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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 12965. (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 12965 | . . . . . 6 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
3 | f1ocnv 6835 | . . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → ◡𝐹:ℂ–1-1-onto→(ℝ × ℝ)) | |
4 | f1of 6823 | . . . . . 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 6950 | . . 3 ⊢ (⊤ → ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧))) |
8 | 7 | mptru 1540 | . 2 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) |
9 | df-ov 7404 | . . . . . . 7 ⊢ ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | |
10 | recl 15053 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ) | |
11 | imcl 15054 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ) | |
12 | oveq1 7408 | . . . . . . . . 9 ⊢ (𝑥 = (ℜ‘𝑧) → (𝑥 + (i · 𝑦)) = ((ℜ‘𝑧) + (i · 𝑦))) | |
13 | oveq2 7409 | . . . . . . . . . 10 ⊢ (𝑦 = (ℑ‘𝑧) → (i · 𝑦) = (i · (ℑ‘𝑧))) | |
14 | 13 | oveq2d 7417 | . . . . . . . . 9 ⊢ (𝑦 = (ℑ‘𝑧) → ((ℜ‘𝑧) + (i · 𝑦)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
15 | ovex 7434 | . . . . . . . . 9 ⊢ ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ V | |
16 | 12, 14, 1, 15 | ovmpo 7560 | . . . . . . . 8 ⊢ (((ℜ‘𝑧) ∈ ℝ ∧ (ℑ‘𝑧) ∈ ℝ) → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
17 | 10, 11, 16 | syl2anc 583 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
18 | 9, 17 | eqtr3id 2778 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
19 | replim 15059 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) | |
20 | 18, 19 | eqtr4d 2767 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) = 𝑧) |
21 | 20 | fveq2d 6885 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = (◡𝐹‘𝑧)) |
22 | 10, 11 | opelxpd 5705 | . . . . 5 ⊢ (𝑧 ∈ ℂ → 〈(ℜ‘𝑧), (ℑ‘𝑧)〉 ∈ (ℝ × ℝ)) |
23 | f1ocnvfv1 7266 | . . . . 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 586 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
25 | 21, 24 | eqtr3d 2766 | . . 3 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘𝑧) = 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
26 | 25 | mpteq2ia 5241 | . 2 ⊢ (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
27 | 8, 26 | eqtri 2752 | 1 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 〈cop 4626 ↦ cmpt 5221 × cxp 5664 ◡ccnv 5665 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 ∈ cmpo 7403 ℂcc 11103 ℝcr 11104 ici 11107 + caddc 11108 · cmul 11110 ℜcre 15040 ℑcim 15041 |
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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-cj 15042 df-re 15043 df-im 15044 |
This theorem is referenced by: cnrehmeo 24788 cnrehmeoOLD 24789 cnheiborlem 24790 mbfimaopnlem 25494 |
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