Theorem cnre2csqima 32891

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 13385	. . 3 ⊢ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ
2		ioossre 13385	. . 3 ⊢ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ
3		xpinpreima2 32887	. . . 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 2820	. . 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 691	. 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 3965	. . 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 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
8	7	recnd 11242	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9		ax-icn 11169	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
10	9	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
11		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
12	11	recnd 11242	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
13	10, 12	mulcld 11234	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
14	8, 13	addcld 11233	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15		reval 15053	. . . . . . . . 9 ⊢ ((𝑥 + (i · 𝑦)) ∈ ℂ → (ℜ‘(𝑥 + (i · 𝑦))) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
16	14, 15	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
17		crre 15061	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
18	16, 17	eqtr3d 2775	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
19	18	mpoeq3ia 7487	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
20	14	adantl 483	. . . . . . . 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 15047	. . . . . . . . 9 ⊢ ℜ = (𝑧 ∈ ℂ ↦ ((𝑧 + (∗‘𝑧)) / 2))
24	23	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ ((𝑧 + (∗‘𝑧)) / 2)))
25		id 22	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 = (𝑥 + (i · 𝑦)))
26		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
27	25, 26	oveq12d 7427	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
28	27	oveq1d 7424	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
29	20, 22, 24, 28	fmpoco 8081	. . . . . . 7 ⊢ (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
30	29	mptru 1549	. . . . . 6 ⊢ (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31		df1stres 31925	. . . . . 6 ⊢ (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
32	19, 30, 31	3eqtr4ri 2772	. . . . 5 ⊢ (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
33	14	rgen2 3198	. . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
34	21	fnmpo 8055	. . . . . 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 7995	. . . . . 6 ⊢ 1st :V–onto→V
37		fofn 6808	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
38	36, 37	ax-mp 5	. . . . 5 ⊢ 1st Fn V
39		xp1st 8007	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
40	21	rnmpo 7542	. . . . . . . 8 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))}
41		simpr 486	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 = (𝑥 + (i · 𝑦)))
42	14	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → (𝑥 + (i · 𝑦)) ∈ ℂ)
43	41, 42	eqeltrd 2834	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 ∈ ℂ)
44	43	ex 414	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ))
45	44	rexlimivv 3200	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
46	45	abssi 4068	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
47	40, 46	eqsstri 4017	. . . . . . 7 ⊢ ran 𝐹 ⊆ ℂ
48		simpl 484	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
49	47, 48	sselid 3981	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
50		simpr 486	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
51	47, 50	sselid 3981	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
52	49, 51	resubd 15163	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧 − 𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
53	32, 35, 38, 39, 52	cnre2csqlem 32890	. . . 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 15054	. . . . . . . . 9 ⊢ ((𝑥 + (i · 𝑦)) ∈ ℂ → (ℑ‘(𝑥 + (i · 𝑦))) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
55	14, 54	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
56		crim 15062	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
57	55, 56	eqtr3d 2775	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
58	57	mpoeq3ia 7487	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59		df-im 15048	. . . . . . . . 9 ⊢ ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
60	59	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61		fvoveq1 7432	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
62	20, 22, 60, 61	fmpoco 8081	. . . . . . 7 ⊢ (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
63	62	mptru 1549	. . . . . 6 ⊢ (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64		df2ndres 31926	. . . . . 6 ⊢ (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
65	58, 63, 64	3eqtr4ri 2772	. . . . 5 ⊢ (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66		fo2nd 7996	. . . . . 6 ⊢ 2nd :V–onto→V
67		fofn 6808	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
69		xp2nd 8008	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
70	49, 51	imsubd 15164	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧 − 𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
71	65, 35, 68, 69, 70	cnre2csqlem 32890	. . . 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 610	. . 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 241	. 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 241	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)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   class class class wbr 5149   ↦ cmpt 5232   × cxp 5675  ^◡ccnv 5676  ran crn 5678   ↾ cres 5679   “ cima 5680   ∘ ccom 5681   Fn wfn 6539  –onto→wfo 6542  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974  ℂcc 11108  ℝcr 11109  ici 11112   + caddc 11113   · cmul 11115   < clt 11248   − cmin 11444   / cdiv 11871  2c2 12267  ℝ⁺crp 12974  (,)cioo 13324  ∗ccj 15043  ℜcre 15044  ℑcim 15045  abscabs 15181

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-ioo 13328  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183

This theorem is referenced by:  tpr2rico  32892