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Theorem cxplim 27090

Description: A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)

**Assertion**
Ref	Expression
cxplim	⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Distinct variable group: 𝐴,𝑛

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

Step	Hyp	Ref	Expression
1		rpre 13013	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2	1	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
3		rpge0 13018	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
4	3	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
5		rpre 13013	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
6	5	renegcld 11629	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ∈ ℝ)
7	6	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ∈ ℝ)
8		rpcn 13015	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
9		rpne0 13021	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
10	8, 9	negne0d 11555	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ≠ 0)
11	10	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ≠ 0)
12	7, 11	rereccld 12030	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 / -𝐴) ∈ ℝ)
13	2, 4, 12	recxpcld 26842	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ)
14		simprl 782	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℝ⁺)
15	5	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℝ)
16	14, 15	rpcxpcld 26852	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
17	16	rpreccld 13058	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
18	17	rprege0d 13055	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))))
19		absid 15335	. . . . . . . 8 ⊢ (((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
20	18, 19	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
21		simplr 780	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℝ⁺)
22		simprr 784	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)
23		rpreccl 13032	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
24	23	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℝ⁺)
25	24	rpcnd 13050	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℂ)
26	21, 25	cxprecd 26851	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
27		rpcn 13015	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
28	27	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℂ)
29		rpne0 13021	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
30	29	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ≠ 0)
31	28, 30, 25	cxpnegd 26834	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐-(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
32		1cnd 11190	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 1 ∈ ℂ)
33	8	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℂ)
34	9	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ≠ 0)
35	32, 33, 34	divneg2d 11993	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → -(1 / 𝐴) = (1 / -𝐴))
36	35	oveq2d 7416	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐-(1 / 𝐴)) = (𝑥↑_𝑐(1 / -𝐴)))
37	26, 31, 36	3eqtr2d 2806	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (𝑥↑_𝑐(1 / -𝐴)))
38	33, 34	recidd 11974	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝐴 · (1 / 𝐴)) = 1)
39	38	oveq2d 7416	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = (𝑛↑_𝑐1))
40	14, 15, 25	cxpmuld 26856	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
41	14	rpcnd 13050	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℂ)
42	41	cxp1d 26825	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐1) = 𝑛)
43	39, 40, 42	3eqtr3d 2808	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)) = 𝑛)
44	22, 37, 43	3brtr4d 5136	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
45		rpreccl 13032	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
46	45	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ⁺)
47	46	rpred 13048	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ)
48	46	rpge0d 13052	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (1 / 𝑥))
49	16	rpred 13048	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ)
50	16	rpge0d 13052	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (𝑛↑_𝑐𝐴))
51	47, 48, 49, 50, 24	cxplt2d 26845	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥) < (𝑛↑_𝑐𝐴) ↔ ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴))))
52	44, 51	mpbird 260	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) < (𝑛↑_𝑐𝐴))
53	21, 16, 52	ltrec1d 13068	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) < 𝑥)
54	20, 53	eqbrtrd 5126	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)
55	54	expr 461	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ℝ⁺) → ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
56	55	ralrimiva 3157	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
57		breq1 5107	. . . . 5 ⊢ (𝑦 = (𝑥↑_𝑐(1 / -𝐴)) → (𝑦 < 𝑛 ↔ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛))
58	57	rspceaimv 3590	. . . 4 ⊢ (((𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ ∧ ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
59	13, 56, 58	syl2anc 595	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
60	59	ralrimiva 3157	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
61		id 23	. . . . . . 7 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ⁺)
62		rpcxpcl 26795	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
63	61, 5, 62	syl2anr 608	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
64	63	rpreccld 13058	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
65	64	rpcnd 13050	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
66	65	ralrimiva 3157	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
67		rpssre 13012	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
68	67	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
69	66, 68	rlim0lt 15548	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)))
70	60, 69	mpbird 260	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 class class class wbr 5104 ↦ cmpt 5185 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 · cmul 11093 < clt 11231 ≤ cle 11232 -cneg 11430 / cdiv 11859 ℝ⁺crp 13004 abscabs 15273 ⇝_𝑟 crli 15524 ↑_𝑐ccxp 26674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12493 df-z 12580 df-dec 12700 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13524 df-fzo 13671 df-fl 13813 df-mod 13891 df-seq 14026 df-exp 14086 df-fac 14298 df-bc 14327 df-hash 14355 df-shft 15092 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15510 df-clim 15527 df-rlim 15528 df-sum 15726 df-ef 16109 df-sin 16111 df-cos 16112 df-pi 16114 df-struct 17195 df-sets 17212 df-slot 17230 df-ndx 17242 df-base 17258 df-ress 17279 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17544 df-qtop 17549 df-imas 17550 df-xps 17552 df-mre 17626 df-mrc 17627 df-acs 17629 df-mgm 18686 df-sgrp 18765 df-mnd 18781 df-submnd 18830 df-mulg 19122 df-cntz 19375 df-cmn 19840 df-psmet 21471 df-xmet 21472 df-met 21473 df-bl 21474 df-mopn 21475 df-fbas 21476 df-fg 21477 df-cnfld 21480 df-top 23008 df-topon 23025 df-topsp 23047 df-bases 23060 df-cld 23133 df-ntr 23134 df-cls 23135 df-nei 23212 df-lp 23250 df-perf 23251 df-cn 23341 df-cnp 23342 df-haus 23429 df-tx 23676 df-hmeo 23869 df-fil 23960 df-fm 24052 df-flim 24053 df-flf 24054 df-xms 24434 df-ms 24435 df-tms 24436 df-cncf 24994 df-limc 25982 df-dv 25983 df-log 26675 df-cxp 26676

This theorem is referenced by: sqrtlim 27091 signsplypnf 34849

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