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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 10271 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | 1 | oveq2d 5660 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · (ℜ‘𝐴)) = (2 · ((𝐴 + (∗‘𝐴)) / 2))) |
3 | cjcl 10270 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | addcl 7457 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (𝐴 + (∗‘𝐴)) ∈ ℂ) | |
5 | 3, 4 | mpdan 412 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℂ) |
6 | 2cn 8483 | . . . 4 ⊢ 2 ∈ ℂ | |
7 | 2ap0 8505 | . . . 4 ⊢ 2 # 0 | |
8 | divcanap2 8137 | . . . 4 ⊢ (((𝐴 + (∗‘𝐴)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → (2 · ((𝐴 + (∗‘𝐴)) / 2)) = (𝐴 + (∗‘𝐴))) | |
9 | 6, 7, 8 | mp3an23 1265 | . . 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 2121 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ∈ wcel 1438 class class class wbr 3843 ‘cfv 5010 (class class class)co 5644 ℂcc 7338 0cc0 7340 + caddc 7343 · cmul 7345 # cap 8048 / cdiv 8129 2c2 8463 ∗ccj 10261 ℜcre 10262 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-mulrcl 7434 ax-addcom 7435 ax-mulcom 7436 ax-addass 7437 ax-mulass 7438 ax-distr 7439 ax-i2m1 7440 ax-0lt1 7441 ax-1rid 7442 ax-0id 7443 ax-rnegex 7444 ax-precex 7445 ax-cnre 7446 ax-pre-ltirr 7447 ax-pre-ltwlin 7448 ax-pre-lttrn 7449 ax-pre-apti 7450 ax-pre-ltadd 7451 ax-pre-mulgt0 7452 ax-pre-mulext 7453 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-mpt 3899 df-id 4118 df-po 4121 df-iso 4122 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-pnf 7514 df-mnf 7515 df-xr 7516 df-ltxr 7517 df-le 7518 df-sub 7645 df-neg 7646 df-reap 8042 df-ap 8049 df-div 8130 df-2 8471 df-cj 10264 df-re 10265 |
This theorem is referenced by: addcji 10349 addcjd 10379 |
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