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Theorem dchrinvcl 26599

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))
dchrmulid2.t	⊢ · = (+_g‘𝐺)
dchrmulid2.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		dchrmulid2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
7		dchrmhm.b	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
8	2, 7	dchrrcl 26586	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10		fveq2 6842	. . . . 5 ⊢ (𝑘 = 𝑥 → (𝑋‘𝑘) = (𝑋‘𝑥))
11	10	oveq2d 7372	. . . 4 ⊢ (𝑘 = 𝑥 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑥)))
12		fveq2 6842	. . . . 5 ⊢ (𝑘 = 𝑦 → (𝑋‘𝑘) = (𝑋‘𝑦))
13	12	oveq2d 7372	. . . 4 ⊢ (𝑘 = 𝑦 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑦)))
14		fveq2 6842	. . . . 5 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑘) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
15	14	oveq2d 7372	. . . 4 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))))
16		fveq2 6842	. . . . 5 ⊢ (𝑘 = (1_r‘𝑍) → (𝑋‘𝑘) = (𝑋‘(1_r‘𝑍)))
17	16	oveq2d 7372	. . . 4 ⊢ (𝑘 = (1_r‘𝑍) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(1_r‘𝑍))))
18	2, 3, 7, 4, 6	dchrf 26588	. . . . . 6 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
19	4, 5	unitss 20087	. . . . . . 7 ⊢ 𝑈 ⊆ 𝐵
20	19	sseli 3940	. . . . . 6 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝐵)
21		ffvelcdm 7032	. . . . . 6 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
22	18, 20, 21	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
23		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
24	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ 𝐷)
25	20	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐵)
26	2, 3, 7, 4, 5, 24, 25	dchrn0 26596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
27	23, 26	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
28	22, 27	reccld 11923	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (1 / (𝑋‘𝑘)) ∈ ℂ)
29		1t1e1 12314	. . . . . . . 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 26587	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
33	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ 𝐷)
34	32, 33	sselid 3942	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
35		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
36	19, 35	sselid 3942	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝐵)
37		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
38	19, 37	sselid 3942	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
39		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
40	39, 4	mgpbas 19900	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
41		eqid 2736	. . . . . . . . 9 ⊢ (._r‘𝑍) = (._r‘𝑍)
42	39, 41	mgpplusg 19898	. . . . . . . 8 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
43		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
44		cnfldmul 20800	. . . . . . . . 9 ⊢ · = (._r‘ℂ_fld)
45	43, 44	mgpplusg 19898	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
46	40, 42, 45	mhmlin 18608	. . . . . . 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 1371	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
48	31, 47	oveq12d 7374	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
49		1cnd 11149	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℂ)
50	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋:𝐵⟶ℂ)
51	50, 36	ffvelcdmd 7035	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ∈ ℂ)
52	50, 38	ffvelcdmd 7035	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ∈ ℂ)
53	2, 3, 7, 4, 5, 33, 36	dchrn0 26596	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
54	35, 53	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ≠ 0)
55	2, 3, 7, 4, 5, 33, 38	dchrn0 26596	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑋‘𝑦) ≠ 0 ↔ 𝑦 ∈ 𝑈))
56	37, 55	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ≠ 0)
57	49, 51, 49, 52, 54, 56	divmuldivd 11971	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
58	48, 57	eqtr4d 2779	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))))
59	32, 6	sselid 3942	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
60		eqid 2736	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
61	39, 60	ringidval 19913	. . . . . . . 8 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
62		cnfld1 20820	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
63	43, 62	ringidval 19913	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
64	61, 63	mhm0 18609	. . . . . . 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 7372	. . . . 5 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = (1 / 1))
67		1div1e1 11844	. . . . 5 ⊢ (1 / 1) = 1
68	66, 67	eqtrdi 2792	. . . 4 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = 1)
69	2, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 58, 68	dchrelbasd 26585	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ∈ 𝐷)
70	1, 69	eqeltrid 2842	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
71		dchrmulid2.t	. . . 4 ⊢ · = (+_g‘𝐺)
72	2, 3, 7, 71, 70, 6	dchrmul 26594	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) = (𝐾 ∘_f · 𝑋))
73	4	fvexi 6856	. . . . . 6 ⊢ 𝐵 ∈ V
74	73	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
75		ovex 7389	. . . . . . 7 ⊢ (1 / (𝑋‘𝑘)) ∈ V
76		c0ex 11148	. . . . . . 7 ⊢ 0 ∈ V
77	75, 76	ifex 4536	. . . . . 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 7034	. . . . 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 6910	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
82	74, 78, 79, 80, 81	offval2 7636	. . . 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 7453	. . . . . . 7 ⊢ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘)))
85	79	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
86	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐷)
87		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
88	2, 3, 7, 4, 5, 86, 87	dchrn0 26596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
89	88	biimpar 478	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
90	85, 89	recid2d 11926	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)) = 1)
91	90	ifeq1da 4517	. . . . . . . 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 11352	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 · (𝑋‘𝑘)) = 0)
93	92	ifeq2d 4506	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
94	91, 93	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
95	84, 94	eqtrid 2788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, 1, 0))
96	95	mpteq2dva 5205	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)))
97	83, 96	eqtr4id 2795	. . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))))
98	82, 97	eqtr4d 2779	. . 3 ⊢ (𝜑 → (𝐾 ∘_f · 𝑋) = 1 )
99	72, 98	eqtrd 2776	. 2 ⊢ (𝜑 → (𝐾 · 𝑋) = 1 )
100	70, 99	jca 512	1 ⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ifcif 4486 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 ∘_f cof 7614 ℂcc 11048 0cc0 11050 1c1 11051 · cmul 11055 / cdiv 11811 ℕcn 12152 Basecbs 17082 +_gcplusg 17132 ._rcmulr 17133 MndHom cmhm 18598 mulGrpcmgp 19894 1_rcur 19911 Unitcui 20066 ℂ_fldccnfld 20794 ℤ/nℤczn 20901 DChrcdchr 26578

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-addf 11129 ax-mulf 11130

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-ec 8649 df-qs 8653 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-inf 9378 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-0g 17322 df-imas 17389 df-qus 17390 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-grp 18750 df-minusg 18751 df-sbg 18752 df-subg 18923 df-nsg 18924 df-eqg 18925 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-cring 19965 df-oppr 20047 df-dvdsr 20068 df-unit 20069 df-invr 20099 df-subrg 20218 df-lmod 20322 df-lss 20391 df-lsp 20431 df-sra 20631 df-rgmod 20632 df-lidl 20633 df-rsp 20634 df-2idl 20700 df-cnfld 20795 df-zring 20868 df-zn 20905 df-dchr 26579

This theorem is referenced by: dchrabl 26600

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