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Theorem rlimuni 15498

Description: A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)

**Hypotheses**
Ref	Expression
rlimuni.1	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
rlimuni.2	⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
rlimuni.3	⊢ (𝜑 → 𝐹 ⇝_𝑟 𝐵)
rlimuni.4	⊢ (𝜑 → 𝐹 ⇝_𝑟 𝐶)

**Assertion**
Ref	Expression
rlimuni	⊢ (𝜑 → 𝐵 = 𝐶)

Proof of Theorem rlimuni Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimuni.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⇝_𝑟 𝐵)
2		rlimcl 15451	. . . . . . . . . . . 12 ⊢ (𝐹 ⇝_𝑟 𝐵 → 𝐵 ∈ ℂ)
3	1, 2	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4	3	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
5		rlimuni.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⇝_𝑟 𝐶)
6		rlimcl 15451	. . . . . . . . . . . 12 ⊢ (𝐹 ⇝_𝑟 𝐶 → 𝐶 ∈ ℂ)
7	5, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℂ)
8	7	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
9	4, 8	subcld 11575	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (𝐵 − 𝐶) ∈ ℂ)
10	9	abscld 15387	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (abs‘(𝐵 − 𝐶)) ∈ ℝ)
11	10	ltnrd 11352	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → ¬ (abs‘(𝐵 − 𝐶)) < (abs‘(𝐵 − 𝐶)))
12		rlimuni.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
13	12	ffvelcdmda 7085	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℂ)
14	13	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℂ)
15	14, 4	abssubd 15404	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (abs‘((𝐹‘𝑘) − 𝐵)) = (abs‘(𝐵 − (𝐹‘𝑘))))
16	15	breq1d 5157	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ↔ (abs‘(𝐵 − (𝐹‘𝑘))) < ((abs‘(𝐵 − 𝐶)) / 2)))
17	16	anbi1d 628	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)) ↔ ((abs‘(𝐵 − (𝐹‘𝑘))) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))))
18		abs3lem 15289	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘(𝐵 − 𝐶)) ∈ ℝ)) → (((abs‘(𝐵 − (𝐹‘𝑘))) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)) → (abs‘(𝐵 − 𝐶)) < (abs‘(𝐵 − 𝐶))))
19	4, 8, 14, 10, 18	syl22anc 835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (((abs‘(𝐵 − (𝐹‘𝑘))) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)) → (abs‘(𝐵 − 𝐶)) < (abs‘(𝐵 − 𝐶))))
20	17, 19	sylbid 239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → (((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)) → (abs‘(𝐵 − 𝐶)) < (abs‘(𝐵 − 𝐶))))
21	20	imim2d 57	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → ((𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))) → (𝑗 ≤ 𝑘 → (abs‘(𝐵 − 𝐶)) < (abs‘(𝐵 − 𝐶)))))
22	21	impcomd 410	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → ((𝑗 ≤ 𝑘 ∧ (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))) → (abs‘(𝐵 − 𝐶)) < (abs‘(𝐵 − 𝐶))))
23	11, 22	mtod 197	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑘 ∈ 𝐴) → ¬ (𝑗 ≤ 𝑘 ∧ (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
24	23	nrexdv 3147	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ¬ ∃𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 ∧ (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
25		r19.29r 3114	. . . . 5 ⊢ ((∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))) → ∃𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 ∧ (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
26	24, 25	nsyl 140	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ¬ (∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
27	26	nrexdv 3147	. . 3 ⊢ (𝜑 → ¬ ∃𝑗 ∈ ℝ (∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
28		rlimuni.2	. . . . 5 ⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
29	12	fdmd 6727	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
30		rlimss 15450	. . . . . . . . 9 ⊢ (𝐹 ⇝_𝑟 𝐵 → dom 𝐹 ⊆ ℝ)
31	1, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
32	29, 31	eqsstrrd 4020	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
33		ressxr 11262	. . . . . . 7 ⊢ ℝ ⊆ ℝ^*
34	32, 33	sstrdi 3993	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
35		supxrunb1 13302	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑗 ∈ ℝ ∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ↔ sup(𝐴, ℝ^*, < ) = +∞))
36	34, 35	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑗 ∈ ℝ ∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ↔ sup(𝐴, ℝ^*, < ) = +∞))
37	28, 36	mpbird 256	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ ℝ ∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘)
38		r19.29 3112	. . . . 5 ⊢ ((∀𝑗 ∈ ℝ ∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))) → ∃𝑗 ∈ ℝ (∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
39	38	ex 411	. . . 4 ⊢ (∀𝑗 ∈ ℝ ∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))) → ∃𝑗 ∈ ℝ (∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))))))
40	37, 39	syl 17	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))) → ∃𝑗 ∈ ℝ (∃𝑘 ∈ 𝐴 𝑗 ≤ 𝑘 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))))))
41	27, 40	mtod 197	. 2 ⊢ (𝜑 → ¬ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))))
42	12	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹:𝐴⟶ℂ)
43		ffvelcdm 7082	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ℂ)
44	43	ralrimiva 3144	. . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ ℂ)
45	42, 44	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ ℂ)
46	3	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ∈ ℂ)
47	7	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐶 ∈ ℂ)
48	46, 47	subcld 11575	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐶) ∈ ℂ)
49		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐵 ≠ 𝐶)
50	46, 47, 49	subne0d 11584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝐵 − 𝐶) ≠ 0)
51	48, 50	absrpcld 15399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (abs‘(𝐵 − 𝐶)) ∈ ℝ⁺)
52	51	rphalfcld 13032	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ((abs‘(𝐵 − 𝐶)) / 2) ∈ ℝ⁺)
53	42	feqmptd 6959	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
54	1	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 ⇝_𝑟 𝐵)
55	53, 54	eqbrtrrd 5171	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ⇝_𝑟 𝐵)
56	45, 52, 55	rlimi 15461	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2)))
57	5	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐹 ⇝_𝑟 𝐶)
58	53, 57	eqbrtrrd 5171	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ⇝_𝑟 𝐶)
59	45, 52, 58	rlimi 15461	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))
60	32	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → 𝐴 ⊆ ℝ)
61		rexanre 15297	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2)) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
62	60, 61	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))) ↔ (∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2)) ∧ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
63	56, 59, 62	mpbir2and 709	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 𝐶) → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))))
64	63	ex 411	. . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2)))))
65	64	necon1bd 2956	. 2 ⊢ (𝜑 → (¬ ∃𝑗 ∈ ℝ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → ((abs‘((𝐹‘𝑘) − 𝐵)) < ((abs‘(𝐵 − 𝐶)) / 2) ∧ (abs‘((𝐹‘𝑘) − 𝐶)) < ((abs‘(𝐵 − 𝐶)) / 2))) → 𝐵 = 𝐶))
66	41, 65	mpd 15	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 supcsup 9437 ℂcc 11110 ℝcr 11111 +∞cpnf 11249 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 2c2 12271 abscabs 15185 ⇝_𝑟 crli 15433

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-rlim 15437

This theorem is referenced by: rlimdm 15499 rlimdmafv 46183 rlimdmafv2 46264

W3C validator