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Theorem cnre2csqima 33872
Description: Image of a centered square by the canonical bijection from (ℝ × ℝ) to . (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 ∈ ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 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.
StepHypRef Expression
1 ioossre 13445 . . 3 (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) ⊆ ℝ
2 ioossre 13445 . . 3 (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)) ⊆ ℝ
3 xpinpreima2 33868 . . . 4 (((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) ⊆ ℝ ∧ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)) ⊆ ℝ) → ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) = (((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∩ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)))))
43eleq2d 2825 . . 3 (((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) ⊆ ℝ ∧ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)) ⊆ ℝ) → (𝑌 ∈ ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) ↔ 𝑌 ∈ (((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∩ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))))))
51, 2, 4mp2an 692 . 2 (𝑌 ∈ ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) ↔ 𝑌 ∈ (((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∩ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)))))
6 elin 3979 . . 3 (𝑌 ∈ (((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∩ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)))) ↔ (𝑌 ∈ ((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∧ 𝑌 ∈ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)))))
7 simpl 482 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
87recnd 11287 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9 ax-icn 11212 . . . . . . . . . . . 12 i ∈ ℂ
109a1i 11 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
11 simpr 484 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1211recnd 11287 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
1310, 12mulcld 11279 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
148, 13addcld 11278 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15 reval 15142 . . . . . . . . 9 ((𝑥 + (i · 𝑦)) ∈ ℂ → (ℜ‘(𝑥 + (i · 𝑦))) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
1614, 15syl 17 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
17 crre 15150 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
1816, 17eqtr3d 2777 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
1918mpoeq3ia 7511 . . . . . 6 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
2014adantl 481 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + (i · 𝑦)) ∈ ℂ)
21 cnre2csqima.1 . . . . . . . . 9 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
2221a1i 11 . . . . . . . 8 (⊤ → 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))))
23 df-re 15136 . . . . . . . . 9 ℜ = (𝑧 ∈ ℂ ↦ ((𝑧 + (∗‘𝑧)) / 2))
2423a1i 11 . . . . . . . 8 (⊤ → ℜ = (𝑧 ∈ ℂ ↦ ((𝑧 + (∗‘𝑧)) / 2)))
25 id 22 . . . . . . . . . 10 (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 = (𝑥 + (i · 𝑦)))
26 fveq2 6907 . . . . . . . . . 10 (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
2725, 26oveq12d 7449 . . . . . . . . 9 (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
2827oveq1d 7446 . . . . . . . 8 (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
2920, 22, 24, 28fmpoco 8119 . . . . . . 7 (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
3029mptru 1544 . . . . . 6 (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31 df1stres 32719 . . . . . 6 (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
3219, 30, 313eqtr4ri 2774 . . . . 5 (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
3314rgen2 3197 . . . . . 6 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
3421fnmpo 8093 . . . . . 6 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
3533, 34ax-mp 5 . . . . 5 𝐹 Fn (ℝ × ℝ)
36 fo1st 8033 . . . . . 6 1st :V–onto→V
37 fofn 6823 . . . . . 6 (1st :V–onto→V → 1st Fn V)
3836, 37ax-mp 5 . . . . 5 1st Fn V
39 xp1st 8045 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
4021rnmpo 7566 . . . . . . . 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)))
4214adantr 480 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → (𝑥 + (i · 𝑦)) ∈ ℂ)
4341, 42eqeltrd 2839 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 ∈ ℂ)
4443ex 412 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ))
4544rexlimivv 3199 . . . . . . . . 9 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
4645abssi 4080 . . . . . . . 8 {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
4740, 46eqsstri 4030 . . . . . . 7 ran 𝐹 ⊆ ℂ
48 simpl 482 . . . . . . 7 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
4947, 48sselid 3993 . . . . . 6 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
50 simpr 484 . . . . . . 7 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
5147, 50sselid 3993 . . . . . 6 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
5249, 51resubd 15252 . . . . 5 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
5332, 35, 38, 39, 52cnre2csqlem 33871 . . . 4 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) → (abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))
54 imval 15143 . . . . . . . . 9 ((𝑥 + (i · 𝑦)) ∈ ℂ → (ℑ‘(𝑥 + (i · 𝑦))) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
5514, 54syl 17 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
56 crim 15151 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
5755, 56eqtr3d 2777 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
5857mpoeq3ia 7511 . . . . . 6 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59 df-im 15137 . . . . . . . . 9 ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
6059a1i 11 . . . . . . . 8 (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61 fvoveq1 7454 . . . . . . . 8 (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
6220, 22, 60, 61fmpoco 8119 . . . . . . 7 (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
6362mptru 1544 . . . . . 6 (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64 df2ndres 32720 . . . . . 6 (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
6558, 63, 643eqtr4ri 2774 . . . . 5 (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66 fo2nd 8034 . . . . . 6 2nd :V–onto→V
67 fofn 6823 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
6866, 67ax-mp 5 . . . . 5 2nd Fn V
69 xp2nd 8046 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
7049, 51imsubd 15253 . . . . 5 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
7165, 35, 68, 69, 70cnre2csqlem 33871 . . . 4 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) → (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))
7253, 71anim12d 609 . . 3 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → ((𝑌 ∈ ((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∧ 𝑌 ∈ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)))) → ((abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷)))
736, 72biimtrid 242 . 2 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) ∩ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)))) → ((abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷)))
745, 73biimtrid 242 1 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wtru 1538  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963   class class class wbr 5148  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cres 5691  cima 5692  ccom 5693   Fn wfn 6558  ontowfo 6561  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  cc 11151  cr 11152  ici 11155   + caddc 11156   · cmul 11158   < clt 11293  cmin 11490   / cdiv 11918  2c2 12319  +crp 13032  (,)cioo 13384  ccj 15132  cre 15133  cim 15134  abscabs 15270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-ioo 13388  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272
This theorem is referenced by:  tpr2rico  33873
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