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Mirrors > Home > MPE Home > Th. List > cnrecnv | Structured version Visualization version GIF version |
Description: The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 12917. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
cnrecnv.1 | ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) |
Ref | Expression |
---|---|
cnrecnv | ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnrecnv.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) | |
2 | 1 | cnref1o 12917 | . . . . . 6 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
3 | f1ocnv 6801 | . . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → ◡𝐹:ℂ–1-1-onto→(ℝ × ℝ)) | |
4 | f1of 6789 | . . . . . 6 ⊢ (◡𝐹:ℂ–1-1-onto→(ℝ × ℝ) → ◡𝐹:ℂ⟶(ℝ × ℝ)) | |
5 | 2, 3, 4 | mp2b 10 | . . . . 5 ⊢ ◡𝐹:ℂ⟶(ℝ × ℝ) |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → ◡𝐹:ℂ⟶(ℝ × ℝ)) |
7 | 6 | feqmptd 6915 | . . 3 ⊢ (⊤ → ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧))) |
8 | 7 | mptru 1549 | . 2 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) |
9 | df-ov 7365 | . . . . . . 7 ⊢ ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = (𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) | |
10 | recl 15002 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ) | |
11 | imcl 15003 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ) | |
12 | oveq1 7369 | . . . . . . . . 9 ⊢ (𝑥 = (ℜ‘𝑧) → (𝑥 + (i · 𝑦)) = ((ℜ‘𝑧) + (i · 𝑦))) | |
13 | oveq2 7370 | . . . . . . . . . 10 ⊢ (𝑦 = (ℑ‘𝑧) → (i · 𝑦) = (i · (ℑ‘𝑧))) | |
14 | 13 | oveq2d 7378 | . . . . . . . . 9 ⊢ (𝑦 = (ℑ‘𝑧) → ((ℜ‘𝑧) + (i · 𝑦)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
15 | ovex 7395 | . . . . . . . . 9 ⊢ ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ V | |
16 | 12, 14, 1, 15 | ovmpo 7520 | . . . . . . . 8 ⊢ (((ℜ‘𝑧) ∈ ℝ ∧ (ℑ‘𝑧) ∈ ℝ) → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
17 | 10, 11, 16 | syl2anc 585 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
18 | 9, 17 | eqtr3id 2791 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
19 | replim 15008 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) | |
20 | 18, 19 | eqtr4d 2780 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) = 𝑧) |
21 | 20 | fveq2d 6851 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)) = (◡𝐹‘𝑧)) |
22 | 10, 11 | opelxpd 5676 | . . . . 5 ⊢ (𝑧 ∈ ℂ → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ ∈ (ℝ × ℝ)) |
23 | f1ocnvfv1 7227 | . . . . 5 ⊢ ((𝐹:(ℝ × ℝ)–1-1-onto→ℂ ∧ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ ∈ (ℝ × ℝ)) → (◡𝐹‘(𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)) = ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) | |
24 | 2, 22, 23 | sylancr 588 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)) = ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
25 | 21, 24 | eqtr3d 2779 | . . 3 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘𝑧) = ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
26 | 25 | mpteq2ia 5213 | . 2 ⊢ (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
27 | 8, 26 | eqtri 2765 | 1 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ⟨cop 4597 ↦ cmpt 5193 × cxp 5636 ◡ccnv 5637 ⟶wf 6497 –1-1-onto→wf1o 6500 ‘cfv 6501 (class class class)co 7362 ∈ cmpo 7364 ℂcc 11056 ℝcr 11057 ici 11060 + caddc 11061 · cmul 11063 ℜcre 14989 ℑcim 14990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-2 12223 df-cj 14991 df-re 14992 df-im 14993 |
This theorem is referenced by: cnrehmeo 24332 cnheiborlem 24333 mbfimaopnlem 25035 |
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