elbigolo1 - Mathbox for Alexander van der Vekens

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Theorem elbigolo1 49062

Description: A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.) (Proof shortened by II, 16-Feb-2023.)

**Assertion**
Ref	Expression
elbigolo1	⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /_f 𝐺) ∈ ≤𝑂(1)))

Proof of Theorem elbigolo1 Dummy variables 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ⁺ → 𝐹:𝐴⟶ℝ⁺)
2		rpssre 12945	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ⁺ → ℝ⁺ ⊆ ℝ)
4	1, 3	fssd 6676	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℝ⁺ → 𝐹:𝐴⟶ℝ)
5	4	3ad2ant3 1142	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐹:𝐴⟶ℝ)
6	5	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐹:𝐴⟶ℝ)
7	6	ffvelcdmda 7029	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
8		simplrr 784	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑚 ∈ ℝ)
9		simpl2 1200	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐺:𝐴⟶ℝ⁺)
10	9	ffvelcdmda 7029	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ ℝ⁺)
11	10	rpregt0d 12987	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) ∈ ℝ ∧ 0 < (𝐺‘𝑦)))
12	7, 8, 11	3jca 1135	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ ((𝐺‘𝑦) ∈ ℝ ∧ 0 < (𝐺‘𝑦))))
13		ledivmul2 12030	. . . . . . . 8 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ ((𝐺‘𝑦) ∈ ℝ ∧ 0 < (𝐺‘𝑦))) → (((𝐹‘𝑦) / (𝐺‘𝑦)) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
14	13	bicomd 225	. . . . . . 7 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ ((𝐺‘𝑦) ∈ ℝ ∧ 0 < (𝐺‘𝑦))) → ((𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)) ↔ ((𝐹‘𝑦) / (𝐺‘𝑦)) ≤ 𝑚))
15	12, 14	syl 17	. . . . . 6 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)) ↔ ((𝐹‘𝑦) / (𝐺‘𝑦)) ≤ 𝑚))
16		id 22	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶ℝ⁺ → 𝐺:𝐴⟶ℝ⁺)
17	2	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶ℝ⁺ → ℝ⁺ ⊆ ℝ)
18	16, 17	fssd 6676	. . . . . . . . . . . 12 ⊢ (𝐺:𝐴⟶ℝ⁺ → 𝐺:𝐴⟶ℝ)
19	18	3ad2ant2 1141	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐺:𝐴⟶ℝ)
20		reex 11124	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
21	20	ssex 5252	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
22	21	3ad2ant1 1140	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ∈ V)
23	5, 19, 22	3jca 1135	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
24	23	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
25	24	adantr 482	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
26		ffun 6662	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → Fun 𝐺)
27	26	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → Fun 𝐺)
28	21	anim1ci 623	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V))
29		fex 7174	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V) → 𝐺 ∈ V)
30	28, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 𝐺 ∈ V)
31		0red 11142	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∈ ℝ)
32		frn 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐴⟶ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
33		0nrp 12974	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 0 ∈ ℝ⁺
34		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
35	34	ssneld 3919	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝐺 ⊆ ℝ⁺ → (¬ 0 ∈ ℝ⁺ → ¬ 0 ∈ ran 𝐺))
36	33, 35	mpi 20	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ¬ 0 ∈ ran 𝐺)
37		df-nel 3041	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∉ ran 𝐺 ↔ ¬ 0 ∈ ran 𝐺)
38	36, 37	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝐺 ⊆ ℝ⁺ → 0 ∉ ran 𝐺)
39	32, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → 0 ∉ ran 𝐺)
40	39	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∉ ran 𝐺)
41		suppdm 49015	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 0 ∈ ℝ) ∧ 0 ∉ ran 𝐺) → (𝐺 supp 0) = dom 𝐺)
42	27, 30, 31, 40, 41	syl31anc 1382	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = dom 𝐺)
43		fdm 6668	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴⟶ℝ⁺ → dom 𝐺 = 𝐴)
44	43	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → dom 𝐺 = 𝐴)
45	42, 44	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
46	45	3adant3 1139	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
47	46	eqcomd 2747	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 = (𝐺 supp 0))
48	47	adantr 482	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐴 = (𝐺 supp 0))
49	48	eleq2d 2827	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ (𝐺 supp 0)))
50	49	biimpa 478	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (𝐺 supp 0))
51		refdivmptfv 49051	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V) ∧ 𝑦 ∈ (𝐺 supp 0)) → ((𝐹 /_f 𝐺)‘𝑦) = ((𝐹‘𝑦) / (𝐺‘𝑦)))
52	25, 50, 51	syl2anc 591	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → ((𝐹 /_f 𝐺)‘𝑦) = ((𝐹‘𝑦) / (𝐺‘𝑦)))
53	52	breq1d 5085	. . . . . 6 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚 ↔ ((𝐹‘𝑦) / (𝐺‘𝑦)) ≤ 𝑚))
54	15, 53	bitr4d 284	. . . . 5 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)) ↔ ((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚))
55	54	imbi2d 342	. . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) ↔ (𝑥 ≤ 𝑦 → ((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚)))
56	55	ralbidva 3162	. . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚)))
57	56	2rexbidva 3204	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚)))
58		simp1 1143	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ ℝ)
59		ssidd 3940	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ 𝐴)
60		elbigo2 49057	. . 3 ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ 𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))))
61	19, 58, 5, 59, 60	syl22anc 845	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))))
62		refdivmptf 49047	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V) → (𝐹 /_f 𝐺):(𝐺 supp 0)⟶ℝ)
63	23, 62	syl 17	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 /_f 𝐺):(𝐺 supp 0)⟶ℝ)
64	47	feq2d 6643	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → ((𝐹 /_f 𝐺):𝐴⟶ℝ ↔ (𝐹 /_f 𝐺):(𝐺 supp 0)⟶ℝ))
65	63, 64	mpbird 259	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 /_f 𝐺):𝐴⟶ℝ)
66		ello12 15473	. . 3 ⊢ (((𝐹 /_f 𝐺):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((𝐹 /_f 𝐺) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚)))
67	65, 58, 66	syl2anc 591	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → ((𝐹 /_f 𝐺) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((𝐹 /_f 𝐺)‘𝑦) ≤ 𝑚)))
68	57, 61, 67	3bitr4d 313	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /_f 𝐺) ∈ ≤𝑂(1)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∉ wnel 3040 ∀wral 3055 ∃wrex 3065 Vcvv 3433 ⊆ wss 3885 class class class wbr 5075 dom cdm 5621 ran crn 5622 Fun wfun 6483 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 supp csupp 8104 ℝcr 11032 0cc0 11033 · cmul 11038 < clt 11174 ≤ cle 11175 / cdiv 11802 ℝ⁺crp 12937 ≤𝑂(1)clo1 15444 /_f cfdiv 49042 Οcbigo 49052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-supp 8105 df-er 8637 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-rp 12938 df-ico 13299 df-lo1 15448 df-fdiv 49043 df-bigo 49053

This theorem is referenced by: (None)

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