Theorem cnre2csqima 33947

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 13429	. . 3 ⊢ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ
2		ioossre 13429	. . 3 ⊢ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ
3		xpinpreima2 33943	. . . 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 2821	. . 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 692	. 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 3947	. . 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 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
8	7	recnd 11268	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9		ax-icn 11193	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
10	9	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
11		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
12	11	recnd 11268	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
13	10, 12	mulcld 11260	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
14	8, 13	addcld 11259	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15		reval 15130	. . . . . . . . 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 15138	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
18	16, 17	eqtr3d 2773	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
19	18	mpoeq3ia 7490	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
20	14	adantl 481	. . . . . . . 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 15124	. . . . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
27	25, 26	oveq12d 7428	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
28	27	oveq1d 7425	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
29	20, 22, 24, 28	fmpoco 8099	. . . . . . 7 ⊢ (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
30	29	mptru 1547	. . . . . 6 ⊢ (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31		df1stres 32686	. . . . . 6 ⊢ (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
32	19, 30, 31	3eqtr4ri 2770	. . . . 5 ⊢ (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
33	14	rgen2 3185	. . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
34	21	fnmpo 8073	. . . . . 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 8013	. . . . . 6 ⊢ 1st :V–onto→V
37		fofn 6797	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
38	36, 37	ax-mp 5	. . . . 5 ⊢ 1st Fn V
39		xp1st 8025	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
40	21	rnmpo 7545	. . . . . . . 8 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))}
41		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 = (𝑥 + (i · 𝑦)))
42	14	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → (𝑥 + (i · 𝑦)) ∈ ℂ)
43	41, 42	eqeltrd 2835	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 ∈ ℂ)
44	43	ex 412	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ))
45	44	rexlimivv 3187	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
46	45	abssi 4050	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
47	40, 46	eqsstri 4010	. . . . . . 7 ⊢ ran 𝐹 ⊆ ℂ
48		simpl 482	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
49	47, 48	sselid 3961	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
50		simpr 484	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
51	47, 50	sselid 3961	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
52	49, 51	resubd 15240	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧 − 𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
53	32, 35, 38, 39, 52	cnre2csqlem 33946	. . . 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 15131	. . . . . . . . 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 15139	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
57	55, 56	eqtr3d 2773	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
58	57	mpoeq3ia 7490	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59		df-im 15125	. . . . . . . . 9 ⊢ ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
60	59	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61		fvoveq1 7433	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
62	20, 22, 60, 61	fmpoco 8099	. . . . . . 7 ⊢ (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
63	62	mptru 1547	. . . . . 6 ⊢ (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64		df2ndres 32687	. . . . . 6 ⊢ (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
65	58, 63, 64	3eqtr4ri 2770	. . . . 5 ⊢ (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66		fo2nd 8014	. . . . . 6 ⊢ 2nd :V–onto→V
67		fofn 6797	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
69		xp2nd 8026	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
70	49, 51	imsubd 15241	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧 − 𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
71	65, 35, 68, 69, 70	cnre2csqlem 33946	. . . 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 609	. . 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 242	. 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 242	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931   class class class wbr 5124   ↦ cmpt 5206   × cxp 5657  ^◡ccnv 5658  ran crn 5660   ↾ cres 5661   “ cima 5662   ∘ ccom 5663   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7991  2^nd c2nd 7992  ℂcc 11132  ℝcr 11133  ici 11136   + caddc 11137   · cmul 11139   < clt 11274   − cmin 11471   / cdiv 11899  2c2 12300  ℝ⁺crp 13013  (,)cioo 13367  ∗ccj 15120  ℜcre 15121  ℑcim 15122  abscabs 15258

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9459  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-ioo 13371  df-seq 14025  df-exp 14085  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260

This theorem is referenced by:  tpr2rico  33948