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Theorem dchrinvcl 27595

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 27582	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
9	6, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10		fveq2 6896	. . . . 5 ⊢ (𝑘 = 𝑥 → (𝑋‘𝑘) = (𝑋‘𝑥))
11	10	oveq2d 7446	. . . 4 ⊢ (𝑘 = 𝑥 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑥)))
12		fveq2 6896	. . . . 5 ⊢ (𝑘 = 𝑦 → (𝑋‘𝑘) = (𝑋‘𝑦))
13	12	oveq2d 7446	. . . 4 ⊢ (𝑘 = 𝑦 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑦)))
14		fveq2 6896	. . . . 5 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑘) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
15	14	oveq2d 7446	. . . 4 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))))
16		fveq2 6896	. . . . 5 ⊢ (𝑘 = (1_r‘𝑍) → (𝑋‘𝑘) = (𝑋‘(1_r‘𝑍)))
17	16	oveq2d 7446	. . . 4 ⊢ (𝑘 = (1_r‘𝑍) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(1_r‘𝑍))))
18	2, 3, 7, 4, 6	dchrf 27584	. . . . . 6 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
19	4, 5	unitss 20621	. . . . . . 7 ⊢ 𝑈 ⊆ 𝐵
20	19	sseli 3909	. . . . . 6 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝐵)
21		ffvelcdm 7092	. . . . . 6 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
22	18, 20, 21	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
23		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
24	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ 𝐷)
25	20	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐵)
26	2, 3, 7, 4, 5, 24, 25	dchrn0 27592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
27	23, 26	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
28	22, 27	reccld 12099	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (1 / (𝑋‘𝑘)) ∈ ℂ)
29		1t1e1 12517	. . . . . . . 8 ⊢ (1 · 1) = 1
30	29	eqcomi 2772	. . . . . . 7 ⊢ 1 = (1 · 1)
31	30	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 = (1 · 1))
32	2, 3, 7	dchrmhm 27583	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
33	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ 𝐷)
34	32, 33	sselid 3911	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
35		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
36	19, 35	sselid 3911	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝐵)
37		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
38	19, 37	sselid 3911	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
39		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
40	39, 4	mgpbas 20380	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
41		eqid 2763	. . . . . . . . 9 ⊢ (._r‘𝑍) = (._r‘𝑍)
42	39, 41	mgpplusg 20379	. . . . . . . 8 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
43		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
44		cnfldmul 21701	. . . . . . . . 9 ⊢ · = (._r‘ℂ_fld)
45	43, 44	mgpplusg 20379	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
46	40, 42, 45	mhmlin 19003	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘(𝑥(._r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)))
48	31, 47	oveq12d 7448	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
49		1cnd 11315	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℂ)
50	18	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋:𝐵⟶ℂ)
51	50, 36	ffvelcdmd 7096	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ∈ ℂ)
52	50, 38	ffvelcdmd 7096	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ∈ ℂ)
53	2, 3, 7, 4, 5, 33, 36	dchrn0 27592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
54	35, 53	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ≠ 0)
55	2, 3, 7, 4, 5, 33, 38	dchrn0 27592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑋‘𝑦) ≠ 0 ↔ 𝑦 ∈ 𝑈))
56	37, 55	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ≠ 0)
57	49, 51, 49, 52, 54, 56	divmuldivd 12147	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
58	48, 57	eqtr4d 2801	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 / (𝑋‘𝑥)) · (1 / (𝑋‘𝑦))))
59	32, 6	sselid 3911	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
60		eqid 2763	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
61	39, 60	ringidval 20424	. . . . . . . 8 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
62		cnfld1 21718	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
63	43, 62	ringidval 20424	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
64	61, 63	mhm0 19004	. . . . . . 7 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑋‘(1_r‘𝑍)) = 1)
65	59, 64	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑋‘(1_r‘𝑍)) = 1)
66	65	oveq2d 7446	. . . . 5 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = (1 / 1))
67		1div1e1 12020	. . . . 5 ⊢ (1 / 1) = 1
68	66, 67	eqtrdi 2814	. . . 4 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = 1)
69	2, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 58, 68	dchrelbasd 27581	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ∈ 𝐷)
70	1, 69	eqeltrid 2868	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
71		dchrmullid.t	. . . 4 ⊢ · = (+_g‘𝐺)
72	2, 3, 7, 71, 70, 6	dchrmul 27590	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) = (𝐾 ∘_f · 𝑋))
73	4	fvexi 6910	. . . . . 6 ⊢ 𝐵 ∈ V
74	73	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
75		ovex 7463	. . . . . . 7 ⊢ (1 / (𝑋‘𝑘)) ∈ V
76		c0ex 11313	. . . . . . 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 7095	. . . . 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 6964	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
82	74, 78, 79, 80, 81	offval2 7723	. . . 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 7528	. . . . . . 7 ⊢ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘)))
85	79	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
86	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐷)
87		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
88	2, 3, 7, 4, 5, 86, 87	dchrn0 27592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
89	88	biimpar 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
90	85, 89	recid2d 12102	. . . . . . . . 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 11521	. . . . . . . . 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 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘))) = if(𝑘 ∈ 𝑈, 1, 0))
95	84, 94	eqtrid 2810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, 1, 0))
96	95	mpteq2dva 5201	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)))
97	83, 96	eqtr4id 2817	. . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘))))
98	82, 97	eqtr4d 2801	. . 3 ⊢ (𝜑 → (𝐾 ∘_f · 𝑋) = 1 )
99	72, 98	eqtrd 2798	. 2 ⊢ (𝜑 → (𝐾 · 𝑋) = 1 )
100	70, 99	jca 521	1 ⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 Vcvv 3454 ifcif 4485 ↦ cmpt 5189 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ∘_f cof 7701 ℂcc 11211 0cc0 11213 1c1 11214 · cmul 11218 / cdiv 11986 ℕcn 12348 Basecbs 17402 +_gcplusg 17443 ._rcmulr 17444 MndHom cmhm 18991 mulGrpcmgp 20375 1_rcur 20422 Unitcui 20600 ℂ_fldccnfld 21693 ℤ/nℤczn 21823 DChrcdchr 27574

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-addf 11292 ax-mulf 11293

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-tpos 8257 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-er 8731 df-ec 8733 df-qs 8737 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-sup 9447 df-inf 9448 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-fz 13653 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-0g 17627 df-imas 17695 df-qus 17696 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-mhm 18993 df-grp 19162 df-minusg 19163 df-sbg 19164 df-subg 19348 df-nsg 19349 df-eqg 19350 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-cring 20477 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-subrng 20813 df-subrg 20837 df-lmod 21152 df-lss 21222 df-lsp 21262 df-sra 21463 df-rgmod 21464 df-lidl 21501 df-rsp 21502 df-2idl 21558 df-cnfld 21694 df-zring 21768 df-zn 21827 df-dchr 27575

This theorem is used by: dchrabl 27596

W3C validator