Theorem cnre2csqima 33924

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 13307	. . 3 ⊢ (((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) ⊆ ℝ
2		ioossre 13307	. . 3 ⊢ (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷)) ⊆ ℝ
3		xpinpreima2 33920	. . . 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 2817	. . 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 3913	. . 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 11140	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9		ax-icn 11065	. . . . . . . . . . . 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 11140	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
13	10, 12	mulcld 11132	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
14	8, 13	addcld 11131	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15		reval 15013	. . . . . . . . 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 15021	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
18	16, 17	eqtr3d 2768	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
19	18	mpoeq3ia 7424	. . . . . 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 15007	. . . . . . . . 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 6822	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
27	25, 26	oveq12d 7364	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
28	27	oveq1d 7361	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
29	20, 22, 24, 28	fmpoco 8025	. . . . . . 7 ⊢ (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
30	29	mptru 1548	. . . . . 6 ⊢ (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31		df1stres 32685	. . . . . 6 ⊢ (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
32	19, 30, 31	3eqtr4ri 2765	. . . . 5 ⊢ (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
33	14	rgen2 3172	. . . . . 6 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
34	21	fnmpo 8001	. . . . . 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 7941	. . . . . 6 ⊢ 1st :V–onto→V
37		fofn 6737	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
38	36, 37	ax-mp 5	. . . . 5 ⊢ 1st Fn V
39		xp1st 7953	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
40	21	rnmpo 7479	. . . . . . . 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 2831	. . . . . . . . . . 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 3174	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
46	45	abssi 4015	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
47	40, 46	eqsstri 3976	. . . . . . 7 ⊢ ran 𝐹 ⊆ ℂ
48		simpl 482	. . . . . . 7 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
49	47, 48	sselid 3927	. . . . . 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 3927	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
52	49, 51	resubd 15123	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧 − 𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
53	32, 35, 38, 39, 52	cnre2csqlem 33923	. . . 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 15014	. . . . . . . . 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 15022	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
57	55, 56	eqtr3d 2768	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
58	57	mpoeq3ia 7424	. . . . . 6 ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59		df-im 15008	. . . . . . . . 9 ⊢ ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
60	59	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61		fvoveq1 7369	. . . . . . . 8 ⊢ (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
62	20, 22, 60, 61	fmpoco 8025	. . . . . . 7 ⊢ (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
63	62	mptru 1548	. . . . . 6 ⊢ (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64		df2ndres 32686	. . . . . 6 ⊢ (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
65	58, 63, 64	3eqtr4ri 2765	. . . . 5 ⊢ (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66		fo2nd 7942	. . . . . 6 ⊢ 2nd :V–onto→V
67		fofn 6737	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
69		xp2nd 7954	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
70	49, 51	imsubd 15124	. . . . 5 ⊢ ((𝑧 ∈ ran 𝐹 ∧ 𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧 − 𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
71	65, 35, 68, 69, 70	cnre2csqlem 33923	. . . 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 1541  ⊤wtru 1542   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897   class class class wbr 5089   ↦ cmpt 5170   × cxp 5612  ^◡ccnv 5613  ran crn 5615   ↾ cres 5616   “ cima 5617   ∘ ccom 5618   Fn wfn 6476  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  ℂcc 11004  ℝcr 11005  ici 11008   + caddc 11009   · cmul 11011   < clt 11146   − cmin 11344   / cdiv 11774  2c2 12180  ℝ⁺crp 12890  (,)cioo 13245  ∗ccj 15003  ℜcre 15004  ℑcim 15005  abscabs 15141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-ioo 13249  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143

This theorem is referenced by:  tpr2rico  33925