neglimc - Mathbox for Glauco Siliprandi

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Theorem neglimc 45064

Description: Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
neglimc.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
neglimc.g	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -𝐵)
neglimc.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
neglimc.c	⊢ (𝜑 → 𝐶 ∈ (𝐹 lim_ℂ 𝐷))

**Assertion**
Ref	Expression
neglimc	⊢ (𝜑 → -𝐶 ∈ (𝐺 lim_ℂ 𝐷))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥) 𝐹(𝑥) 𝐺(𝑥)

Proof of Theorem neglimc Dummy variables 𝑣 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 25824	. . . 4 ⊢ (𝐹 lim_ℂ 𝐷) ⊆ ℂ
2		neglimc.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐹 lim_ℂ 𝐷))
3	1, 2	sselid 3980	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4	3	negcld 11596	. 2 ⊢ (𝜑 → -𝐶 ∈ ℂ)
5		neglimc.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6		neglimc.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	5, 6	fmptd 7129	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
8	6, 5, 2	limcmptdm 45052	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
9		limcrcl 25823	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝐹 lim_ℂ 𝐷) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ))
10	2, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐷 ∈ ℂ))
11	10	simp3d 1141	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ)
12	7, 8, 11	ellimc3 25828	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐹 lim_ℂ 𝐷) ↔ (𝐶 ∈ ℂ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦))))
13	2, 12	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦)))
14	13	simprd 494	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦))
15	14	r19.21bi 3246	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦))
16		simplll 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) → 𝜑)
17	16	3ad2ant1 1130	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) ∧ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤)) → 𝜑)
18		simp1r 1195	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) ∧ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤)) → 𝑣 ∈ 𝐴)
19		simp3 1135	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) ∧ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤)) → (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤))
20		simp2 1134	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) ∧ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤)) → ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦))
21	19, 20	mpd 15	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) ∧ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤)) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦)
22		nfv 1909	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑣 ∈ 𝐴)
23		neglimc.g	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -𝐵)
24		nfmpt1 5260	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ -𝐵)
25	23, 24	nfcxfr 2897	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝐺
26		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑣
27	25, 26	nffv 6912	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝐺‘𝑣)
28		nfmpt1 5260	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
29	6, 28	nfcxfr 2897	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐹
30	29, 26	nffv 6912	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝐹‘𝑣)
31	30	nfneg 11494	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥-(𝐹‘𝑣)
32	27, 31	nfeq 2913	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝐺‘𝑣) = -(𝐹‘𝑣)
33	22, 32	nfim 1891	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐺‘𝑣) = -(𝐹‘𝑣))
34		eleq1w 2812	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → (𝑥 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
35	34	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑣 ∈ 𝐴)))
36		fveq2 6902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → (𝐺‘𝑥) = (𝐺‘𝑣))
37		fveq2 6902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑣 → (𝐹‘𝑥) = (𝐹‘𝑣))
38	37	negeqd 11492	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → -(𝐹‘𝑥) = -(𝐹‘𝑣))
39	36, 38	eqeq12d 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → ((𝐺‘𝑥) = -(𝐹‘𝑥) ↔ (𝐺‘𝑣) = -(𝐹‘𝑣)))
40	35, 39	imbi12d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = -(𝐹‘𝑥)) ↔ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐺‘𝑣) = -(𝐹‘𝑣))))
41		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
42	5	negcld 11596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
43	23	fvmpt2 7021	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ -𝐵 ∈ ℂ) → (𝐺‘𝑥) = -𝐵)
44	41, 42, 43	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = -𝐵)
45	6	fvmpt2 7021	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → (𝐹‘𝑥) = 𝐵)
46	41, 5, 45	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
47	46	negeqd 11492	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐹‘𝑥) = -𝐵)
48	44, 47	eqtr4d 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = -(𝐹‘𝑥))
49	33, 40, 48	chvarfv 2228	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐺‘𝑣) = -(𝐹‘𝑣))
50	49	oveq1d 7441	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐺‘𝑣) − -𝐶) = (-(𝐹‘𝑣) − -𝐶))
51	7	ffvelcdmda 7099	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ ℂ)
52	3	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝐶 ∈ ℂ)
53	51, 52	negsubdi3d 44704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → -((𝐹‘𝑣) − 𝐶) = (-(𝐹‘𝑣) − -𝐶))
54	50, 53	eqtr4d 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐺‘𝑣) − -𝐶) = -((𝐹‘𝑣) − 𝐶))
55	54	fveq2d 6906	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐺‘𝑣) − -𝐶)) = (abs‘-((𝐹‘𝑣) − 𝐶)))
56	51, 52	subcld 11609	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑣) − 𝐶) ∈ ℂ)
57	56	absnegd 15436	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘-((𝐹‘𝑣) − 𝐶)) = (abs‘((𝐹‘𝑣) − 𝐶)))
58	55, 57	eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (abs‘((𝐺‘𝑣) − -𝐶)) = (abs‘((𝐹‘𝑣) − 𝐶)))
59	58	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) → (abs‘((𝐺‘𝑣) − -𝐶)) = (abs‘((𝐹‘𝑣) − 𝐶)))
60		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦)
61	59, 60	eqbrtrd 5174	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐴) ∧ (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦)
62	17, 18, 21, 61	syl21anc 836	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) ∧ ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) ∧ (𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤)) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦)
63	62	3exp 1116	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ 𝐴) → (((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) → ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦)))
64	63	ralimdva 3164	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) → (∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) → ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦)))
65	64	reximdva 3165	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐹‘𝑣) − 𝐶)) < 𝑦) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦)))
66	15, 65	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦))
67	66	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦))
68	42, 23	fmptd 7129	. . 3 ⊢ (𝜑 → 𝐺:𝐴⟶ℂ)
69	68, 8, 11	ellimc3 25828	. 2 ⊢ (𝜑 → (-𝐶 ∈ (𝐺 lim_ℂ 𝐷) ↔ (-𝐶 ∈ ℂ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ 𝐴 ((𝑣 ≠ 𝐷 ∧ (abs‘(𝑣 − 𝐷)) < 𝑤) → (abs‘((𝐺‘𝑣) − -𝐶)) < 𝑦))))
70	4, 67, 69	mpbir2and 711	1 ⊢ (𝜑 → -𝐶 ∈ (𝐺 lim_ℂ 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5682 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 < clt 11286 − cmin 11482 -cneg 11483 ℝ⁺crp 13014 abscabs 15221 lim_ℂ climc 25811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-rest 17411 df-topn 17412 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cnp 23152 df-xms 24246 df-ms 24247 df-limc 25815

This theorem is referenced by: sublimc 45069 reclimc 45070

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