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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 15085 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
2 | recj 15104 | . . . . 5 ⊢ ((∗‘𝐴) ∈ ℂ → (ℜ‘(∗‘(∗‘𝐴))) = (ℜ‘(∗‘𝐴))) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘(∗‘𝐴))) = (ℜ‘(∗‘𝐴))) |
4 | recj 15104 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) | |
5 | 3, 4 | eqtrd 2768 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘(∗‘𝐴))) = (ℜ‘𝐴)) |
6 | imcj 15112 | . . . . . 6 ⊢ ((∗‘𝐴) ∈ ℂ → (ℑ‘(∗‘(∗‘𝐴))) = -(ℑ‘(∗‘𝐴))) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘(∗‘𝐴))) = -(ℑ‘(∗‘𝐴))) |
8 | imcj 15112 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) | |
9 | 8 | negeqd 11485 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → -(ℑ‘(∗‘𝐴)) = --(ℑ‘𝐴)) |
10 | imcl 15091 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
11 | 10 | recnd 11273 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
12 | 11 | negnegd 11593 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → --(ℑ‘𝐴) = (ℑ‘𝐴)) |
13 | 9, 12 | eqtrd 2768 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -(ℑ‘(∗‘𝐴)) = (ℑ‘𝐴)) |
14 | 7, 13 | eqtrd 2768 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘(∗‘𝐴))) = (ℑ‘𝐴)) |
15 | 14 | oveq2d 7436 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘(∗‘(∗‘𝐴)))) = (i · (ℑ‘𝐴))) |
16 | 5, 15 | oveq12d 7438 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℜ‘(∗‘(∗‘𝐴))) + (i · (ℑ‘(∗‘(∗‘𝐴))))) = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) |
17 | cjcl 15085 | . . 3 ⊢ ((∗‘𝐴) ∈ ℂ → (∗‘(∗‘𝐴)) ∈ ℂ) | |
18 | replim 15096 | . . 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 15096 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
21 | 16, 19, 20 | 3eqtr4d 2778 | 1 ⊢ (𝐴 ∈ ℂ → (∗‘(∗‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 ici 11141 + caddc 11142 · cmul 11144 -cneg 11476 ∗ccj 15076 ℜcre 15077 ℑcim 15078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-cj 15079 df-re 15080 df-im 15081 |
This theorem is referenced by: cjmulrcl 15124 cjreim2 15141 cj11 15142 cjcji 15151 cjcjd 15179 abscj 15259 sqabsadd 15262 sqabssub 15263 cnsrng 21333 plycjlem 26224 dipassr2 30670 his52 30910 cnvbramul 31938 |
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