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Mirrors > Home > ILE Home > Th. List > addcj | GIF version |
Description: A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) (Revised by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
addcj | ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reval 10589 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | 1 | oveq2d 5758 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · (ℜ‘𝐴)) = (2 · ((𝐴 + (∗‘𝐴)) / 2))) |
3 | cjcl 10588 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | addcl 7713 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (𝐴 + (∗‘𝐴)) ∈ ℂ) | |
5 | 3, 4 | mpdan 417 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℂ) |
6 | 2cn 8759 | . . . 4 ⊢ 2 ∈ ℂ | |
7 | 2ap0 8781 | . . . 4 ⊢ 2 # 0 | |
8 | divcanap2 8408 | . . . 4 ⊢ (((𝐴 + (∗‘𝐴)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → (2 · ((𝐴 + (∗‘𝐴)) / 2)) = (𝐴 + (∗‘𝐴))) | |
9 | 6, 7, 8 | mp3an23 1292 | . . 3 ⊢ ((𝐴 + (∗‘𝐴)) ∈ ℂ → (2 · ((𝐴 + (∗‘𝐴)) / 2)) = (𝐴 + (∗‘𝐴))) |
10 | 5, 9 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · ((𝐴 + (∗‘𝐴)) / 2)) = (𝐴 + (∗‘𝐴))) |
11 | 2, 10 | eqtr2d 2151 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 ℂcc 7586 0cc0 7588 + caddc 7591 · cmul 7593 # cap 8311 / cdiv 8400 2c2 8739 ∗ccj 10579 ℜcre 10580 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-2 8747 df-cj 10582 df-re 10583 |
This theorem is referenced by: addcji 10667 addcjd 10697 |
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