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Theorem dchrinvcl 27252

Description: Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrmhm.g	⊢ 𝐺 = (DChr‘𝑁)
dchrmhm.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrmhm.b	⊢ 𝐷 = (Base‘𝐺)
dchrn0.b	⊢ 𝐵 = (Base‘𝑍)
dchrn0.u	⊢ 𝑈 = (Unit‘𝑍)
dchr1cl.o	⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))
dchrmullid.t	⊢ · = (+_g‘𝐺)
dchrmullid.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrinvcl.n	⊢ 𝐾 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0))

**Assertion**
Ref	Expression
dchrinvcl	⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Distinct variable groups: 𝐵,𝑘 𝑈,𝑘 𝑘,𝑁 𝜑,𝑘 𝑘,𝑋 𝑘,𝑍 Allowed substitution hints: 𝐷(𝑘) · (𝑘) 1 (𝑘) 𝐺(𝑘) 𝐾(𝑘)

Proof of Theorem dchrinvcl Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrinvcl.n	. . 3 ⊢ 𝐾 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0))
2		dchrmhm.g	. . . 4 ⊢ 𝐺 = (DChr‘𝑁)
3		dchrmhm.z	. . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
4		dchrn0.b	. . . 4 ⊢ 𝐵 = (Base‘𝑍)
5		dchrn0.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑍)
6		dchrmullid.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
7		dchrmhm.b	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
8	2, 7	dchrrcl 27239	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10		fveq2 6887	. . . . 5 ⊢ (𝑘 = 𝑥 → (𝑋‘𝑘) = (𝑋‘𝑥))
11	10	oveq2d 7430	. . . 4 ⊢ (𝑘 = 𝑥 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑥)))
12		fveq2 6887	. . . . 5 ⊢ (𝑘 = 𝑦 → (𝑋‘𝑘) = (𝑋‘𝑦))
13	12	oveq2d 7430	. . . 4 ⊢ (𝑘 = 𝑦 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑦)))
14		fveq2 6887	. . . . 5 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑘) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
15	14	oveq2d 7430	. . . 4 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))))
16		fveq2 6887	. . . . 5 ⊢ (𝑘 = (1_r‘𝑍) → (𝑋‘𝑘) = (𝑋‘(1_r‘𝑍)))
17	16	oveq2d 7430	. . . 4 ⊢ (𝑘 = (1_r‘𝑍) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(1_r‘𝑍))))
18	2, 3, 7, 4, 6	dchrf 27241	. . . . . 6 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
19	4, 5	unitss 20349	. . . . . . 7 ⊢ 𝑈 ⊆ 𝐵
20	19	sseli 3961	. . . . . 6 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝐵)
21		ffvelcdm 7082	. . . . . 6 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
22	18, 20, 21	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
23		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
24	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ 𝐷)
25	20	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐵)
26	2, 3, 7, 4, 5, 24, 25	dchrn0 27249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
27	23, 26	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
28	22, 27	reccld 12019	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (1 / (𝑋‘𝑘)) ∈ ℂ)
29		1t1e1 12411	. . . . . . . 8 ⊢ (1 · 1) = 1
30	29	eqcomi 2743	. . . . . . 7 ⊢ 1 = (1 · 1)
31	30	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 = (1 · 1))
32	2, 3, 7	dchrmhm 27240	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
33	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ 𝐷)
34	32, 33	sselid 3963	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
35		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
36	19, 35	sselid 3963	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝐵)
37		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
38	19, 37	sselid 3963	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
39		eqid 2734	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
40	39, 4	mgpbas 20115	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
41		eqid 2734	. . . . . . . . 9 ⊢ (._r‘𝑍) = (._r‘𝑍)
42	39, 41	mgpplusg 20114	. . . . . . . 8 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
43		eqid 2734	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
44		cnfldmul 21339	. . . . . . . . 9 ⊢ · = (._r‘ℂ_fld)
45	43, 44	mgpplusg 20114	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
46	40, 42, 45	mhmlin 18780	. . . . . . 7 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
47	34, 36, 38, 46	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
48	31, 47	oveq12d 7432	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
49		1cnd 11239	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℂ)
50	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋:𝐵⟶ℂ)
51	50, 36	ffvelcdmd 7086	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ∈ ℂ)
52	50, 38	ffvelcdmd 7086	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ∈ ℂ)
53	2, 3, 7, 4, 5, 33, 36	dchrn0 27249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
54	35, 53	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ≠ 0)
55	2, 3, 7, 4, 5, 33, 38	dchrn0 27249	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑋‘𝑦) ≠ 0 ↔ 𝑦 ∈ 𝑈))
56	37, 55	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ≠ 0)
