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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 12970. (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 12970 | . . . . . 6 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
3 | f1ocnv 6838 | . . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → ◡𝐹:ℂ–1-1-onto→(ℝ × ℝ)) | |
4 | f1of 6826 | . . . . . 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 6953 | . . 3 ⊢ (⊤ → ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧))) |
8 | 7 | mptru 1540 | . 2 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) |
9 | df-ov 7407 | . . . . . . 7 ⊢ ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = (𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) | |
10 | recl 15060 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ) | |
11 | imcl 15061 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ) | |
12 | oveq1 7411 | . . . . . . . . 9 ⊢ (𝑥 = (ℜ‘𝑧) → (𝑥 + (i · 𝑦)) = ((ℜ‘𝑧) + (i · 𝑦))) | |
13 | oveq2 7412 | . . . . . . . . . 10 ⊢ (𝑦 = (ℑ‘𝑧) → (i · 𝑦) = (i · (ℑ‘𝑧))) | |
14 | 13 | oveq2d 7420 | . . . . . . . . 9 ⊢ (𝑦 = (ℑ‘𝑧) → ((ℜ‘𝑧) + (i · 𝑦)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
15 | ovex 7437 | . . . . . . . . 9 ⊢ ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ V | |
16 | 12, 14, 1, 15 | ovmpo 7563 | . . . . . . . 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 2780 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
19 | replim 15066 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) | |
20 | 18, 19 | eqtr4d 2769 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) = 𝑧) |
21 | 20 | fveq2d 6888 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)) = (◡𝐹‘𝑧)) |
22 | 10, 11 | opelxpd 5708 | . . . . 5 ⊢ (𝑧 ∈ ℂ → ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩ ∈ (ℝ × ℝ)) |
23 | f1ocnvfv1 7269 | . . . . 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 2768 | . . 3 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘𝑧) = ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
26 | 25 | mpteq2ia 5244 | . 2 ⊢ (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
27 | 8, 26 | eqtri 2754 | 1 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ⟨cop 4629 ↦ cmpt 5224 × cxp 5667 ◡ccnv 5668 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 ℂcc 11107 ℝcr 11108 ici 11111 + caddc 11112 · cmul 11114 ℜcre 15047 ℑcim 15048 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-2 12276 df-cj 15049 df-re 15050 df-im 15051 |
This theorem is referenced by: cnrehmeo 24828 cnrehmeoOLD 24829 cnheiborlem 24830 mbfimaopnlem 25534 |
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