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Theorem cnre2csqima 31040
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 12788 . . 3 (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) ⊆ ℝ
2 ioossre 12788 . . 3 (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)) ⊆ ℝ
3 xpinpreima2 31036 . . . 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 2903 . . 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 688 . 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 4173 . . 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 483 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
87recnd 10658 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9 ax-icn 10585 . . . . . . . . . . . 12 i ∈ ℂ
109a1i 11 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
11 simpr 485 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1211recnd 10658 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
1310, 12mulcld 10650 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
148, 13addcld 10649 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15 reval 14455 . . . . . . . . 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 14463 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
1816, 17eqtr3d 2863 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
1918mpoeq3ia 7224 . . . . . 6 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
2014adantl 482 . . . . . . . 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 14449 . . . . . . . . 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 6667 . . . . . . . . . 10 (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
2725, 26oveq12d 7166 . . . . . . . . 9 (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
2827oveq1d 7163 . . . . . . . 8 (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
2920, 22, 24, 28fmpoco 7781 . . . . . . 7 (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
3029mptru 1537 . . . . . 6 (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31 df1stres 30352 . . . . . 6 (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
3219, 30, 313eqtr4ri 2860 . . . . 5 (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
3314rgen2a 3234 . . . . . 6 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
3421fnmpo 7758 . . . . . 6 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
3533, 34ax-mp 5 . . . . 5 𝐹 Fn (ℝ × ℝ)
36 fo1st 7700 . . . . . 6 1st :V–onto→V
37 fofn 6589 . . . . . 6 (1st :V–onto→V → 1st Fn V)
3836, 37ax-mp 5 . . . . 5 1st Fn V
39 xp1st 7712 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
4021rnmpo 7274 . . . . . . . 8 ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))}
41 simpr 485 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 = (𝑥 + (i · 𝑦)))
4214adantr 481 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → (𝑥 + (i · 𝑦)) ∈ ℂ)
4341, 42eqeltrd 2918 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 = (𝑥 + (i · 𝑦))) → 𝑧 ∈ ℂ)
4443ex 413 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ))
4544rexlimivv 3297 . . . . . . . . 9 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
4645abssi 4050 . . . . . . . 8 {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
4740, 46eqsstri 4005 . . . . . . 7 ran 𝐹 ⊆ ℂ
48 simpl 483 . . . . . . 7 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
4947, 48sseldi 3969 . . . . . 6 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
50 simpr 485 . . . . . . 7 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
5147, 50sseldi 3969 . . . . . 6 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
5249, 51resubd 14565 . . . . 5 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
5332, 35, 38, 39, 52cnre2csqlem 31039 . . . 4 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) → (abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))
54 imval 14456 . . . . . . . . 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 14464 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
5755, 56eqtr3d 2863 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
5857mpoeq3ia 7224 . . . . . 6 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59 df-im 14450 . . . . . . . . 9 ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
6059a1i 11 . . . . . . . 8 (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61 fvoveq1 7171 . . . . . . . 8 (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
6220, 22, 60, 61fmpoco 7781 . . . . . . 7 (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
6362mptru 1537 . . . . . 6 (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64 df2ndres 30353 . . . . . 6 (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
6558, 63, 643eqtr4ri 2860 . . . . 5 (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66 fo2nd 7701 . . . . . 6 2nd :V–onto→V
67 fofn 6589 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
6866, 67ax-mp 5 . . . . 5 2nd Fn V
69 xp2nd 7713 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
7049, 51imsubd 14566 . . . . 5 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
7165, 35, 68, 69, 70cnre2csqlem 31039 . . . 4 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((2nd ↾ (ℝ × ℝ)) “ (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) → (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))
7253, 71anim12d 608 . . 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, 72syl5bi 243 . 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, 73syl5bi 243 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 207  wa 396  w3a 1081   = wceq 1530  wtru 1531  wcel 2107  {cab 2804  wral 3143  wrex 3144  Vcvv 3500  cin 3939  wss 3940   class class class wbr 5063  cmpt 5143   × cxp 5552  ccnv 5553  ran crn 5555  cres 5556  cima 5557  ccom 5558   Fn wfn 6347  ontowfo 6350  cfv 6352  (class class class)co 7148  cmpo 7150  1st c1st 7678  2nd c2nd 7679  cc 10524  cr 10525  ici 10528   + caddc 10529   · cmul 10531   < clt 10664  cmin 10859   / cdiv 11286  2c2 11681  +crp 12379  (,)cioo 12728  ccj 14445  cre 14446  cim 14447  abscabs 14583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-ioo 12732  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585
This theorem is referenced by:  tpr2rico  31041
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