Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnrecnv | GIF version |
Description: The inverse to the canonical bijection from (ℝ × ℝ) to ℂ from cnref1o 9582. (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 9582 | . . . . . 6 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ |
3 | f1ocnv 5442 | . . . . . 6 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ → ◡𝐹:ℂ–1-1-onto→(ℝ × ℝ)) | |
4 | f1of 5429 | . . . . . 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 5536 | . . 3 ⊢ (⊤ → ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧))) |
8 | 7 | mptru 1351 | . 2 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) |
9 | df-ov 5842 | . . . . . . 7 ⊢ ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) | |
10 | recl 10789 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ) | |
11 | imcl 10790 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ) | |
12 | 10 | recnd 7921 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ) |
13 | ax-icn 7842 | . . . . . . . . . . 11 ⊢ i ∈ ℂ | |
14 | 13 | a1i 9 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ) |
15 | 11 | recnd 7921 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ) |
16 | 14, 15 | mulcld 7913 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (i · (ℑ‘𝑧)) ∈ ℂ) |
17 | 12, 16 | addcld 7912 | . . . . . . . 8 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) + (i · (ℑ‘𝑧))) ∈ ℂ) |
18 | oveq1 5846 | . . . . . . . . 9 ⊢ (𝑥 = (ℜ‘𝑧) → (𝑥 + (i · 𝑦)) = ((ℜ‘𝑧) + (i · 𝑦))) | |
19 | oveq2 5847 | . . . . . . . . . 10 ⊢ (𝑦 = (ℑ‘𝑧) → (i · 𝑦) = (i · (ℑ‘𝑧))) | |
20 | 19 | oveq2d 5855 | . . . . . . . . 9 ⊢ (𝑦 = (ℑ‘𝑧) → ((ℜ‘𝑧) + (i · 𝑦)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
21 | 18, 20, 1 | ovmpog 5970 | . . . . . . . 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 1227 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧)𝐹(ℑ‘𝑧)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
23 | 9, 22 | eqtr3id 2211 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) |
24 | replim 10795 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧)))) | |
25 | 23, 24 | eqtr4d 2200 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉) = 𝑧) |
26 | 25 | fveq2d 5487 | . . . 4 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘(𝐹‘〈(ℜ‘𝑧), (ℑ‘𝑧)〉)) = (◡𝐹‘𝑧)) |
27 | opelxpi 4633 | . . . . . 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 5742 | . . . . 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 2199 | . . 3 ⊢ (𝑧 ∈ ℂ → (◡𝐹‘𝑧) = 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
32 | 31 | mpteq2ia 4065 | . 2 ⊢ (𝑧 ∈ ℂ ↦ (◡𝐹‘𝑧)) = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
33 | 8, 32 | eqtri 2185 | 1 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ⊤wtru 1343 ∈ wcel 2135 〈cop 3576 ↦ cmpt 4040 × cxp 4599 ◡ccnv 4600 ⟶wf 5181 –1-1-onto→wf1o 5184 ‘cfv 5185 (class class class)co 5839 ∈ cmpo 5841 ℂcc 7745 ℝcr 7746 ici 7749 + caddc 7750 · cmul 7752 ℜcre 10776 ℑcim 10777 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-id 4268 df-po 4271 df-iso 4272 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-2 8910 df-cj 10778 df-re 10779 df-im 10780 |
This theorem is referenced by: cnrehmeocntop 13191 |
Copyright terms: Public domain | W3C validator |