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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 15150 | . 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 2108 ‘cfv 6561 ℂcc 11153 ℝcr 11154 ℑcim 15137 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140 |
| This theorem is referenced by: rlimrecl 15616 resincl 16176 sin01bnd 16221 recld2 24836 mbfeqa 25678 mbfss 25681 mbfmulc2re 25683 mbfadd 25696 mbfmulc2 25698 mbflim 25703 mbfmul 25761 iblcn 25834 itgcnval 25835 itgre 25836 itgim 25837 iblneg 25838 itgneg 25839 ibladd 25856 itgadd 25860 iblabs 25864 itgmulc2 25869 bddiblnc 25877 aaliou2b 26383 efif1olem3 26586 eff1olem 26590 logimclad 26614 abslogimle 26615 logrnaddcl 26616 lognegb 26632 logcj 26648 efiarg 26649 cosargd 26650 argregt0 26652 argrege0 26653 argimgt0 26654 argimlt0 26655 logimul 26656 abslogle 26660 tanarg 26661 logcnlem2 26685 logcnlem3 26686 logcnlem4 26687 logcnlem5 26688 logcn 26689 dvloglem 26690 logf1o2 26692 efopnlem1 26698 efopnlem2 26699 cxpsqrtlem 26744 abscxpbnd 26796 ang180lem2 26853 lawcos 26859 isosctrlem1 26861 isosctrlem2 26862 asinneg 26929 asinsinlem 26934 atanlogaddlem 26956 atanlogsublem 26958 atanlogsub 26959 basellem3 27126 re0cj 32753 constrconj 33786 sqsscirc2 33908 ibladdnc 37684 itgaddnc 37687 iblabsnc 37691 iblmulc2nc 37692 itgmulc2nc 37695 ftc1anclem2 37701 ftc1anclem6 37705 ftc1anclem8 37707 cntotbnd 37803 isosctrlem1ALT 44954 dstregt0 45293 absimnre 45487 absimlere 45490 cnrefiisplem 45844 sigarim 46866 readdcnnred 47315 resubcnnred 47316 cndivrenred 47318 |
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