r1plmhm - Mathbox for Thierry Arnoux

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Theorem r1plmhm 34211

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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑅 ∈ Ring)
4		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → 𝑓 ∈ 𝑈)
5		r1plmhm.10	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑁)
6	5	adantr 486	. . . . 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 26507	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝑈 ∧ 𝑀 ∈ 𝑁) → (𝑓𝐸𝑀) ∈ 𝑈)
11	3, 4, 6, 10	syl3anc 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → (𝑓𝐸𝑀) ∈ 𝑈)
12		r1plmhm.6	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))
13	11, 12	fmptd 7138	. . 3 ⊢ (𝜑 → 𝐹:𝑈⟶𝑈)
14		eqid 2763	. . 3 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
15		anass 474	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ↔ (𝜑 ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)))
16	2	ad6antr 749	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑅 ∈ Ring)
17		simp-6r 800	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑎 ∈ 𝑈)
18	5	ad6antr 749	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑀 ∈ 𝑁)
19		simplr 781	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝐹‘𝑝))
20		oveq1 7450	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝑓𝐸𝑀) = (𝑎𝐸𝑀))
21		ovexd 7478	. . . . . . . . . . . . 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 7040	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
23		oveq1 7450	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑝 → (𝑓𝐸𝑀) = (𝑝𝐸𝑀))
24		simp-4r 796	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑝 ∈ 𝑈)
25		ovexd 7478	. . . . . . . . . . . . 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 7040	. . . . . . . . . . . 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 2806	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝑎𝐸𝑀) = (𝑝𝐸𝑀))
28		simp-5r 798	. . . . . . . . . . 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 34210	. . . . . . . . . 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 7450	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑎(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀))
31	8	ply1ring 22595	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
32	2, 31	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Ring)
33	32	ringgrpd 20499	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ Grp)
34	33	ad6antr 749	. . . . . . . . . . . 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 19187	. . . . . . . . . . 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 7478	. . . . . . . . . . 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 7040	. . . . . . . . . 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 7450	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑝(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
39	1, 14, 34, 24, 28	grpcld 19187	. . . . . . . . . . 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 7478	. . . . . . . . . . 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 7040	. . . . . . . . . 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 2808	. . . . . . . . 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 20542	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Abel)
44	43	ad6antr 749	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑃 ∈ Abel)
45	1, 14	ablcom 20043	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Abel ∧ 𝑝 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) → (𝑝(+_g‘𝑃)𝑏) = (𝑏(+_g‘𝑃)𝑝))
46	44, 24, 28, 45	syl3anc 1399	. . . . . . . . . 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 6912	. . . . . . . . 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 2798	. . . . . . . 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 490	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝐹‘𝑞))
50		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (𝑓𝐸𝑀) = (𝑏𝐸𝑀))
51		ovexd 7478	. . . . . . . . . . . 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 7040	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
53		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑞 → (𝑓𝐸𝑀) = (𝑞𝐸𝑀))
54		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → 𝑞 ∈ 𝑈)
55		ovexd 7478	. . . . . . . . . . . 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 7040	. . . . . . . . . . 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 2806	. . . . . . . . . 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 34210	. . . . . . . . 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 7450	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀))
60	1, 14, 34, 28, 24	grpcld 19187	. . . . . . . . . 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 7478	. . . . . . . . . 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 7040	. . . . . . . . 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 7450	. . . . . . . . . 10 ⊢ (𝑓 = (𝑞(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
64	1, 14, 34, 54, 24	grpcld 19187	. . . . . . . . . 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 7478	. . . . . . . . . 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 7040	. . . . . . . . 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 2808	. . . . . . . 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 20043	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Abel ∧ 𝑞 ∈ 𝑈 ∧ 𝑝 ∈ 𝑈) → (𝑞(+_g‘𝑃)𝑝) = (𝑝(+_g‘𝑃)𝑞))
69	44, 54, 24, 68	syl3anc 1399	. . . . . . . . 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 6912	. . . . . . . 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 2802	. . . . . . 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 463	. . . . . 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 472	. . . . 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 594	. . . 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 1127	. . 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 2763	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
77		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
78		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝐹‘𝑏))
79		simpr2 1214	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑈)
80		ovexd 7478	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) ∈ V)
81	12, 20, 79, 80	fvmptd3 7040	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
82		simpr3 1215	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
83		ovexd 7478	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑏𝐸𝑀) ∈ V)
84	12, 50, 82, 83	fvmptd3 7040	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
85	78, 81, 84	3eqtr3d 2806	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) = (𝑏𝐸𝑀))
86	85	oveq2d 7459	. . . . . . 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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑅 ∈ Ring)
88	5	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑀 ∈ 𝑁)
89		eqid 2763	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
90		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
91		simpr1 1213	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
92	8	ply1sca 22600	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
93	2, 92	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
94	93	fveq2d 6912	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
95	94	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
96	91, 95	eleqtrrd 2867	. . . . . . . 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 34207	. . . . . . 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 34207	. . . . . . 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 2808	. . . . . 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 7450	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑎) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀))
101	8	ply1lmod 22599	. . . . . . . . 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 21184	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑎) ∈ 𝑈)
104		ovexd 7478	. . . . . . 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 7040	. . . . . 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 7450	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
107	1, 76, 89, 77, 102, 91, 82	lmodvscld 21184	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑏) ∈ 𝑈)
108		ovexd 7478	. . . . . . 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 7040	. . . . . 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 2808	. . . . 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 665	. . . 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 418	. . 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 33988	. 2 ⊢ (𝜑 → ((𝐹 “_s 𝑃) ∈ LMod ∧ 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃))))
115	114	simprd 501	1 ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “_s 𝑃)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ↦ cmpt 5189 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 +_gcplusg 17457 Scalarcsca 17460 ·_𝑠 cvsca 17461 “_s cimas 17705 Grpcgrp 19173 Abelcabl 20025 Ringcrg 20489 LModclmod 21165 LMHom clmhm 21324 Poly₁cpl1 22525 Unic_1pcuc1p 26475 rem_1pcr1p 26477

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 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

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 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-ofr 7718 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-fzo 13816 df-seq 14172 df-hash 14502 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-0g 17641 df-gsum 17642 df-prds 17647 df-pws 17649 df-imas 17709 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-ghm 19457 df-cntz 19561 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-subrng 20828 df-subrg 20852 df-rlreg 20976 df-lmod 21167 df-lss 21237 df-lmhm 21327 df-cnfld 21709 df-psr 22247 df-mvr 22248 df-mpl 22249 df-opsr 22251 df-psr1 22528 df-vr1 22529 df-ply1 22530 df-coe1 22531 df-mdeg 26403 df-deg1 26404 df-uc1p 26480 df-q1p 26481 df-r1p 26482

This theorem is used by: r1pquslmic 34212 algextdeglem8 34426

W3C validator