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Theorem dchrinvcl 27216

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 27203	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10		fveq2 6840	. . . . 5 ⊢ (𝑘 = 𝑥 → (𝑋‘𝑘) = (𝑋‘𝑥))
11	10	oveq2d 7383	. . . 4 ⊢ (𝑘 = 𝑥 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑥)))
12		fveq2 6840	. . . . 5 ⊢ (𝑘 = 𝑦 → (𝑋‘𝑘) = (𝑋‘𝑦))
13	12	oveq2d 7383	. . . 4 ⊢ (𝑘 = 𝑦 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑦)))
14		fveq2 6840	. . . . 5 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑘) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
15	14	oveq2d 7383	. . . 4 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))))
16		fveq2 6840	. . . . 5 ⊢ (𝑘 = (1_r‘𝑍) → (𝑋‘𝑘) = (𝑋‘(1_r‘𝑍)))
17	16	oveq2d 7383	. . . 4 ⊢ (𝑘 = (1_r‘𝑍) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(1_r‘𝑍))))
18	2, 3, 7, 4, 6	dchrf 27205	. . . . . 6 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
19	4, 5	unitss 20356	. . . . . . 7 ⊢ 𝑈 ⊆ 𝐵
20	19	sseli 3917	. . . . . 6 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝐵)
21		ffvelcdm 7033	. . . . . 6 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
22	18, 20, 21	syl2an 597	. . . . 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 27213	. . . . . 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 11924	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (1 / (𝑋‘𝑘)) ∈ ℂ)
29		1t1e1 12338	. . . . . . . 8 ⊢ (1 · 1) = 1
30	29	eqcomi 2745	. . . . . . 7 ⊢ 1 = (1 · 1)
31	30	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 = (1 · 1))
32	2, 3, 7	dchrmhm 27204	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
33	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ 𝐷)
34	32, 33	sselid 3919	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
35		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
36	19, 35	sselid 3919	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝐵)
37		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
38	19, 37	sselid 3919	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
39		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
40	39, 4	mgpbas 20126	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
41		eqid 2736	. . . . . . . . 9 ⊢ (._r‘𝑍) = (._r‘𝑍)
42	39, 41	mgpplusg 20125	. . . . . . . 8 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
43		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
44		cnfldmul 21360	. . . . . . . . 9 ⊢ · = (._r‘ℂ_fld)
45	43, 44	mgpplusg 20125	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
46	40, 42, 45	mhmlin 18761	. . . . . . 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 1374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
48	31, 47	oveq12d 7385	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
49		1cnd 11139	. . . . . 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 7037	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ∈ ℂ)
52	50, 38	ffvelcdmd 7037	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ∈ ℂ)
53	2, 3, 7, 4, 5, 33, 36	dchrn0 27213	. . . . . . 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 27213	. . . . . . 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 11972	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
58	48, 57	eqtr4d 2774	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))))
59	32, 6	sselid 3919	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
60		eqid 2736	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
61	39, 60	ringidval 20164	. . . . . . . 8 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
62		cnfld1 21377	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
63	43, 62	ringidval 20164	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
64	61, 63	mhm0 18762	. . . . . . 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 7383	. . . . 5 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = (1 / 1))
67		1div1e1 11845	. . . . 5 ⊢ (1 / 1) = 1
68	66, 67	eqtrdi 2787	. . . 4 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = 1)
69	2, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 58, 68	dchrelbasd 27202	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ∈ 𝐷)
70	1, 69	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
71		dchrmullid.t	. . . 4 ⊢ · = (+_g‘𝐺)
72	2, 3, 7, 71, 70, 6	dchrmul 27211	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) = (𝐾 ∘_f · 𝑋))
73	4	fvexi 6854	. . . . . 6 ⊢ 𝐵 ∈ V
74	73	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
75		ovex 7400	. . . . . . 7 ⊢ (1 / (𝑋‘𝑘)) ∈ V
76		c0ex 11138	. . . . . . 7 ⊢ 0 ∈ V
77	75, 76	ifex 4517	. . . . . 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 7036	. . . . 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 6908	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
82	74, 78, 79, 80, 81	offval2 7651	. . . 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 7465	. . . . . . 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 27213	. . . . . . . . . . 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 11927	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)) = 1)
91	90	ifeq1da 4498	. . . . . . . 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 11344	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 · (𝑋‘𝑘)) = 0)
93	92	ifeq2d 4487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
94	91, 93	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
95	84, 94	eqtrid 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, 1, 0))
96	95	mpteq2dva 5178	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)))
97	83, 96	eqtr4id 2790	. . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))))
98	82, 97	eqtr4d 2774	. . 3 ⊢ (𝜑 → (𝐾 ∘_f · 𝑋) = 1 )
99	72, 98	eqtrd 2771	. 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 1542 ∈ wcel 2114 ≠ wne 2932 Vcvv 3429 ifcif 4466 ↦ cmpt 5166 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 ℂcc 11036 0cc0 11038 1c1 11039 · cmul 11043 / cdiv 11807 ℕcn 12174 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 MndHom cmhm 18749 mulGrpcmgp 20121 1_rcur 20162 Unitcui 20335 ℂ_fldccnfld 21352 ℤ/nℤczn 21482 DChrcdchr 27195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-nsg 19100 df-eqg 19101 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-subrng 20523 df-subrg 20547 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zn 21486 df-dchr 27196

This theorem is referenced by: dchrabl 27217

W3C validator