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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 14463 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 10663 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
3 | ax-icn 10590 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | imcl 14464 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
5 | 4 | recnd 10663 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
6 | mulcl 10615 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
7 | 3, 5, 6 | sylancr 589 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
8 | 2, 7 | negsubd 10997 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + -(i · (ℑ‘𝐴))) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) |
9 | mulneg2 11071 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) | |
10 | 3, 5, 9 | sylancr 589 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) |
11 | 10 | oveq2d 7166 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + (i · -(ℑ‘𝐴))) = ((ℜ‘𝐴) + -(i · (ℑ‘𝐴)))) |
12 | remim 14470 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) − (i · (ℑ‘𝐴)))) | |
13 | 8, 11, 12 | 3eqtr4rd 2867 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) = ((ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) |
14 | 13 | fveq2d 6668 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘((ℜ‘𝐴) + (i · -(ℑ‘𝐴))))) |
15 | 4 | renegcld 11061 | . . 3 ⊢ (𝐴 ∈ ℂ → -(ℑ‘𝐴) ∈ ℝ) |
16 | crre 14467 | . . 3 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ -(ℑ‘𝐴) ∈ ℝ) → (ℜ‘((ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) = (ℜ‘𝐴)) | |
17 | 1, 15, 16 | syl2anc 586 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘((ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) = (ℜ‘𝐴)) |
18 | 14, 17 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘(∗‘𝐴)) = (ℜ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 ici 10533 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 ∗ccj 14449 ℜcre 14450 ℑcim 14451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-2 11694 df-cj 14452 df-re 14453 df-im 14454 |
This theorem is referenced by: cjcj 14493 ipcnval 14496 recji 14528 recjd 14557 |
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