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Mirrors > Home > ILE Home > Th. List > crim | GIF version |
Description: The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Ref | Expression |
---|---|
crim | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7014 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | ax-icn 6979 | . . . . 5 ⊢ i ∈ ℂ | |
3 | recn 7014 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | mulcl 7008 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (i · 𝐵) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 393 | . . . 4 ⊢ (𝐵 ∈ ℝ → (i · 𝐵) ∈ ℂ) |
6 | addcl 7006 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (𝐴 + (i · 𝐵)) ∈ ℂ) | |
7 | 1, 5, 6 | syl2an 273 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (i · 𝐵)) ∈ ℂ) |
8 | imval 9450 | . . 3 ⊢ ((𝐴 + (i · 𝐵)) ∈ ℂ → (ℑ‘(𝐴 + (i · 𝐵))) = (ℜ‘((𝐴 + (i · 𝐵)) / i))) | |
9 | 7, 8 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + (i · 𝐵))) = (ℜ‘((𝐴 + (i · 𝐵)) / i))) |
10 | 2, 4 | mpan 400 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (i · 𝐵) ∈ ℂ) |
11 | iap0 8148 | . . . . . . 7 ⊢ i # 0 | |
12 | divdirap 7674 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ ∧ (i ∈ ℂ ∧ i # 0)) → ((𝐴 + (i · 𝐵)) / i) = ((𝐴 / i) + ((i · 𝐵) / i))) | |
13 | 12 | 3expa 1104 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) ∧ (i ∈ ℂ ∧ i # 0)) → ((𝐴 + (i · 𝐵)) / i) = ((𝐴 / i) + ((i · 𝐵) / i))) |
14 | 2, 11, 13 | mpanr12 415 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → ((𝐴 + (i · 𝐵)) / i) = ((𝐴 / i) + ((i · 𝐵) / i))) |
15 | 10, 14 | sylan2 270 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) / i) = ((𝐴 / i) + ((i · 𝐵) / i))) |
16 | divrecap2 7668 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ ∧ i # 0) → (𝐴 / i) = ((1 / i) · 𝐴)) | |
17 | 2, 11, 16 | mp3an23 1224 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 / i) = ((1 / i) · 𝐴)) |
18 | irec 9352 | . . . . . . . . 9 ⊢ (1 / i) = -i | |
19 | 18 | oveq1i 5522 | . . . . . . . 8 ⊢ ((1 / i) · 𝐴) = (-i · 𝐴) |
20 | 19 | a1i 9 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((1 / i) · 𝐴) = (-i · 𝐴)) |
21 | mulneg12 7394 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴)) | |
22 | 2, 21 | mpan 400 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) = (i · -𝐴)) |
23 | 17, 20, 22 | 3eqtrd 2076 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 / i) = (i · -𝐴)) |
24 | divcanap3 7675 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ i ∈ ℂ ∧ i # 0) → ((i · 𝐵) / i) = 𝐵) | |
25 | 2, 11, 24 | mp3an23 1224 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → ((i · 𝐵) / i) = 𝐵) |
26 | 23, 25 | oveqan12d 5531 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 / i) + ((i · 𝐵) / i)) = ((i · -𝐴) + 𝐵)) |
27 | negcl 7211 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
28 | mulcl 7008 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ -𝐴 ∈ ℂ) → (i · -𝐴) ∈ ℂ) | |
29 | 2, 27, 28 | sylancr 393 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · -𝐴) ∈ ℂ) |
30 | addcom 7150 | . . . . . 6 ⊢ (((i · -𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((i · -𝐴) + 𝐵) = (𝐵 + (i · -𝐴))) | |
31 | 29, 30 | sylan 267 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((i · -𝐴) + 𝐵) = (𝐵 + (i · -𝐴))) |
32 | 15, 26, 31 | 3eqtrrd 2077 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + (i · -𝐴)) = ((𝐴 + (i · 𝐵)) / i)) |
33 | 1, 3, 32 | syl2an 273 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + (i · -𝐴)) = ((𝐴 + (i · 𝐵)) / i)) |
34 | 33 | fveq2d 5182 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℜ‘(𝐵 + (i · -𝐴))) = (ℜ‘((𝐴 + (i · 𝐵)) / i))) |
35 | id 19 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ) | |
36 | renegcl 7272 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
37 | crre 9457 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (ℜ‘(𝐵 + (i · -𝐴))) = 𝐵) | |
38 | 35, 36, 37 | syl2anr 274 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℜ‘(𝐵 + (i · -𝐴))) = 𝐵) |
39 | 9, 34, 38 | 3eqtr2d 2078 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 ℝcr 6888 0cc0 6889 1c1 6890 ici 6891 + caddc 6892 · cmul 6894 -cneg 7183 # cap 7572 / cdiv 7651 ℜcre 9440 ℑcim 9441 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-2 7973 df-cj 9442 df-re 9443 df-im 9444 |
This theorem is referenced by: replim 9459 reim0 9461 remullem 9471 imcj 9475 imneg 9476 imadd 9477 imi 9500 crimi 9537 crimd 9577 absreimsq 9665 |
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