Theorem rhmmpl 41663

Description: Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20403. TODO: Currently mhmvlin 22250 would have to be moved up. Investigate the usefulness of surrounding theorems like mndvcl 22244 and the difference between mhmvlin 22250, ofco 7689, and ofco2 22304. (Contributed by SN, 8-Feb-2025.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem rhmmpl

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

Step	Hyp	Ref	Expression
1		rhmmpl.b	. 2 ⊢ 𝐵 = (Base‘𝑃)
2		eqid 2726	. 2 ⊢ (1r‘𝑃) = (1r‘𝑃)
3		eqid 2726	. 2 ⊢ (1r‘𝑄) = (1r‘𝑄)
4		eqid 2726	. 2 ⊢ (.r‘𝑃) = (.r‘𝑃)
5		eqid 2726	. 2 ⊢ (.r‘𝑄) = (.r‘𝑄)
6		rhmmpl.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
7		rhmmpl.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8		rhmmpl.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
9		rhmrcl1 20376	. . . 4 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
11	6, 7, 10	mplringd 41655	. 2 ⊢ (𝜑 → 𝑃 ∈ Ring)
12		rhmmpl.q	. . 3 ⊢ 𝑄 = (𝐼 mPoly 𝑆)
13		rhmrcl2 20377	. . . 4 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
14	8, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ Ring)
15	12, 7, 14	mplringd 41655	. 2 ⊢ (𝜑 → 𝑄 ∈ Ring)
16		eqid 2726	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
17		eqid 2726	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
18		eqid 2726	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
19	6, 16, 17, 18, 2, 7, 10	mpl1 21909	. . . . 5 ⊢ (𝜑 → (1r‘𝑃) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))))
20	19	coeq2d 5855	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (1r‘𝑃)) = (𝐻 ∘ (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))))
21		eqid 2726	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2726	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
23	21, 22	rhmf 20385	. . . . . 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 20163	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
26	10, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
27	21, 17	ring0cl 20164	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
28	10, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
29	26, 28	ifcld 4569	. . . . . 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 7125	. . . 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 6900	. . . . . 6 ⊢ (𝐻‘if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) = if(𝑑 = (𝐼 × {0}), (𝐻‘(1r‘𝑅)), (𝐻‘(0g‘𝑅)))
33		eqid 2726	. . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆)
34	18, 33	rhm1 20389	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
35	8, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
36		rhmghm 20384	. . . . . . . 8 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
37		eqid 2726	. . . . . . . . 9 ⊢ (0g‘𝑆) = (0g‘𝑆)
38	17, 37	ghmid 19145	. . . . . . . 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 4544	. . . . . 6 ⊢ (𝜑 → if(𝑑 = (𝐼 × {0}), (𝐻‘(1r‘𝑅)), (𝐻‘(0g‘𝑅))) = if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆)))
41	32, 40	eqtrid 2778	. . . . 5 ⊢ (𝜑 → (𝐻‘if(𝑑 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) = if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆)))
42	41	mpteq2dv 5243	. . . 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 2770	. . 3 ⊢ (𝜑 → (𝐻 ∘ (1r‘𝑃)) = (𝑑 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑆), (0g‘𝑆))))
44		rhmmpl.f	. . . 4 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
45		coeq2 5851	. . . 4 ⊢ (𝑝 = (1r‘𝑃) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (1r‘𝑃)))
46	1, 2	ringidcl 20163	. . . . 5 ⊢ (𝑃 ∈ Ring → (1r‘𝑃) ∈ 𝐵)
47	11, 46	syl 17	. . . 4 ⊢ (𝜑 → (1r‘𝑃) ∈ 𝐵)
48	8, 47	coexd 41590	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (1r‘𝑃)) ∈ V)
49	44, 45, 47, 48	fvmptd3 7014	. . 3 ⊢ (𝜑 → (𝐹‘(1r‘𝑃)) = (𝐻 ∘ (1r‘𝑃)))
50	12, 16, 37, 33, 3, 7, 14	mpl1 21909	. . 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 2776	. 2 ⊢ (𝜑 → (𝐹‘(1r‘𝑃)) = (1r‘𝑄))
52		eqid 2726	. . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄)
53	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ 𝑉)
54	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐻 ∈ (𝑅 RingHom 𝑆))
55		simprl 768	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
56		simprr 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
