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Mirrors > Home > MPE Home > Th. List > recld | Structured version Visualization version GIF version |
Description: The real 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 |
---|---|
recld | ⊢ (𝜑 → (ℜ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | recl 13850 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (ℜ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ‘cfv 5888 ℂcc 9934 ℝcr 9935 ℜcre 13837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 |
This theorem is referenced by: abstri 14070 sqreulem 14099 eqsqrt2d 14108 rlimrege0 14310 recoscl 14871 cos01bnd 14916 cnsubrg 19806 mbfeqa 23410 mbfss 23413 mbfmulc2re 23415 mbfadd 23428 mbfmulc2 23430 mbflim 23435 mbfmul 23493 iblcn 23565 itgcnval 23566 itgre 23567 itgim 23568 iblneg 23569 itgneg 23570 iblss 23571 itgeqa 23580 iblconst 23584 ibladd 23587 itgadd 23591 iblabs 23595 iblabsr 23596 iblmulc2 23597 itgmulc2 23600 itgabs 23601 itgsplit 23602 dvlip 23756 tanregt0 24285 efif1olem4 24291 eff1olem 24294 lognegb 24336 relog 24343 efiarg 24353 cosarg0d 24355 argregt0 24356 argrege0 24357 abslogle 24364 logcnlem4 24391 cxpsqrtlem 24448 cxpcn3lem 24488 abscxpbnd 24494 cosangneg2d 24537 angrtmuld 24538 lawcoslem1 24545 isosctrlem1 24548 asinlem3a 24597 asinlem3 24598 asinneg 24613 asinsinlem 24618 asinsin 24619 acosbnd 24627 atanlogaddlem 24640 atanlogadd 24641 atanlogsublem 24642 atanlogsub 24643 atantan 24650 o1cxp 24701 cxploglim2 24705 zetacvg 24741 lgamgulmlem2 24756 sqsscirc2 29955 ibladdnc 33467 itgaddnc 33470 iblabsnc 33474 iblmulc2nc 33475 itgmulc2nc 33478 itgabsnc 33479 bddiblnc 33480 ftc1anclem2 33486 ftc1anclem5 33489 ftc1anclem6 33490 ftc1anclem8 33492 cntotbnd 33595 isosctrlem1ALT 39170 iblsplit 40182 |
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