Theorem cnre2csqima 34043

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 13349	. . 3 ⊢ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ
2		ioossre 13349	. . 3 ⊢ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ
3		xpinpreima2 34039	. . . 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 693	. 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 3901	. . 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 11162	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9		ax-icn 11086	. . . . . . . . . . . 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 11162	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
13	10, 12	mulcld 11154	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
14	8, 13	addcld 11153	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15		reval 15057	. . . . . . . . 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 15065	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
18	16, 17	eqtr3d 2772	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
19	18	mpoeq3ia 7434	. . . . . 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 15051	. . . . . . . . 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 6829	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
27	25, 26	oveq12d 7374	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
28	27	oveq1d 7371	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
29	20, 22, 24, 28	fmpoco 8034	. . . . . . 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 32765	. . . . . 6 ⊢ (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
32	19, 30, 31	3eqtr4ri 2769	. . . . 5 ⊢ (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
33	14	rgen2 3175	. . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
34	21	fnmpo 8011	. . . . . 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 7951	. . . . . 6 ⊢ 1st :V–onto→V
37		fofn 6743	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
38	36, 37	ax-mp 5	. . . . 5 ⊢ 1st Fn V
39		xp1st 7963	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
40	21	rnmpo 7489	. . . . . . . 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 3177	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
46	45	abssi 4001	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
47	40, 46	eqsstri 3963	. . . . . . 7 ⊢ ran 𝐹 ⊆ ℂ
48		simpl 482	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
49	47, 48	sselid 3915	. . . . . 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 3915	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
52	49, 51	resubd 15167	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧 − 𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
53	32, 35, 38, 39, 52	cnre2csqlem 34042	. . . 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 15058	. . . . . . . . 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 15066	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
57	55, 56	eqtr3d 2772	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
58	57	mpoeq3ia 7434	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59		df-im 15052	. . . . . . . . 9 ⊢ ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
60	59	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61		fvoveq1 7379	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
62	20, 22, 60, 61	fmpoco 8034	. . . . . . 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 32766	. . . . . 6 ⊢ (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
65	58, 63, 64	3eqtr4ri 2769	. . . . 5 ⊢ (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66		fo2nd 7952	. . . . . 6 ⊢ 2nd :V–onto→V
67		fofn 6743	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
69		xp2nd 7964	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
70	49, 51	imsubd 15168	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧 − 𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
71	65, 35, 68, 69, 70	cnre2csqlem 34042	. . . 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 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 1087   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  {cab 2713  ∀wral 3049  ∃wrex 3059  Vcvv 3427   ∩ cin 3884   ⊆ wss 3885   class class class wbr 5074   ↦ cmpt 5155   × cxp 5618  ^◡ccnv 5619  ran crn 5621   ↾ cres 5622   “ cima 5623   ∘ ccom 5624   Fn wfn 6482  –onto→wfo 6485  ‘cfv 6487  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11025  ℝcr 11026  ici 11029   + caddc 11030   · cmul 11032   < clt 11168   − cmin 11366   / cdiv 11796  2c2 12225  ℝ⁺crp 12931  (,)cioo 13287  ∗ccj 15047  ℜcre 15048  ℑcim 15049  abscabs 15185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-sup 9344  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12164  df-2 12233  df-3 12234  df-n0 12427  df-z 12514  df-uz 12778  df-rp 12932  df-ioo 13291  df-seq 13953  df-exp 14013  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187

This theorem is referenced by:  tpr2rico  34044