elbigolo1 - Mathbox for Alexander van der Vekens

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Theorem elbigolo1 48546

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 12959	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ⁺ → ℝ⁺ ⊆ ℝ)
4	1, 3	fssd 6705	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℝ⁺ → 𝐹:𝐴⟶ℝ)
5	4	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐹:𝐴⟶ℝ)
6	5	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐹:𝐴⟶ℝ)
7	6	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
8		simplrr 777	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑚 ∈ ℝ)
9		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐺:𝐴⟶ℝ⁺)
10	9	ffvelcdmda 7056	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ ℝ⁺)
11	10	rpregt0d 13001	. . . . . . . 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 1128	. . . . . . 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 12062	. . . . . . . 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 223	. . . . . . 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 6705	. . . . . . . . . . . 12 ⊢ (𝐺:𝐴⟶ℝ⁺ → 𝐺:𝐴⟶ℝ)
19	18	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐺:𝐴⟶ℝ)
20		reex 11159	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
21	20	ssex 5276	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
22	21	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ∈ V)
23	5, 19, 22	3jca 1128	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
24	23	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
25	24	adantr 480	. . . . . . . 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 6691	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → Fun 𝐺)
27	26	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → Fun 𝐺)
28	21	anim1ci 616	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V))
29		fex 7200	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V) → 𝐺 ∈ V)
30	28, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 𝐺 ∈ V)
31		0red 11177	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∈ ℝ)
32		frn 6695	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐴⟶ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
33		0nrp 12988	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 0 ∈ ℝ⁺
34		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
35	34	ssneld 3948	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝐺 ⊆ ℝ⁺ → (¬ 0 ∈ ℝ⁺ → ¬ 0 ∈ ran 𝐺))
36	33, 35	mpi 20	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ¬ 0 ∈ ran 𝐺)
37		df-nel 3030	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∉ ran 𝐺 ↔ ¬ 0 ∈ ran 𝐺)
38	36, 37	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝐺 ⊆ ℝ⁺ → 0 ∉ ran 𝐺)
39	32, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → 0 ∉ ran 𝐺)
40	39	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∉ ran 𝐺)
41		suppdm 48499	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 0 ∈ ℝ) ∧ 0 ∉ ran 𝐺) → (𝐺 supp 0) = dom 𝐺)
42	27, 30, 31, 40, 41	syl31anc 1375	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = dom 𝐺)
43		fdm 6697	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴⟶ℝ⁺ → dom 𝐺 = 𝐴)
44	43	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → dom 𝐺 = 𝐴)
45	42, 44	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
46	45	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
47	46	eqcomd 2735	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 = (𝐺 supp 0))
48	47	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐴 = (𝐺 supp 0))
49	48	eleq2d 2814	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ (𝐺 supp 0)))
50	49	biimpa 476	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (𝐺 supp 0))
51		refdivmptfv 48535	. . . . . . . 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 584	. . . . . . 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 5117	. . . . . 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 282	. . . . 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 340	. . . 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 3154	. . 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 3200	. 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 1136	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ ℝ)
59		ssidd 3970	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ 𝐴)
60		elbigo2 48541	. . 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 838	. 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 48531	. . . . 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 6672	. . . 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 257	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 /_f 𝐺):𝐴⟶ℝ)
66		ello12 15482	. . 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 584	. 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 311	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∉ wnel 3029 ∀wral 3044 ∃wrex 3053 Vcvv 3447 ⊆ wss 3914 class class class wbr 5107 dom cdm 5638 ran crn 5639 Fun wfun 6505 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supp csupp 8139 ℝcr 11067 0cc0 11068 · cmul 11073 < clt 11208 ≤ cle 11209 / cdiv 11835 ℝ⁺crp 12951 ≤𝑂(1)clo1 15453 /_f cfdiv 48526 Οcbigo 48536

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-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

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-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 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-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-supp 8140 df-er 8671 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 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-rp 12952 df-ico 13312 df-lo1 15457 df-fdiv 48527 df-bigo 48537

This theorem is referenced by: (None)

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