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Mirrors > Home > ILE Home > Th. List > cnrecnv | GIF version |
Description: The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 9469. (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 9469 | . . . . . 6 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
3 | f1ocnv 5388 | . . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → ◡𝐹:ℂ–1-1-onto→(ℝ × ℝ)) | |
4 | f1of 5375 | . . . . . 6 ⊢ (◡𝐹:ℂ–1-1-onto→(ℝ × ℝ) → ◡𝐹:ℂ⟶(ℝ × ℝ)) | |
5 | 2, 3, 4 | mp2b 8 | . . . . 5 ⊢ ◡𝐹:ℂ⟶(ℝ × ℝ) |
6 | 5 | a1i 9 | . . . 4 ⊢ (⊤ → ◡𝐹:ℂ⟶(ℝ × ℝ)) |
7 | 6 | feqmptd 5482 | . . 3 ⊢ (⊤ → ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧))) |
8 | 7 | mptru 1341 | . 2 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) |
9 | df-ov 5785 | . . . . . . 7 ⊢ ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | |
10 | recl 10657 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ) | |
11 | imcl 10658 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ) | |
12 | 10 | recnd 7818 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ) |
13 | ax-icn 7739 | . . . . . . . . . . 11 ⊢ i ∈ ℂ | |
14 | 13 | a1i 9 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ) |
15 | 11 | recnd 7818 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ) |
16 | 14, 15 | mulcld 7810 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (i · (ℑ‘𝑧)) ∈ ℂ) |
17 | 12, 16 | addcld 7809 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ ℂ) |
18 | oveq1 5789 | . . . . . . . . 9 ⊢ (𝑥 = (ℜ‘𝑧) → (𝑥 + (i · 𝑦)) = ((ℜ‘𝑧) + (i · 𝑦))) | |
19 | oveq2 5790 | . . . . . . . . . 10 ⊢ (𝑦 = (ℑ‘𝑧) → (i · 𝑦) = (i · (ℑ‘𝑧))) | |
20 | 19 | oveq2d 5798 | . . . . . . . . 9 ⊢ (𝑦 = (ℑ‘𝑧) → ((ℜ‘𝑧) + (i · 𝑦)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
21 | 18, 20, 1 | ovmpog 5913 | . . . . . . . 8 ⊢ (((ℜ‘𝑧) ∈ ℝ ∧ (ℑ‘𝑧) ∈ ℝ ∧ ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ ℂ) → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
22 | 10, 11, 17, 21 | syl3anc 1217 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
23 | 9, 22 | syl5eqr 2187 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
24 | replim 10663 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) | |
25 | 23, 24 | eqtr4d 2176 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) = 𝑧) |
26 | 25 | fveq2d 5433 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = (◡𝐹‘𝑧)) |
27 | opelxpi 4579 | . . . . . 6 ⊢ (((ℜ‘𝑧) ∈ ℝ ∧ (ℑ‘𝑧) ∈ ℝ) → 〈(ℜ‘𝑧), (ℑ‘𝑧)〉 ∈ (ℝ × ℝ)) | |
28 | 10, 11, 27 | syl2anc 409 | . . . . 5 ⊢ (𝑧 ∈ ℂ → 〈(ℜ‘𝑧), (ℑ‘𝑧)〉 ∈ (ℝ × ℝ)) |
29 | f1ocnvfv1 5686 | . . . . 5 ⊢ ((𝐹:(ℝ × ℝ)–1-1-onto→ℂ ∧ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉 ∈ (ℝ × ℝ)) → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | |
30 | 2, 28, 29 | sylancr 411 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
31 | 26, 30 | eqtr3d 2175 | . . 3 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘𝑧) = 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
32 | 31 | mpteq2ia 4022 | . 2 ⊢ (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
33 | 8, 32 | eqtri 2161 | 1 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ⊤wtru 1333 ∈ wcel 1481 〈cop 3535 ↦ cmpt 3997 × cxp 4545 ◡ccnv 4546 ⟶wf 5127 –1-1-onto→wf1o 5130 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 ℂcc 7642 ℝcr 7643 ici 7646 + caddc 7647 · cmul 7649 ℜcre 10644 ℑcim 10645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-2 8803 df-cj 10646 df-re 10647 df-im 10648 |
This theorem is referenced by: cnrehmeocntop 12801 |
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