elbigolo1 - Mathbox for Alexander van der Vekens

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Theorem elbigolo1 45903

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 12737	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ⁺ → ℝ⁺ ⊆ ℝ)
4	1, 3	fssd 6618	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℝ⁺ → 𝐹:𝐴⟶ℝ)
5	4	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐹:𝐴⟶ℝ)
6	5	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐹:𝐴⟶ℝ)
7	6	ffvelrnda 6961	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
8		simplrr 775	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑚 ∈ ℝ)
9		simpl2 1191	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐺:𝐴⟶ℝ⁺)
10	9	ffvelrnda 6961	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ ℝ⁺)
11	10	rpregt0d 12778	. . . . . . . 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 1127	. . . . . . 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 11854	. . . . . . . 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 222	. . . . . . 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 6618	. . . . . . . . . . . 12 ⊢ (𝐺:𝐴⟶ℝ⁺ → 𝐺:𝐴⟶ℝ)
19	18	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐺:𝐴⟶ℝ)
20		reex 10962	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
21	20	ssex 5245	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
22	21	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ∈ V)
23	5, 19, 22	3jca 1127	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
24	23	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
25	24	adantr 481	. . . . . . . 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 6603	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → Fun 𝐺)
27	26	adantl 482	. . . . . . . . . . . . . . 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 7102	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V) → 𝐺 ∈ V)
30	28, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 𝐺 ∈ V)
31		0red 10978	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∈ ℝ)
32		frn 6607	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐴⟶ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
33		0nrp 12765	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 0 ∈ ℝ⁺
34		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
35	34	ssneld 3923	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝐺 ⊆ ℝ⁺ → (¬ 0 ∈ ℝ⁺ → ¬ 0 ∈ ran 𝐺))
36	33, 35	mpi 20	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ¬ 0 ∈ ran 𝐺)
37		df-nel 3050	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∉ ran 𝐺 ↔ ¬ 0 ∈ ran 𝐺)
38	36, 37	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝐺 ⊆ ℝ⁺ → 0 ∉ ran 𝐺)
39	32, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → 0 ∉ ran 𝐺)
40	39	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∉ ran 𝐺)
41		suppdm 45851	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 0 ∈ ℝ) ∧ 0 ∉ ran 𝐺) → (𝐺 supp 0) = dom 𝐺)
42	27, 30, 31, 40, 41	syl31anc 1372	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = dom 𝐺)
43		fdm 6609	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴⟶ℝ⁺ → dom 𝐺 = 𝐴)
44	43	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → dom 𝐺 = 𝐴)
45	42, 44	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
46	45	3adant3 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
47	46	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 = (𝐺 supp 0))
48	47	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐴 = (𝐺 supp 0))
49	48	eleq2d 2824	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ (𝐺 supp 0)))
50	49	biimpa 477	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (𝐺 supp 0))
51		refdivmptfv 45892	. . . . . . . 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 5084	. . . . . 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 281	. . . . 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 341	. . . 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 3111	. . 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 3228	. 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 1135	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ ℝ)
59		ssidd 3944	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ 𝐴)
60		elbigo2 45898	. . 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 836	. 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 45888	. . . . 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 6586	. . . 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 256	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 /_f 𝐺):𝐴⟶ℝ)
66		ello12 15225	. . 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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∉ wnel 3049 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 dom cdm 5589 ran crn 5590 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supp csupp 7977 ℝcr 10870 0cc0 10871 · cmul 10876 < clt 11009 ≤ cle 11010 / cdiv 11632 ℝ⁺crp 12730 ≤𝑂(1)clo1 15196 /_f cfdiv 45883 Οcbigo 45893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-supp 7978 df-er 8498 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-rp 12731 df-ico 13085 df-lo1 15200 df-fdiv 45884 df-bigo 45894

This theorem is referenced by: (None)

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