r1plmhm - Mathbox for Thierry Arnoux

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Theorem r1plmhm 33575

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 26064	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝑈 ∧ 𝑀 ∈ 𝑁) → (𝑓𝐸𝑀) ∈ 𝑈)
11	3, 4, 6, 10	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → (𝑓𝐸𝑀) ∈ 𝑈)
12		r1plmhm.6	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))
13	11, 12	fmptd 7086	. . 3 ⊢ (𝜑 → 𝐹:𝑈⟶𝑈)
14		eqid 2729	. . 3 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
15		anass 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ↔ (𝜑 ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)))
16	2	ad6antr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑅 ∈ Ring)
17		simp-6r 787	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑎 ∈ 𝑈)
18	5	ad6antr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑀 ∈ 𝑁)
19		simplr 768	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝐹‘𝑝))
20		oveq1 7394	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝑓𝐸𝑀) = (𝑎𝐸𝑀))
21		ovexd 7422	. . . . . . . . . . . . 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 6991	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
23		oveq1 7394	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑝 → (𝑓𝐸𝑀) = (𝑝𝐸𝑀))
24		simp-4r 783	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑝 ∈ 𝑈)
25		ovexd 7422	. . . . . . . . . . . . 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 6991	. . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑎𝐸𝑀) = (𝑝𝐸𝑀))
28		simp-5r 785	. . . . . . . . . . 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 33573	. . . . . . . . . 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 7394	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑎(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀))
31	8	ply1ring 22132	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
32	2, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Ring)
33	32	ringgrpd 20151	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ Grp)
34	33	ad6antr 736	. . . . . . . . . . . 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 18879	. . . . . . . . . . 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 7422	. . . . . . . . . . 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 6991	. . . . . . . . . 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 7394	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑝(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
39	1, 14, 34, 24, 28	grpcld 18879	. . . . . . . . . . 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 7422	. . . . . . . . . . 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 6991	. . . . . . . . . 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 2774	. . . . . . . . 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 20192	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Abel)
44	43	ad6antr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑃 ∈ Abel)
45	1, 14	ablcom 19729	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Abel ∧ 𝑝 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) → (𝑝(+_g‘𝑃)𝑏) = (𝑏(+_g‘𝑃)𝑝))
46	44, 24, 28, 45	syl3anc 1373	. . . . . . . . . 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 6862	. . . . . . . . 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 2764	. . . . . . . 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 7394	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (𝑓𝐸𝑀) = (𝑏𝐸𝑀))
51		ovexd 7422	. . . . . . . . . . . 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 6991	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
53		oveq1 7394	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑞 → (𝑓𝐸𝑀) = (𝑞𝐸𝑀))
54		simpllr 775	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑞 ∈ 𝑈)
55		ovexd 7422	. . . . . . . . . . . 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 6991	. . . . . . . . . . 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 2772	. . . . . . . . . 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 33573	. . . . . . . . 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 7394	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀))
60	1, 14, 34, 28, 24	grpcld 18879	. . . . . . . . . 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 7422	. . . . . . . . . 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 6991	. . . . . . . . 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 7394	. . . . . . . . . 10 ⊢ (𝑓 = (𝑞(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
64	1, 14, 34, 54, 24	grpcld 18879	. . . . . . . . . 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 7422	. . . . . . . . . 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 6991	. . . . . . . . 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 2774	. . . . . . . 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 19729	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Abel ∧ 𝑞 ∈ 𝑈 ∧ 𝑝 ∈ 𝑈) → (𝑞(+_g‘𝑃)𝑝) = (𝑝(+_g‘𝑃)𝑞))
69	44, 54, 24, 68	syl3anc 1373	. . . . . . . . 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 6862	. . . . . . . 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 2768	. . . . . . 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 582	. . . 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 1109	. . 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 2729	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
77		eqid 2729	. . 3 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
78		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝐹‘𝑏))
79		simpr2 1196	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑈)
80		ovexd 7422	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) ∈ V)
81	12, 20, 79, 80	fvmptd3 6991	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
82		simpr3 1197	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
83		ovexd 7422	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑏𝐸𝑀) ∈ V)
84	12, 50, 82, 83	fvmptd3 6991	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
85	78, 81, 84	3eqtr3d 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) = (𝑏𝐸𝑀))
86	85	oveq2d 7403	. . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑅 ∈ Ring)
88	5	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑀 ∈ 𝑁)
89		eqid 2729	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
90		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
91		simpr1 1195	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
92	8	ply1sca 22137	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
93	2, 92	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
94	93	fveq2d 6862	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
95	94	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
96	91, 95	eleqtrrd 2831	. . . . . . . 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 33570	. . . . . . 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 33570	. . . . . . 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 2774	. . . . . 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 7394	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑎) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀))
101	8	ply1lmod 22136	. . . . . . . . 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 20785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑎) ∈ 𝑈)
104		ovexd 7422	. . . . . . 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 6991	. . . . . 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 7394	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
107	1, 76, 89, 77, 102, 91, 82	lmodvscld 20785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑏) ∈ 𝑈)
108		ovexd 7422	. . . . . . 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 6991	. . . . . 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 2774	. . . . 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 652	. . . 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 33328	. 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 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 +_gcplusg 17220 Scalarcsca 17223 ·_𝑠 cvsca 17224 “_s cimas 17467 Grpcgrp 18865 Abelcabl 19711 Ringcrg 20142 LModclmod 20766 LMHom clmhm 20926 Poly₁cpl1 22061 Unic_1pcuc1p 26032 rem_1pcr1p 26034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 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-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 17471 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-lmod 20768 df-lss 20838 df-lmhm 20929 df-cnfld 21265 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-coe1 22067 df-mdeg 25960 df-deg1 25961 df-uc1p 26037 df-q1p 26038 df-r1p 26039

This theorem is referenced by: r1pquslmic 33576 algextdeglem8 33714

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