elbigolo1 - Mathbox for Alexander van der Vekens

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Theorem elbigolo1 49669

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 23	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ⁺ → 𝐹:𝐴⟶ℝ⁺)
2		rpssre 13128	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ⊆ ℝ
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ⁺ → ℝ⁺ ⊆ ℝ)
4	1, 3	fssd 6727	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℝ⁺ → 𝐹:𝐴⟶ℝ)
5	4	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐹:𝐴⟶ℝ)
6	5	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐹:𝐴⟶ℝ)
7	6	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
8		simplrr 790	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑚 ∈ ℝ)
9		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐺:𝐴⟶ℝ⁺)
10	9	ffvelcdmda 7084	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ ℝ⁺)
11	10	rpregt0d 13170	. . . . . . . 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 1146	. . . . . . 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 12196	. . . . . . . 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 226	. . . . . . 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 18	. . . . . 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 23	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶ℝ⁺ → 𝐺:𝐴⟶ℝ⁺)
17	2	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶ℝ⁺ → ℝ⁺ ⊆ ℝ)
18	16, 17	fssd 6727	. . . . . . . . . . . 12 ⊢ (𝐺:𝐴⟶ℝ⁺ → 𝐺:𝐴⟶ℝ)
19	18	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐺:𝐴⟶ℝ)
20		reex 11291	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
21	20	ssex 5281	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
22	21	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ∈ V)
23	5, 19, 22	3jca 1146	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
24	23	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V))
25	24	adantr 486	. . . . . . . 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 6712	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → Fun 𝐺)
27	26	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → Fun 𝐺)
28	21	anim1ci 628	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V))
29		fex 7232	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:𝐴⟶ℝ⁺ ∧ 𝐴 ∈ V) → 𝐺 ∈ V)
30	28, 29	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 𝐺 ∈ V)
31		0red 11311	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∈ ℝ)
32		frn 6717	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐴⟶ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
33		0nrp 13157	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 0 ∈ ℝ⁺
34		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ran 𝐺 ⊆ ℝ⁺)
35	34	ssneld 3910	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran 𝐺 ⊆ ℝ⁺ → (¬ 0 ∈ ℝ⁺ → ¬ 0 ∈ ran 𝐺))
36	33, 35	mpi 21	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝐺 ⊆ ℝ⁺ → ¬ 0 ∈ ran 𝐺)
37		df-nel 3062	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∉ ran 𝐺 ↔ ¬ 0 ∈ ran 𝐺)
38	36, 37	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝐺 ⊆ ℝ⁺ → 0 ∉ ran 𝐺)
39	32, 38	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶ℝ⁺ → 0 ∉ ran 𝐺)
40	39	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → 0 ∉ ran 𝐺)
41		suppdm 49622	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 0 ∈ ℝ) ∧ 0 ∉ ran 𝐺) → (𝐺 supp 0) = dom 𝐺)
42	27, 30, 31, 40, 41	syl31anc 1400	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = dom 𝐺)
43		fdm 6719	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴⟶ℝ⁺ → dom 𝐺 = 𝐴)
44	43	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → dom 𝐺 = 𝐴)
45	42, 44	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
46	45	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐺 supp 0) = 𝐴)
47	46	eqcomd 2766	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 = (𝐺 supp 0))
48	47	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → 𝐴 = (𝐺 supp 0))
49	48	eleq2d 2846	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ (𝐺 supp 0)))
50	49	biimpa 482	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) ∧ (𝑥 ∈ ℝ ∧ 𝑚 ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (𝐺 supp 0))
51		refdivmptfv 49658	. . . . . . . 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 596	. . . . . . 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 5112	. . . . . 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 285	. . . . 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 343	. . . 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 3183	. . 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 3225	. 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 1154	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ ℝ)
59		ssidd 3931	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → 𝐴 ⊆ 𝐴)
60		elbigo2 49664	. . 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 852	. 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 49654	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴 ∈ V) → (𝐹 /_f 𝐺):(𝐺 supp 0)⟶ℝ)
63	23, 62	syl 18	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 /_f 𝐺):(𝐺 supp 0)⟶ℝ)
64	47	feq2d 6693	. . . 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 260	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 /_f 𝐺):𝐴⟶ℝ)
66		ello12 15683	. . 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 596	. 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 314	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:𝐴⟶ℝ⁺ ∧ 𝐹:𝐴⟶ℝ⁺) → (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 /_f 𝐺) ∈ ≤𝑂(1)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∉ wnel 3061 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 class class class wbr 5102 dom cdm 5650 ran crn 5651 Fun wfun 6532 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 supp csupp 8177 ℝcr 11199 0cc0 11200 · cmul 11205 < clt 11343 ≤ cle 11344 / cdiv 11973 ℝ⁺crp 13120 ≤𝑂(1)clo1 15654 /_f cfdiv 49649 Οcbigo 49659

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-po 5558 df-so 5559 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-supp 8178 df-er 8717 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-rp 13121 df-ico 13482 df-lo1 15658 df-fdiv 49650 df-bigo 49660

This theorem is used by: (None)

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