r1plmhm - Mathbox for Thierry Arnoux

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Theorem r1plmhm 34187

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 26482	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝑈 ∧ 𝑀 ∈ 𝑁) → (𝑓𝐸𝑀) ∈ 𝑈)
11	3, 4, 6, 10	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑈) → (𝑓𝐸𝑀) ∈ 𝑈)
12		r1plmhm.6	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))
13	11, 12	fmptd 7114	. . 3 ⊢ (𝜑 → 𝐹:𝑈⟶𝑈)
14		eqid 2760	. . 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 7427	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝑓𝐸𝑀) = (𝑎𝐸𝑀))
21		ovexd 7455	. . . . . . . . . . . . 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 7017	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝑎𝐸𝑀))
23		oveq1 7427	. . . . . . . . . . . . 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 7455	. . . . . . . . . . . . 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 7017	. . . . . . . . . . . 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 2803	. . . . . . . . . . 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 34186	. . . . . . . . . 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 7427	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑎(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑎(+_g‘𝑃)𝑏)𝐸𝑀))
31	8	ply1ring 22570	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
32	2, 31	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ Ring)
33	32	ringgrpd 20474	. . . . . . . . . . . . 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 19163	. . . . . . . . . . 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 7455	. . . . . . . . . . 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 7017	. . . . . . . . . 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 7427	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑝(+_g‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑝(+_g‘𝑃)𝑏)𝐸𝑀))
39	1, 14, 34, 24, 28	grpcld 19163	. . . . . . . . . . 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 7455	. . . . . . . . . . 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 7017	. . . . . . . . . 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 2805	. . . . . . . . 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 20517	. . . . . . . . . . . 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 20018	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Abel ∧ 𝑝 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) → (𝑝(+_g‘𝑃)𝑏) = (𝑏(+_g‘𝑃)𝑝))
46	44, 24, 28, 45	syl3anc 1398	. . . . . . . . . 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 6889	. . . . . . . . 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 2795	. . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑏 → (𝑓𝐸𝑀) = (𝑏𝐸𝑀))
51		ovexd 7455	. . . . . . . . . . . 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 7017	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑎 ∈ 𝑈) ∧ 𝑏 ∈ 𝑈) ∧ 𝑝 ∈ 𝑈) ∧ 𝑞 ∈ 𝑈) ∧ (𝐹‘𝑎) = (𝐹‘𝑝)) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
53		oveq1 7427	. . . . . . . . . . . 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 7455	. . . . . . . . . . . 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 7017	. . . . . . . . . . 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 2803	. . . . . . . . . 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 34186	. . . . . . . . 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 7427	. . . . . . . . . 10 ⊢ (𝑓 = (𝑏(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑏(+_g‘𝑃)𝑝)𝐸𝑀))
60	1, 14, 34, 28, 24	grpcld 19163	. . . . . . . . . 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 7455	. . . . . . . . . 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 7017	. . . . . . . . 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 7427	. . . . . . . . . 10 ⊢ (𝑓 = (𝑞(+_g‘𝑃)𝑝) → (𝑓𝐸𝑀) = ((𝑞(+_g‘𝑃)𝑝)𝐸𝑀))
64	1, 14, 34, 54, 24	grpcld 19163	. . . . . . . . . 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 7455	. . . . . . . . . 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 7017	. . . . . . . . 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 2805	. . . . . . . 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 20018	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Abel ∧ 𝑞 ∈ 𝑈 ∧ 𝑝 ∈ 𝑈) → (𝑞(+_g‘𝑃)𝑝) = (𝑝(+_g‘𝑃)𝑞))
69	44, 54, 24, 68	syl3anc 1398	. . . . . . . . 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 6889	. . . . . . . 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 2799	. . . . . . 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 2760	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
77		eqid 2760	. . 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 7455	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) ∈ V)
81	12, 20, 79, 80	fvmptd3 7017	. . . . . . . . 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 7455	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑏𝐸𝑀) ∈ V)
84	12, 50, 82, 83	fvmptd3 7017	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝐹‘𝑏) = (𝑏𝐸𝑀))
85	78, 81, 84	3eqtr3d 2803	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑎𝐸𝑀) = (𝑏𝐸𝑀))
86	85	oveq2d 7436	. . . . . . 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 2760	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
90		eqid 2760	. . . . . . . 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 22575	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
93	2, 92	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
94	93	fveq2d 6889	. . . . . . . . . 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 2863	. . . . . . . 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 34183	. . . . . . 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 34183	. . . . . . 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 2805	. . . . . 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 7427	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑎) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑎)𝐸𝑀))
101	8	ply1lmod 22574	. . . . . . . . 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 21159	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑎) ∈ 𝑈)
104		ovexd 7455	. . . . . . 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 7017	. . . . . 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 7427	. . . . . . 7 ⊢ (𝑓 = (𝑘( ·_𝑠 ‘𝑃)𝑏) → (𝑓𝐸𝑀) = ((𝑘( ·_𝑠 ‘𝑃)𝑏)𝐸𝑀))
107	1, 76, 89, 77, 102, 91, 82	lmodvscld 21159	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑘( ·_𝑠 ‘𝑃)𝑏) ∈ 𝑈)
108		ovexd 7455	. . . . . . 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 7017	. . . . . 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 2805	. . . . 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 33964	. 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 1569 ∈ wcel 2144 Vcvv 3450 ↦ cmpt 5185 ‘cfv 6538 (class class class)co 7420 Basecbs 17392 +_gcplusg 17433 Scalarcsca 17436 ·_𝑠 cvsca 17437 “_s cimas 17681 Grpcgrp 19149 Abelcabl 20000 Ringcrg 20464 LModclmod 21140 LMHom clmhm 21299 Poly₁cpl1 22500 Unic_1pcuc1p 26450 rem_1pcr1p 26452

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 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

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 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 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 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-tpos 8246 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-sup 9437 df-inf 9438 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-fz 13643 df-fzo 13792 df-seq 14148 df-hash 14478 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-0g 17617 df-gsum 17618 df-prds 17623 df-pws 17625 df-imas 17685 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-mhm 18983 df-submnd 18984 df-grp 19152 df-minusg 19153 df-sbg 19154 df-mulg 19283 df-subg 19338 df-ghm 19433 df-cntz 19536 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-ring 20466 df-cring 20467 df-oppr 20572 df-dvdsr 20592 df-unit 20593 df-invr 20623 df-subrng 20803 df-subrg 20827 df-rlreg 20951 df-lmod 21142 df-lss 21212 df-lmhm 21302 df-cnfld 21684 df-psr 22222 df-mvr 22223 df-mpl 22224 df-opsr 22226 df-psr1 22503 df-vr1 22504 df-ply1 22505 df-coe1 22506 df-mdeg 26378 df-deg1 26379 df-uc1p 26455 df-q1p 26456 df-r1p 26457

This theorem is used by: r1pquslmic 34188 algextdeglem8 34402

W3C validator