s 𝑃) ∈ LMod ∧ 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃))))
115	114	simprd 495	1 ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 +_gcplusg 17189 Scalarcsca 17192 ·_𝑠 cvsca 17193 “_s cimas 17437 Grpcgrp 18875 Abelcabl 19722 Ringcrg 20180 LModclmod 20823 LMHom clmhm 20983 Poly₁cpl1 22129 Unic_1pcuc1p 26100 rem_1pcr1p 26102

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-imas 17441 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-lmod 20825 df-lss 20895 df-lmhm 20986 df-cnfld 21322 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-mdeg 26028 df-deg1 26029 df-uc1p 26105 df-q1p 26106 df-r1p 26107

This theorem is referenced by: r1pquslmic 33703 algextdeglem8 33901

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