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Mirrors > Home > MPE Home > Th. List > cjcj | Structured version Visualization version GIF version |
Description: The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
cjcj | ⊢ (𝐴 ∈ ℂ → (∗‘(∗‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjcl 14456 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
2 | recj 14475 | . . . . 5 ⊢ ((∗‘𝐴) ∈ ℂ → (ℜ‘(∗‘(∗‘𝐴))) = (ℜ‘(∗‘𝐴))) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘(∗‘𝐴))) = (ℜ‘(∗‘𝐴))) |
4 | recj 14475 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) | |
5 | 3, 4 | eqtrd 2833 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘(∗‘𝐴))) = (ℜ‘𝐴)) |
6 | imcj 14483 | . . . . . 6 ⊢ ((∗‘𝐴) ∈ ℂ → (ℑ‘(∗‘(∗‘𝐴))) = -(ℑ‘(∗‘𝐴))) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘(∗‘𝐴))) = -(ℑ‘(∗‘𝐴))) |
8 | imcj 14483 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) | |
9 | 8 | negeqd 10869 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → -(ℑ‘(∗‘𝐴)) = --(ℑ‘𝐴)) |
10 | imcl 14462 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
11 | 10 | recnd 10658 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
12 | 11 | negnegd 10977 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → --(ℑ‘𝐴) = (ℑ‘𝐴)) |
13 | 9, 12 | eqtrd 2833 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -(ℑ‘(∗‘𝐴)) = (ℑ‘𝐴)) |
14 | 7, 13 | eqtrd 2833 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘(∗‘𝐴))) = (ℑ‘𝐴)) |
15 | 14 | oveq2d 7151 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘(∗‘(∗‘𝐴)))) = (i · (ℑ‘𝐴))) |
16 | 5, 15 | oveq12d 7153 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℜ‘(∗‘(∗‘𝐴))) + (i · (ℑ‘(∗‘(∗‘𝐴))))) = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) |
17 | cjcl 14456 | . . 3 ⊢ ((∗‘𝐴) ∈ ℂ → (∗‘(∗‘𝐴)) ∈ ℂ) | |
18 | replim 14467 | . . 3 ⊢ ((∗‘(∗‘𝐴)) ∈ ℂ → (∗‘(∗‘𝐴)) = ((ℜ‘(∗‘(∗‘𝐴))) + (i · (ℑ‘(∗‘(∗‘𝐴)))))) | |
19 | 1, 17, 18 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ℂ → (∗‘(∗‘𝐴)) = ((ℜ‘(∗‘(∗‘𝐴))) + (i · (ℑ‘(∗‘(∗‘𝐴)))))) |
20 | replim 14467 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
21 | 16, 19, 20 | 3eqtr4d 2843 | 1 ⊢ (𝐴 ∈ ℂ → (∗‘(∗‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ici 10528 + caddc 10529 · cmul 10531 -cneg 10860 ∗ccj 14447 ℜcre 14448 ℑcim 14449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-cj 14450 df-re 14451 df-im 14452 |
This theorem is referenced by: cjmulrcl 14495 cjreim2 14512 cj11 14513 cjcji 14522 cjcjd 14550 abscj 14631 sqabsadd 14634 sqabssub 14635 cnsrng 20125 plycjlem 24873 dipassr2 28630 his52 28870 cnvbramul 29898 |
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