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Theorem cxplim 26882

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 12960	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2	1	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
3		rpge0 12965	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
4	3	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
5		rpre 12960	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
6	5	renegcld 11605	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ∈ ℝ)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ∈ ℝ)
8		rpcn 12962	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
9		rpne0 12968	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
10	8, 9	negne0d 11531	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ≠ 0)
11	10	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ≠ 0)
12	7, 11	rereccld 12009	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 / -𝐴) ∈ ℝ)
13	2, 4, 12	recxpcld 26632	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ)
14		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℝ⁺)
15	5	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℝ)
16	14, 15	rpcxpcld 26642	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
17	16	rpreccld 13005	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
18	17	rprege0d 13002	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))))
19		absid 15262	. . . . . . . 8 ⊢ (((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
20	18, 19	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
21		simplr 768	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℝ⁺)
22		simprr 772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)
23		rpreccl 12979	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
24	23	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℝ⁺)
25	24	rpcnd 12997	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℂ)
26	21, 25	cxprecd 26641	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
27		rpcn 12962	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
28	27	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℂ)
29		rpne0 12968	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
30	29	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ≠ 0)
31	28, 30, 25	cxpnegd 26624	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐-(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
32		1cnd 11169	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 1 ∈ ℂ)
33	8	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℂ)
34	9	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ≠ 0)
35	32, 33, 34	divneg2d 11972	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → -(1 / 𝐴) = (1 / -𝐴))
36	35	oveq2d 7403	. . . . . . . . . . 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 2770	. . . . . . . . . 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 11953	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝐴 · (1 / 𝐴)) = 1)
39	38	oveq2d 7403	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = (𝑛↑_𝑐1))
40	14, 15, 25	cxpmuld 26646	. . . . . . . . . . 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 12997	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℂ)
42	41	cxp1d 26615	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐1) = 𝑛)
43	39, 40, 42	3eqtr3d 2772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)) = 𝑛)
44	22, 37, 43	3brtr4d 5139	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
45		rpreccl 12979	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
46	45	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ⁺)
47	46	rpred 12995	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ)
48	46	rpge0d 12999	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (1 / 𝑥))
49	16	rpred 12995	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ)
50	16	rpge0d 12999	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (𝑛↑_𝑐𝐴))
51	47, 48, 49, 50, 24	cxplt2d 26635	. . . . . . . . 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 257	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) < (𝑛↑_𝑐𝐴))
53	21, 16, 52	ltrec1d 13015	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) < 𝑥)
54	20, 53	eqbrtrd 5129	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)
55	54	expr 456	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ℝ⁺) → ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
56	55	ralrimiva 3125	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
57		breq1 5110	. . . . 5 ⊢ (𝑦 = (𝑥↑_𝑐(1 / -𝐴)) → (𝑦 < 𝑛 ↔ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛))
58	57	rspceaimv 3594	. . . 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 584	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
60	59	ralrimiva 3125	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
61		id 22	. . . . . . 7 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ⁺)
62		rpcxpcl 26585	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
63	61, 5, 62	syl2anr 597	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
64	63	rpreccld 13005	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
65	64	rpcnd 12997	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
66	65	ralrimiva 3125	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
67		rpssre 12959	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
68	67	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
69	66, 68	rlim0lt 15475	. 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 257	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⊆ wss 3914 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 < clt 11208 ≤ cle 11209 -cneg 11406 / cdiv 11835 ℝ⁺crp 12951 abscabs 15200 ⇝_𝑟 crli 15451 ↑_𝑐ccxp 26464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 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-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-cxp 26466

This theorem is referenced by: sqrtlim 26883 signsplypnf 34541

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