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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 9874 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | 1 | oveq2d 5559 | . 2 ⊢ (𝐴 ∈ ℂ → (2 · (ℜ‘𝐴)) = (2 · ((𝐴 + (∗‘𝐴)) / 2))) |
3 | cjcl 9873 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | addcl 7160 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (𝐴 + (∗‘𝐴)) ∈ ℂ) | |
5 | 3, 4 | mpdan 412 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℂ) |
6 | 2cn 8177 | . . . 4 ⊢ 2 ∈ ℂ | |
7 | 2ap0 8199 | . . . 4 ⊢ 2 # 0 | |
8 | divcanap2 7835 | . . . 4 ⊢ (((𝐴 + (∗‘𝐴)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → (2 · ((𝐴 + (∗‘𝐴)) / 2)) = (𝐴 + (∗‘𝐴))) | |
9 | 6, 7, 8 | mp3an23 1261 | . . 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 2115 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) = (2 · (ℜ‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 class class class wbr 3793 ‘cfv 4932 (class class class)co 5543 ℂcc 7041 0cc0 7043 + caddc 7046 · cmul 7048 # cap 7748 / cdiv 7827 2c2 8156 ∗ccj 9864 ℜcre 9865 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-mulrcl 7137 ax-addcom 7138 ax-mulcom 7139 ax-addass 7140 ax-mulass 7141 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-1rid 7145 ax-0id 7146 ax-rnegex 7147 ax-precex 7148 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-apti 7153 ax-pre-ltadd 7154 ax-pre-mulgt0 7155 ax-pre-mulext 7156 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rmo 2357 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-mpt 3849 df-id 4056 df-po 4059 df-iso 4060 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-reap 7742 df-ap 7749 df-div 7828 df-2 8165 df-cj 9867 df-re 9868 |
This theorem is referenced by: addcji 9952 addcjd 9982 |
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