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Mirrors > Home > MPE Home > Th. List > recj | Structured version Visualization version GIF version |
Description: Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
recj | ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recl 14307 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 10522 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
3 | ax-icn 10449 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | imcl 14308 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
5 | 4 | recnd 10522 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
6 | mulcl 10474 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
7 | 3, 5, 6 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
8 | 2, 7 | negsubd 10857 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + -(i · (ℑ‘𝐴))) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) |
9 | mulneg2 10931 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) | |
10 | 3, 5, 9 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) |
11 | 10 | oveq2d 7039 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + (i · -(ℑ‘𝐴))) = ((ℜ‘𝐴) + -(i · (ℑ‘𝐴)))) |
12 | remim 14314 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) | |
13 | 8, 11, 12 | 3eqtr4rd 2844 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) |
14 | 13 | fveq2d 6549 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘((ℜ‘𝐴) + (i · -(ℑ‘𝐴))))) |
15 | 4 | renegcld 10921 | . . 3 ⊢ (𝐴 ∈ ℂ → -(ℑ‘𝐴) ∈ ℝ) |
16 | crre 14311 | . . 3 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ -(ℑ‘𝐴) ∈ ℝ) → (ℜ‘((ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) = (ℜ‘𝐴)) | |
17 | 1, 15, 16 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘((ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) = (ℜ‘𝐴)) |
18 | 14, 17 | eqtrd 2833 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 ici 10392 + caddc 10393 · cmul 10395 − cmin 10723 -cneg 10724 ∗ccj 14293 ℜcre 14294 ℑcim 14295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-2 11554 df-cj 14296 df-re 14297 df-im 14298 |
This theorem is referenced by: cjcj 14337 ipcnval 14340 recji 14372 recjd 14401 |
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