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Mirrors > Home > MPE Home > Th. List > imcld | Structured version Visualization version GIF version |
Description: The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
recld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
imcld | ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | imcl 14462 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ‘cfv 6324 ℂcc 10524 ℝcr 10525 ℑcim 14449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-cj 14450 df-re 14451 df-im 14452 |
This theorem is referenced by: rlimrecl 14929 resincl 15485 sin01bnd 15530 recld2 23419 mbfeqa 24247 mbfss 24250 mbfmulc2re 24252 mbfadd 24265 mbfmulc2 24267 mbflim 24272 mbfmul 24330 iblcn 24402 itgcnval 24403 itgre 24404 itgim 24405 iblneg 24406 itgneg 24407 ibladd 24424 itgadd 24428 iblabs 24432 itgmulc2 24437 bddiblnc 24445 aaliou2b 24937 efif1olem3 25136 eff1olem 25140 logimclad 25164 abslogimle 25165 logrnaddcl 25166 lognegb 25181 logcj 25197 efiarg 25198 cosargd 25199 argregt0 25201 argrege0 25202 argimgt0 25203 argimlt0 25204 logimul 25205 abslogle 25209 tanarg 25210 logcnlem2 25234 logcnlem3 25235 logcnlem4 25236 logcnlem5 25237 logcn 25238 dvloglem 25239 logf1o2 25241 efopnlem1 25247 efopnlem2 25248 cxpsqrtlem 25293 abscxpbnd 25342 ang180lem2 25396 lawcos 25402 isosctrlem1 25404 isosctrlem2 25405 asinneg 25472 asinsinlem 25477 atanlogaddlem 25499 atanlogsublem 25501 atanlogsub 25502 basellem3 25668 sqsscirc2 31262 ibladdnc 35114 itgaddnc 35117 iblabsnc 35121 iblmulc2nc 35122 itgmulc2nc 35125 ftc1anclem2 35131 ftc1anclem6 35135 ftc1anclem8 35137 cntotbnd 35234 isosctrlem1ALT 41640 dstregt0 41912 absimnre 42116 absimlere 42119 cnrefiisplem 42471 sigarim 43465 readdcnnred 43860 resubcnnred 43861 cndivrenred 43863 |
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