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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 13698 | . 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 1977 ‘cfv 5804 ℂcc 9813 ℝcr 9814 ℜcre 13685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 |
This theorem is referenced by: abstri 13918 sqreulem 13947 eqsqrt2d 13956 rlimrege0 14158 recoscl 14710 cos01bnd 14755 cnsubrg 19625 mbfeqa 23216 mbfss 23219 mbfmulc2re 23221 mbfadd 23234 mbfmulc2 23236 mbflim 23241 mbfmul 23299 iblcn 23371 itgcnval 23372 itgre 23373 itgim 23374 iblneg 23375 itgneg 23376 iblss 23377 itgeqa 23386 iblconst 23390 ibladd 23393 itgadd 23397 iblabs 23401 iblabsr 23402 iblmulc2 23403 itgmulc2 23406 itgabs 23407 itgsplit 23408 dvlip 23560 tanregt0 24089 efif1olem4 24095 eff1olem 24098 lognegb 24140 relog 24147 efiarg 24157 cosarg0d 24159 argregt0 24160 argrege0 24161 abslogle 24168 logcnlem4 24191 cxpsqrtlem 24248 cxpcn3lem 24288 abscxpbnd 24294 cosangneg2d 24337 angrtmuld 24338 lawcoslem1 24345 isosctrlem1 24348 asinlem3a 24397 asinlem3 24398 asinneg 24413 asinsinlem 24418 asinsin 24419 acosbnd 24427 atanlogaddlem 24440 atanlogadd 24441 atanlogsublem 24442 atanlogsub 24443 atantan 24450 o1cxp 24501 cxploglim2 24505 zetacvg 24541 lgamgulmlem2 24556 sqsscirc2 29283 ibladdnc 32637 itgaddnc 32640 iblabsnc 32644 iblmulc2nc 32645 itgmulc2nc 32648 itgabsnc 32649 bddiblnc 32650 ftc1anclem2 32656 ftc1anclem5 32659 ftc1anclem6 32660 ftc1anclem8 32662 cntotbnd 32765 isosctrlem1ALT 38192 iblsplit 38858 |
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