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Theorem cnre2csqima 31040
 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 ∈ ((((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  –onto→wfo 6350  ‘cfv 6352  (class class class)co 7148   ∈ cmpo 7150  1st c1st 7678  2nd c2nd 7679  ℂcc 10524  ℝcr 10525  ici 10528   + caddc 10529   · cmul 10531   < clt 10664   − cmin 10859   / cdiv 11286  2c2 11681  ℝ+crp 12379  (,)cioo 12728  ∗ccj 14445  ℜcre 14446  ℑcim 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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