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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 13027. (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 13027 | . . . . . 6 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
| 3 | f1ocnv 6860 | . . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → ◡𝐹:ℂ–1-1-onto→(ℝ × ℝ)) | |
| 4 | f1of 6848 | . . . . . 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 6977 | . . 3 ⊢ (⊤ → ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧))) |
| 8 | 7 | mptru 1547 | . 2 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) |
| 9 | df-ov 7434 | . . . . . . 7 ⊢ ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | |
| 10 | recl 15149 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ) | |
| 11 | imcl 15150 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ) | |
| 12 | oveq1 7438 | . . . . . . . . 9 ⊢ (𝑥 = (ℜ‘𝑧) → (𝑥 + (i · 𝑦)) = ((ℜ‘𝑧) + (i · 𝑦))) | |
| 13 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑦 = (ℑ‘𝑧) → (i · 𝑦) = (i · (ℑ‘𝑧))) | |
| 14 | 13 | oveq2d 7447 | . . . . . . . . 9 ⊢ (𝑦 = (ℑ‘𝑧) → ((ℜ‘𝑧) + (i · 𝑦)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
| 15 | ovex 7464 | . . . . . . . . 9 ⊢ ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ V | |
| 16 | 12, 14, 1, 15 | ovmpo 7593 | . . . . . . . 8 ⊢ (((ℜ‘𝑧) ∈ ℝ ∧ (ℑ‘𝑧) ∈ ℝ) → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
| 17 | 10, 11, 16 | syl2anc 584 | . . . . . . 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 15155 | . . . . . 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 6910 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = (◡𝐹‘𝑧)) |
| 22 | 10, 11 | opelxpd 5724 | . . . . 5 ⊢ (𝑧 ∈ ℂ → 〈(ℜ‘𝑧), (ℑ‘𝑧)〉 ∈ (ℝ × ℝ)) |
| 23 | f1ocnvfv1 7296 | . . . . 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 587 | . . . 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 5245 | . 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 1540 ⊤wtru 1541 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11153 ℝcr 11154 ici 11157 + caddc 11158 · cmul 11160 ℜcre 15136 ℑcim 15137 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140 |
| This theorem is referenced by: cnrehmeo 24984 cnrehmeoOLD 24985 cnheiborlem 24986 mbfimaopnlem 25690 |
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