57	6, 12, 1, 52, 4, 5, 53, 54, 55, 56	rhmcomulmpl 41662	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)) = ((𝐻 ∘ 𝑥)(.r‘𝑄)(𝐻 ∘ 𝑦)))
58		coeq2 5851	. . . 4 ⊢ (𝑝 = (𝑥(.r‘𝑃)𝑦) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)))
59	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Ring)
60	1, 4, 59, 55, 56	ringcld 20160	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑃)𝑦) ∈ 𝐵)
61		ovexd 7439	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑃)𝑦) ∈ V)
62	54, 61	coexd 41590	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)) ∈ V)
63	44, 58, 60, 62	fvmptd3 7014	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(.r‘𝑃)𝑦)) = (𝐻 ∘ (𝑥(.r‘𝑃)𝑦)))
64		coeq2 5851	. . . . 5 ⊢ (𝑝 = 𝑥 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑥))
65	54, 55	coexd 41590	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ 𝑥) ∈ V)
66	44, 64, 55, 65	fvmptd3 7014	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐻 ∘ 𝑥))
67		coeq2 5851	. . . . 5 ⊢ (𝑝 = 𝑦 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑦))
68	54, 56	coexd 41590	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ 𝑦) ∈ V)
69	44, 67, 56, 68	fvmptd3 7014	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) = (𝐻 ∘ 𝑦))
70	66, 69	oveq12d 7422	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)(.r‘𝑄)(𝐹‘𝑦)) = ((𝐻 ∘ 𝑥)(.r‘𝑄)(𝐻 ∘ 𝑦)))
71	57, 63, 70	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(.r‘𝑃)𝑦)) = ((𝐹‘𝑥)(.r‘𝑄)(𝐹‘𝑦)))
72		eqid 2726	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
73		eqid 2726	. 2 ⊢ (+g‘𝑄) = (+g‘𝑄)
74	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐼 ∈ 𝑉)
75		ghmmhm 19149	. . . . . 6 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → 𝐻 ∈ (𝑅 MndHom 𝑆))
76	8, 36, 75	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))
77	76	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐻 ∈ (𝑅 MndHom 𝑆))
78		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵)
79	6, 12, 1, 52, 74, 77, 78	mhmcompl 41658	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝐻 ∘ 𝑝) ∈ (Base‘𝑄))
80	79, 44	fmptd 7108	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑄))
81	76	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐻 ∈ (𝑅 MndHom 𝑆))
82	6, 12, 1, 52, 72, 73, 53, 81, 55, 56	mhmcoaddmpl 41661	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)) = ((𝐻 ∘ 𝑥)(+g‘𝑄)(𝐻 ∘ 𝑦)))
83		coeq2 5851	. . . 4 ⊢ (𝑝 = (𝑥(+g‘𝑃)𝑦) → (𝐻 ∘ 𝑝) = (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)))
84	11	ringgrpd 20145	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Grp)
85	84	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ Grp)
86	1, 72, 85, 55, 56	grpcld 18875	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑃)𝑦) ∈ 𝐵)
87		ovexd 7439	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑃)𝑦) ∈ V)
88	54, 87	coexd 41590	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)) ∈ V)
89	44, 83, 86, 88	fvmptd3 7014	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(+g‘𝑃)𝑦)) = (𝐻 ∘ (𝑥(+g‘𝑃)𝑦)))
90	66, 69	oveq12d 7422	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)(+g‘𝑄)(𝐹‘𝑦)) = ((𝐻 ∘ 𝑥)(+g‘𝑄)(𝐻 ∘ 𝑦)))
91	82, 89, 90	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥(+g‘𝑃)𝑦)) = ((𝐹‘𝑥)(+g‘𝑄)(𝐹‘𝑦)))
92	1, 2, 3, 4, 5, 11, 15, 51, 71, 52, 72, 73, 80, 91	isrhmd 20388	1 ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468  ifcif 4523  {csn 4623   ↦ cmpt 5224   × cxp 5667  ^◡ccnv 5668   “ cima 5672   ∘ ccom 5673  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404   ↑_m cmap 8819  Fincfn 8938  0cc0 11109  ℕcn 12213  ℕ₀cn0 12473  Basecbs 17151  +_gcplusg 17204  ._rcmulr 17205  0_gc0g 17392   MndHom cmhm 18709  Grpcgrp 18861   GrpHom cghm 19136  1_rcur 20084  Ringcrg 20136   RingHom crh 20369   mPoly cmpl 21796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8144  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-pm 8822  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-sup 9436  df-oi 9504  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-fzo 13631  df-seq 13970  df-hash 14294  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-submnd 18712  df-grp 18864  df-minusg 18865  df-mulg 18994  df-subg 19048  df-ghm 19137  df-cntz 19231  df-cmn 19700  df-abl 19701  df-mgp 20038  df-rng 20056  df-ur 20085  df-ring 20138  df-rhm 20372  df-subrng 20444  df-subrg 20469  df-psr 21799  df-mpl 21801

This theorem is referenced by: (None)