Theorem rhmpsr 42529

Description: Provide a ring homomorphism between two power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025.)

Hypotheses

Ref	Expression
rhmpsr.p	⊢ 𝑃 = (𝐼 mPwSer 𝑅)
rhmpsr.q	⊢ 𝑄 = (𝐼 mPwSer 𝑆)
rhmpsr.b	⊢ 𝐵 = (Base‘𝑃)
rhmpsr.f	⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
rhmpsr.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
rhmpsr.h	⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))

Assertion

Ref	Expression
rhmpsr	⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))

Distinct variable groups:   𝐵,𝑝   𝐻,𝑝   𝑃,𝑝   𝑄,𝑝   𝜑,𝑝

Allowed substitution hints:   𝑅(𝑝)   𝑆(𝑝)   𝐹(𝑝)   𝐼(𝑝)   𝑉(𝑝)

Proof of Theorem rhmpsr

Dummy variables 𝑥 𝑦 𝑑 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmpsr.b	. 2 ⊢ 𝐵 = (Base‘𝑃)
2		eqid 2729	. 2 ⊢ (1r‘𝑃) = (1r‘𝑃)
3		eqid 2729	. 2 ⊢ (1r‘𝑄) = (1r‘𝑄)
4		eqid 2729	. 2 ⊢ (.r‘𝑃) = (.r‘𝑃)
5		eqid 2729	. 2 ⊢ (.r‘𝑄) = (.r‘𝑄)
6		rhmpsr.p	. . 3 ⊢ 𝑃 = (𝐼 mPwSer 𝑅)
7		rhmpsr.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8		rhmpsr.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
9		rhmrcl1 20361	. . . 4 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
11	6, 7, 10	psrring 21877	. 2 ⊢ (𝜑 → 𝑃 ∈ Ring)
12		rhmpsr.q	. . 3 ⊢ 𝑄 = (𝐼 mPwSer 𝑆)
13		rhmrcl2 20362	. . . 4 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
14	8, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ Ring)
15	12, 7, 14	psrring 21877	. 2 ⊢ (𝜑 → 𝑄 ∈ Ring)
16		eqid 2729	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
17		eqid 2729	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
18		eqid 2729	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
19	6, 7, 10, 16, 17, 18, 2	psr1 21878	. . . . 5 ⊢ (𝜑 → (1r‘𝑃) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))))
20	19	coeq2d 5805	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (1r‘𝑃)) = (𝐻 ∘ (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))))
21		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
23	21, 22	rhmf 20370	. . . . . 6 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
24	8, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
25	21, 18	ringidcl 20150	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26	10, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
27	21, 17	ring0cl 20152	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	10, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
29	26, 28	ifcld 4523	. . . . . 6 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
30	29	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
31	24, 30	cofmpt 7066	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝐻‘if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))))
32		fvif 6838	. . . . . 6 ⊢ (𝐻‘if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) = if(𝑑 = (𝐼 × {0}), (𝐻‘(1r‘𝑅)), (𝐻‘(0g‘𝑅)))
33		eqid 2729	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
34	18, 33	rhm1 20374	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
35	8, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
36		rhmghm 20369	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
37		eqid 2729	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
38	17, 37	ghmid 19101	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → (𝐻‘(0g‘𝑅)) = (0g‘𝑆))
39	8, 36, 38	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(0g‘𝑅)) = (0g‘𝑆))
40	35, 39	ifeq12d 4498	. . . . . 6 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), (𝐻‘(1r‘𝑅)), (𝐻‘(0g‘𝑅))) = if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆)))
41	32, 40	eqtrid 2776	. . . . 5 ⊢ (𝜑 → (𝐻‘if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) = if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆)))
42	41	mpteq2dv 5186	. . . 4 ⊢ (𝜑 → (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ (𝐻‘if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆))))
43	20, 31, 42	3eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐻 ∘ (1r‘𝑃)) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆))))
44		rhmpsr.f	. . . 4 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
45		coeq2 5801	. . . 4 ⊢ (𝑝 = (1r‘𝑃) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (1r‘𝑃)))
46	1, 2	ringidcl 20150	. . . . 5 ⊢ (𝑃 ∈ Ring → (1r‘𝑃) ∈ 𝐵)
47	11, 46	syl 17	. . . 4 ⊢ (𝜑 → (1r‘𝑃) ∈ 𝐵)
