Theorem cnre2csqima 34613

Description: Image of a centered square by the canonical bijection from (ℝ × ℝ) to ℂ. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Hypothesis

Ref	Expression
cnre2csqima.1	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))

Assertion

Ref	Expression
cnre2csqima	⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐷(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem cnre2csqima

Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossre 13565	. . 3 ⊢ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ
2		ioossre 13565	. . 3 ⊢ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ
3		xpinpreima2 34609	. . . 4 ⊢ (((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ ∧ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ) → ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) = ((◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∩ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)))))
4	3	eleq2d 2850	. . 3 ⊢ (((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ ∧ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) ↔ 𝑌 ∈ ((◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∩ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))))))
5	1, 2, 4	mp2an 705	. 2 ⊢ (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) ↔ 𝑌 ∈ ((◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∩ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)))))
6		elin 4096	. . 3 ⊢ (𝑌 ∈ ((◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∩ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)))) ↔ (𝑌 ∈ (◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∧ 𝑌 ∈ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)))))
7		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
8	7	recnd 11364	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9		ax-icn 11286	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
10	9	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
11		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
12	11	recnd 11364	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
13	10, 12	mulcld 11356	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
14	8, 13	addcld 11355	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15		reval 15302	. . . . . . . . 9 ⊢ ((𝑥 + (i · 𝑦)) ∈ ℂ → (ℜ‘(𝑥 + (i · 𝑦))) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
16	14, 15	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
17		crre 15310	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
18	16, 17	eqtr3d 2800	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
19	18	mpoeq3ia 7521	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
20	14	adantl 487	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + (i · 𝑦)) ∈ ℂ)
21		cnre2csqima.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
22	21	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))))
23		df-re 15296	. . . . . . . . 9 ⊢ ℜ = (𝑧 ∈ ℂ ↦ ((𝑧 + (∗‘𝑧)) / 2))
24	23	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ ((𝑧 + (∗‘𝑧)) / 2)))
25		id 23	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 = (𝑥 + (i · 𝑦)))
26		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
27	25, 26	oveq12d 7461	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
28	27	oveq1d 7458	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
29	20, 22, 24, 28	fmpoco 8132	. . . . . . 7 ⊢ (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
30	29	mptru 1577	. . . . . 6 ⊢ (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31		df1stres 33368	. . . . . 6 ⊢ (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
32	19, 30, 31	3eqtr4ri 2797	. . . . 5 ⊢ (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
33	14	rgen2 3206	. . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
34	21	fnmpo 8105	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
35	33, 34	ax-mp 5	. . . . 5 ⊢ 𝐹 Fn (ℝ × ℝ)
36		fo1st 8045	. . . . . 6 ⊢ 1st :V⟶ontoV
37		fofn 6820	. . . . . 6 ⊢ (1st :V⟶ontoV → 1st Fn V)
38	36, 37	ax-mp 5	. . . . 5 ⊢ 1st Fn V
39		xp1st 8057	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
40	21	rnmpo 7577	. . . . . . . 8 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))}
41		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 = (𝑥 + (i · 𝑦)))
42	14	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → (𝑥 + (i · 𝑦)) ∈ ℂ)
43	41, 42	eqeltrd 2864	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 ∈ ℂ)
44	43	ex 418	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ))
45	44	rexlimivv 3208	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
46	45	abssi 3996	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
47	40, 46	eqsstri 3959	. . . . . . 7 ⊢ ran 𝐹 ⊆ ℂ
48		simpl 488	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
49	47, 48	sselid 3911	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
50		simpr 490	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
51	47, 50	sselid 3911	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
52	49, 51	resubd 15412	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧 − 𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
53	32, 35, 38, 39, 52	cnre2csqlem 34612	. . . 4 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) → (abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))
54		imval 15303	. . . . . . . . 9 ⊢ ((𝑥 + (i · 𝑦)) ∈ ℂ → (ℑ‘(𝑥 + (i · 𝑦))) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
55	14, 54	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
56		crim 15311	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
57	55, 56	eqtr3d 2800	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
58	57	mpoeq3ia 7521	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59		df-im 15297	. . . . . . . . 9 ⊢ ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
60	59	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61		fvoveq1 7466	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
62	20, 22, 60, 61	fmpoco 8132	. . . . . . 7 ⊢ (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
63	62	mptru 1577	. . . . . 6 ⊢ (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64		df2ndres 33369	. . . . . 6 ⊢ (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
65	58, 63, 64	3eqtr4ri 2797	. . . . 5 ⊢ (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66		fo2nd 8046	. . . . . 6 ⊢ 2nd :V⟶ontoV
67		fofn 6820	. . . . . 6 ⊢ (2nd :V⟶ontoV → 2nd Fn V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
69		xp2nd 8058	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
70	49, 51	imsubd 15413	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧 − 𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
71	65, 35, 68, 69, 70	cnre2csqlem 34612	. . . 4 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))
72	53, 71	anim12d 621	. . 3 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → ((𝑌 ∈ (◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∧ 𝑌 ∈ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)))
73	6, 72	biimtrid 245	. 2 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((◡(1st ↾ (ℝ × ℝ)) “ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷))) ∩ (◡(2nd ↾ (ℝ × ℝ)) “ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)))
74	5, 73	biimtrid 245	1 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654  ran crn 5656   ↾ cres 5657   “ cima 5658   ∘ ccom 5659   Fn wfn 6553  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  1^st c1st 8023  2^nd c2nd 8024  ℂcc 11225  ℝcr 11226  ici 11229   + caddc 11230   · cmul 11232   < clt 11370   − cmin 11568   / cdiv 12000  2c2 12424  ℝ⁺crp 13147  (,)cioo 13503  ∗ccj 15292  ℜcre 15293  ℑcim 15294  abscabs 15430

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-sup 9461  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-n0 12634  df-z 12721  df-uz 12993  df-rp 13148  df-ioo 13507  df-seq 14172  df-exp 14232  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432

This theorem is used by:  tpr2rico  34614