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Theorem dchrinvcl 27426

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 27413	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
9	6, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
10		fveq2 6881	. . . . 5 ⊢ (𝑘 = 𝑥 → (𝑋‘𝑘) = (𝑋‘𝑥))
11	10	oveq2d 7426	. . . 4 ⊢ (𝑘 = 𝑥 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑥)))
12		fveq2 6881	. . . . 5 ⊢ (𝑘 = 𝑦 → (𝑋‘𝑘) = (𝑋‘𝑦))
13	12	oveq2d 7426	. . . 4 ⊢ (𝑘 = 𝑦 → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘𝑦)))
14		fveq2 6881	. . . . 5 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (𝑋‘𝑘) = (𝑋‘(𝑥(._r‘𝑍)𝑦)))
15	14	oveq2d 7426	. . . 4 ⊢ (𝑘 = (𝑥(._r‘𝑍)𝑦) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))))
16		fveq2 6881	. . . . 5 ⊢ (𝑘 = (1_r‘𝑍) → (𝑋‘𝑘) = (𝑋‘(1_r‘𝑍)))
17	16	oveq2d 7426	. . . 4 ⊢ (𝑘 = (1_r‘𝑍) → (1 / (𝑋‘𝑘)) = (1 / (𝑋‘(1_r‘𝑍))))
18	2, 3, 7, 4, 6	dchrf 27415	. . . . . 6 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ)
19	4, 5	unitss 20463	. . . . . . 7 ⊢ 𝑈 ⊆ 𝐵
20	19	sseli 3933	. . . . . 6 ⊢ (𝑘 ∈ 𝑈 → 𝑘 ∈ 𝐵)
21		ffvelcdm 7076	. . . . . 6 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ)
22	18, 20, 21	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
23		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝑈)
24	6	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ 𝐷)
25	20	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐵)
26	2, 3, 7, 4, 5, 24, 25	dchrn0 27423	. . . . . 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 11988	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (1 / (𝑋‘𝑘)) ∈ ℂ)
29		1t1e1 12406	. . . . . . . 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 27414	. . . . . . . 8 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
33	6	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ 𝐷)
34	32, 33	sselid 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
35		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈)
36	19, 35	sselid 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝐵)
37		simprr 784	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
38	19, 37	sselid 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝐵)
39		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
40	39, 4	mgpbas 20225	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
41		eqid 2763	. . . . . . . . 9 ⊢ (._r‘𝑍) = (._r‘𝑍)
42	39, 41	mgpplusg 20224	. . . . . . . 8 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
43		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
44		cnfldmul 21539	. . . . . . . . 9 ⊢ · = (._r‘ℂ_fld)
45	43, 44	mgpplusg 20224	. . . . . . . 8 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
46	40, 42, 45	mhmlin 18855	. . . . . . 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 7428	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (1 / (𝑋‘(𝑥(._r‘𝑍)𝑦))) = ((1 · 1) / ((𝑋‘𝑥) · (𝑋‘𝑦))))
49		1cnd 11206	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 ∈ ℂ)
50	18	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑋:𝐵⟶ℂ)
51	50, 36	ffvelcdmd 7080	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑥) ∈ ℂ)
52	50, 38	ffvelcdmd 7080	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑋‘𝑦) ∈ ℂ)
53	2, 3, 7, 4, 5, 33, 36	dchrn0 27423	. . . . . . 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 27423	. . . . . . 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 12036	. . . . 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 3935	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
60		eqid 2763	. . . . . . . . 9 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
61	39, 60	ringidval 20269	. . . . . . . 8 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
62		cnfld1 21556	. . . . . . . . 9 ⊢ 1 = (1_r‘ℂ_fld)
63	43, 62	ringidval 20269	. . . . . . . 8 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
64	61, 63	mhm0 18856	. . . . . . 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 7426	. . . . 5 ⊢ (𝜑 → (1 / (𝑋‘(1_r‘𝑍))) = (1 / 1))
67		1div1e1 11909	. . . . 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 27412	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ∈ 𝐷)
70	1, 69	eqeltrid 2867	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷)
71		dchrmullid.t	. . . 4 ⊢ · = (+_g‘𝐺)
72	2, 3, 7, 71, 70, 6	dchrmul 27421	. . 3 ⊢ (𝜑 → (𝐾 · 𝑋) = (𝐾 ∘_f · 𝑋))
73	4	fvexi 6895	. . . . . 6 ⊢ 𝐵 ∈ V
74	73	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
75		ovex 7443	. . . . . . 7 ⊢ (1 / (𝑋‘𝑘)) ∈ V
76		c0ex 11204	. . . . . . 7 ⊢ 0 ∈ V
77	75, 76	ifex 4538	. . . . . 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 7079	. . . . 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 6949	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘)))
82	74, 78, 79, 80, 81	offval2 7694	. . . 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 7508	. . . . . . 7 ⊢ (if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0) · (𝑋‘𝑘)) = if(𝑘 ∈ 𝑈, ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)), (0 · (𝑋‘𝑘)))
85	79	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ)
86	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑋 ∈ 𝐷)
87		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
88	2, 3, 7, 4, 5, 86, 87	dchrn0 27423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈))
89	88	biimpar 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ≠ 0)
90	85, 89	recid2d 11991	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → ((1 / (𝑋‘𝑘)) · (𝑋‘𝑘)) = 1)
91	90	ifeq1da 4519	. . . . . . . 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 11412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (0 · (𝑋‘𝑘)) = 0)
93	92	ifeq2d 4508	. . . . . . . 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 5204	. . . . 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 520	1 ⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ifcif 4487 ↦ cmpt 5192 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∘_f cof 7672 ℂcc 11102 0cc0 11104 1c1 11105 · cmul 11109 / cdiv 11875 ℕcn 12237 Basecbs 17273 +_gcplusg 17314 ._rcmulr 17315 MndHom cmhm 18843 mulGrpcmgp 20220 1_rcur 20267 Unitcui 20442 ℂ_fldccnfld 21531 ℤ/nℤczn 21661 DChrcdchr 27405

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-addf 11183 ax-mulf 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-ec 8692 df-qs 8696 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-fz 13540 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-0g 17498 df-imas 17566 df-qus 17567 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-mhm 18845 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-nsg 19194 df-eqg 19195 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-cring 20322 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-subrng 20654 df-subrg 20678 df-lmod 20992 df-lss 21062 df-lsp 21102 df-sra 21303 df-rgmod 21304 df-lidl 21341 df-rsp 21342 df-2idl 21398 df-cnfld 21532 df-zring 21606 df-zn 21665 df-dchr 27406

This theorem is used by: dchrabl 27427

W3C validator