48	8, 47	coexd 7864	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (1r‘𝑃)) ∈ V)
49	44, 45, 47, 48	fvmptd3 6953	. . 3 ⊢ (𝜑 → (𝐹‘(1r‘𝑃)) = (𝐻 ∘ (1r‘𝑃)))
50	12, 7, 14, 16, 37, 33, 3	psr1 21878	. . 3 ⊢ (𝜑 → (1r‘𝑄) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆))))
51	43, 49, 50	3eqtr4d 2774	. 2 ⊢ (𝜑 → (𝐹‘(1r‘𝑃)) = (1r‘𝑄))
52		eqid 2729	. . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄)
53	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐻 ∈ (𝑅 RingHom 𝑆))
54		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
55		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
56	6, 12, 1, 52, 4, 5, 53, 54, 55	rhmcomulpsr 42528	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)) = ((𝐻 ∘ 𝑥)(.r‘𝑄)(𝐻 ∘ 𝑦)))
57		coeq2 5801	. . . 4 ⊢ (𝑝 = (𝑥(.r‘𝑃)𝑦) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)))
58	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
59	1, 4, 58, 54, 55	ringcld 20145	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑃)𝑦) ∈ 𝐵)
60	53, 59	coexd 7864	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)) ∈ V)
61	44, 57, 59, 60	fvmptd3 6953	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(.r‘𝑃)𝑦)) = (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)))
62		coeq2 5801	. . . . 5 ⊢ (𝑝 = 𝑥 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑥))
63	53, 54	coexd 7864	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ 𝑥) ∈ V)
64	44, 62, 54, 63	fvmptd3 6953	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐻 ∘ 𝑥))
65		coeq2 5801	. . . . 5 ⊢ (𝑝 = 𝑦 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑦))
66	53, 55	coexd 7864	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ 𝑦) ∈ V)
67	44, 65, 55, 66	fvmptd3 6953	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) = (𝐻 ∘ 𝑦))
68	64, 67	oveq12d 7367	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)(.r‘𝑄)(𝐹‘𝑦)) = ((𝐻 ∘ 𝑥)(.r‘𝑄)(𝐻 ∘ 𝑦)))
69	56, 61, 68	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(.r‘𝑃)𝑦)) = ((𝐹‘𝑥)(.r‘𝑄)(𝐹‘𝑦)))
70		eqid 2729	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
71		eqid 2729	. 2 ⊢ (+g‘𝑄) = (+g‘𝑄)
72		ghmmhm 19105	. . . . . 6 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → 𝐻 ∈ (𝑅 MndHom 𝑆))
73	8, 36, 72	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
74	73	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐻 ∈ (𝑅 MndHom 𝑆))
75		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
76	6, 12, 1, 52, 74, 75	mhmcopsr 42526	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐻 ∘ 𝑝) ∈ (Base‘𝑄))
77	76, 44	fmptd 7048	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑄))
78	53, 36, 72	3syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐻 ∈ (𝑅 MndHom 𝑆))
79	6, 12, 1, 52, 70, 71, 78, 54, 55	mhmcoaddpsr 42527	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)) = ((𝐻 ∘ 𝑥)(+g‘𝑄)(𝐻 ∘ 𝑦)))
80		coeq2 5801	. . . 4 ⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)))
81	58	ringgrpd 20127	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Grp)
82	1, 70, 81, 54, 55	grpcld 18826	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑃)𝑦) ∈ 𝐵)
83	53, 82	coexd 7864	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)) ∈ V)
84	44, 80, 82, 83	fvmptd3 6953	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(+g‘𝑃)𝑦)) = (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)))
85	64, 67	oveq12d 7367	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)(+g‘𝑄)(𝐹‘𝑦)) = ((𝐻 ∘ 𝑥)(+g‘𝑄)(𝐻 ∘ 𝑦)))
86	79, 84, 85	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(+g‘𝑃)𝑦)) = ((𝐹‘𝑥)(+g‘𝑄)(𝐹‘𝑦)))
87	1, 2, 3, 4, 5, 11, 15, 51, 69, 52, 70, 71, 77, 86	isrhmd 20373	1 ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3394  Vcvv 3436  ifcif 4476  {csn 4577   ↦ cmpt 5173   × cxp 5617  ^◡ccnv 5618   “ cima 5622   ∘ ccom 5623  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  Fincfn 8872  0cc0 11009  ℕcn 12128  ℕ₀cn0 12384  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  0_gc0g 17343   MndHom cmhm 18655   GrpHom cghm 19091  1_rcur 20066  Ringcrg 20118   RingHom crh 20354   mPwSer cmps 21811

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-mulg 18947  df-ghm 19092  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-rhm 20357  df-psr 21816

This theorem is referenced by:  rhmpsr1  42530