57	49, 51, 49, 52, 54, 56	divmuldivd 12067	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
58	48, 57	eqtr4d 2772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))))
59	32, 6	sselid 3963	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
60		eqid 2734	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
61	39, 60	ringidval 20153	. . . . . . . 8 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
62		cnfld1 21373	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
63	43, 62	ringidval 20153	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
64	61, 63	mhm0 18781	. . . . . . 7 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑋‘(1_r‘𝑍)) = 1)
65	59, 64	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
66	65	oveq2d 7430	. . . . 5 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = (1 / 1))
67		1div1e1 11941	. . . . 5 ⊢ (1 / 1) = 1
68	66, 67	eqtrdi 2785	. . . 4 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = 1)
69	2, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 58, 68	dchrelbasd 27238	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ∈ 𝐷)
70	1, 69	eqeltrid 2837	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
71		dchrmullid.t	. . . 4 ⊢ · = (+_g‘𝐺)
72	2, 3, 7, 71, 70, 6	dchrmul 27247	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) = (𝐾 ∘_f · 𝑋))
73	4	fvexi 6901	. . . . . 6 ⊢ 𝐵 ∈ V
74	73	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
75		ovex 7447	. . . . . . 7 ⊢ (1 / (𝑋‘𝑘)) ∈ V
76		c0ex 11238	. . . . . . 7 ⊢ 0 ∈ V
77	75, 76	ifex 4558	. . . . . 6 ⊢ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) ∈ V
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) ∈ V)
79	18	ffvelcdmda 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
80	1	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐾 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)))
81	18	feqmptd 6958	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
82	74, 78, 79, 80, 81	offval2 7700	. . . 4 ⊢ (𝜑 → (𝐾 ∘_f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))))
83		dchr1cl.o	. . . . 5 ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))
84		ovif 7514	. . . . . . 7 ⊢ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘)))
85	79	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
86	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐷)
87		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
88	2, 3, 7, 4, 5, 86, 87	dchrn0 27249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
89	88	biimpar 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
90	85, 89	recid2d 12022	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)) = 1)
91	90	ifeq1da 4539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, (0 · (𝑋‘𝑘))))
92	79	mul02d 11442	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 · (𝑋‘𝑘)) = 0)
93	92	ifeq2d 4528	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
94	91, 93	eqtrd 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
95	84, 94	eqtrid 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, 1, 0))
96	95	mpteq2dva 5224	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)))
97	83, 96	eqtr4id 2788	. . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))))
98	82, 97	eqtr4d 2772	. . 3 ⊢ (𝜑 → (𝐾 ∘_f · 𝑋) = 1 )
99	72, 98	eqtrd 2769	. 2 ⊢ (𝜑 → (𝐾 · 𝑋) = 1 )
100	70, 99	jca 511	1 ⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 Vcvv 3464 ifcif 4507 ↦ cmpt 5207 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∘_f cof 7678 ℂcc 11136 0cc0 11138 1c1 11139 · cmul 11143 / cdiv 11903 ℕcn 12249 Basecbs 17230 +_gcplusg 17277 ._rcmulr 17278 MndHom cmhm 18768 mulGrpcmgp 20110 1_rcur 20151 Unitcui 20328 ℂ_fldccnfld 21331 ℤ/nℤczn 21480 DChrcdchr 27231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-tp 4613 df-op 4615 df-uni 4890 df-int 4929 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7871 df-1st 7997 df-2nd 7998 df-tpos 8234 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-er 8728 df-ec 8730 df-qs 8734 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-div 11904 df-nn 12250 df-2 12312 df-3 12313 df-4 12314 df-5 12315 df-6 12316 df-7 12317 df-8 12318 df-9 12319 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12862 df-fz 13531 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17257 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-0g 17462 df-imas 17529 df-qus 17530 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-mhm 18770 df-grp 18928 df-minusg 18929 df-sbg 18930 df-subg 19115 df-nsg 19116 df-eqg 19117 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20307 df-dvdsr 20330 df-unit 20331 df-invr 20361 df-subrng 20519 df-subrg 20543 df-lmod 20833 df-lss 20903 df-lsp 20943 df-sra 21145 df-rgmod 21146 df-lidl 21185 df-rsp 21186 df-2idl 21227 df-cnfld 21332 df-zring 21425 df-zn 21484 df-dchr 27232

This theorem is referenced by: dchrabl 27253

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