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Theorem cnre2csqima 31053
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 12786 . . 3 (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) ⊆ ℝ
2 ioossre 12786 . . 3 (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷)) ⊆ ℝ
3 xpinpreima2 31049 . . . 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 2895 . . 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 4166 . . 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 10657 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
9 ax-icn 10584 . . . . . . . . . . . 12 i ∈ ℂ
109a1i 11 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
11 simpr 485 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
1211recnd 10657 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
1310, 12mulcld 10649 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
148, 13addcld 10648 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
15 reval 14453 . . . . . . . . 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 14461 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘(𝑥 + (i · 𝑦))) = 𝑥)
1816, 17eqtr3d 2855 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2) = 𝑥)
1918mpoeq3ia 7221 . . . . . 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 14447 . . . . . . . . 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 6663 . . . . . . . . . 10 (𝑧 = (𝑥 + (i · 𝑦)) → (∗‘𝑧) = (∗‘(𝑥 + (i · 𝑦))))
2725, 26oveq12d 7163 . . . . . . . . 9 (𝑧 = (𝑥 + (i · 𝑦)) → (𝑧 + (∗‘𝑧)) = ((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))))
2827oveq1d 7160 . . . . . . . 8 (𝑧 = (𝑥 + (i · 𝑦)) → ((𝑧 + (∗‘𝑧)) / 2) = (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
2920, 22, 24, 28fmpoco 7779 . . . . . . 7 (⊤ → (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2)))
3029mptru 1535 . . . . . 6 (ℜ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (((𝑥 + (i · 𝑦)) + (∗‘(𝑥 + (i · 𝑦)))) / 2))
31 df1stres 30365 . . . . . 6 (1st ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥)
3219, 30, 313eqtr4ri 2852 . . . . 5 (1st ↾ (ℝ × ℝ)) = (ℜ ∘ 𝐹)
3314rgen2 3200 . . . . . 6 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
3421fnmpo 7756 . . . . . 6 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
3533, 34ax-mp 5 . . . . 5 𝐹 Fn (ℝ × ℝ)
36 fo1st 7698 . . . . . 6 1st :V–onto→V
37 fofn 6585 . . . . . 6 (1st :V–onto→V → 1st Fn V)
3836, 37ax-mp 5 . . . . 5 1st Fn V
39 xp1st 7710 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
4021rnmpo 7273 . . . . . . . 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 2910 . . . . . . . . . . 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 3289 . . . . . . . . 9 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦)) → 𝑧 ∈ ℂ)
4645abssi 4043 . . . . . . . 8 {𝑧 ∣ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = (𝑥 + (i · 𝑦))} ⊆ ℂ
4740, 46eqsstri 3998 . . . . . . 7 ran 𝐹 ⊆ ℂ
48 simpl 483 . . . . . . 7 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑧 ∈ ran 𝐹)
4947, 48sseldi 3962 . . . . . 6 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
50 simpr 485 . . . . . . 7 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
5147, 50sseldi 3962 . . . . . 6 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → 𝑢 ∈ ℂ)
5249, 51resubd 14563 . . . . 5 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → (ℜ‘(𝑧𝑢)) = ((ℜ‘𝑧) − (ℜ‘𝑢)))
5332, 35, 38, 39, 52cnre2csqlem 31052 . . . 4 ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((1st ↾ (ℝ × ℝ)) “ (((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷))) → (abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))
54 imval 14454 . . . . . . . . 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 14462 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℑ‘(𝑥 + (i · 𝑦))) = 𝑦)
5755, 56eqtr3d 2855 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (ℜ‘((𝑥 + (i · 𝑦)) / i)) = 𝑦)
5857mpoeq3ia 7221 . . . . . 6 (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
59 df-im 14448 . . . . . . . . 9 ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i)))
6059a1i 11 . . . . . . . 8 (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℜ‘(𝑧 / i))))
61 fvoveq1 7168 . . . . . . . 8 (𝑧 = (𝑥 + (i · 𝑦)) → (ℜ‘(𝑧 / i)) = (ℜ‘((𝑥 + (i · 𝑦)) / i)))
6220, 22, 60, 61fmpoco 7779 . . . . . . 7 (⊤ → (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i))))
6362mptru 1535 . . . . . 6 (ℑ ∘ 𝐹) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (ℜ‘((𝑥 + (i · 𝑦)) / i)))
64 df2ndres 30366 . . . . . 6 (2nd ↾ (ℝ × ℝ)) = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦)
6558, 63, 643eqtr4ri 2852 . . . . 5 (2nd ↾ (ℝ × ℝ)) = (ℑ ∘ 𝐹)
66 fo2nd 7699 . . . . . 6 2nd :V–onto→V
67 fofn 6585 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
6866, 67ax-mp 5 . . . . 5 2nd Fn V
69 xp2nd 7711 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
7049, 51imsubd 14564 . . . . 5 ((𝑧 ∈ ran 𝐹𝑢 ∈ ran 𝐹) → (ℑ‘(𝑧𝑢)) = ((ℑ‘𝑧) − (ℑ‘𝑢)))
7165, 35, 68, 69, 70cnre2csqlem 31052 . . . 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 1079   = wceq 1528  wtru 1529  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  cin 3932  wss 3933   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  ran crn 5549  cres 5550  cima 5551  ccom 5552   Fn wfn 6343  ontowfo 6346  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  cc 10523  cr 10524  ici 10527   + caddc 10528   · cmul 10530   < clt 10663  cmin 10858   / cdiv 11285  2c2 11680  +crp 12377  (,)cioo 12726  ccj 14443  cre 14444  cim 14445  abscabs 14581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ioo 12730  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583
This theorem is referenced by:  tpr2rico  31054
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