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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 15039 | . 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 2106 ‘cfv 6532 ℂcc 11090 ℝcr 11091 ℜcre 15026 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-2 12257 df-cj 15028 df-re 15029 |
This theorem is referenced by: abstri 15259 sqreulem 15288 eqsqrt2d 15297 rlimrege0 15505 recoscl 16066 cos01bnd 16111 cnsubrg 20939 mbfeqa 25089 mbfss 25092 mbfmulc2re 25094 mbfadd 25107 mbfmulc2 25109 mbflim 25114 mbfmul 25173 iblcn 25245 itgcnval 25246 itgre 25247 itgim 25248 iblneg 25249 itgneg 25250 iblss 25251 itgeqa 25260 iblconst 25264 ibladd 25267 itgadd 25271 iblabs 25275 iblabsr 25276 iblmulc2 25277 itgmulc2 25280 itgabs 25281 itgsplit 25282 bddiblnc 25288 dvlip 25439 tanregt0 25977 efif1olem4 25983 eff1olem 25986 lognegb 26027 relog 26034 efiarg 26044 cosarg0d 26046 argregt0 26047 argrege0 26048 abslogle 26055 logcnlem4 26082 cxpsqrtlem 26139 cxpcn3lem 26182 abscxpbnd 26188 cosangneg2d 26239 angrtmuld 26240 lawcoslem1 26247 isosctrlem1 26250 asinlem3a 26302 asinlem3 26303 asinneg 26318 asinsinlem 26323 asinsin 26324 acosbnd 26332 atanlogaddlem 26345 atanlogadd 26346 atanlogsublem 26347 atanlogsub 26348 atantan 26355 o1cxp 26406 cxploglim2 26410 zetacvg 26446 lgamgulmlem2 26461 sqsscirc2 32718 ibladdnc 36347 itgaddnc 36350 iblabsnc 36354 iblmulc2nc 36355 itgmulc2nc 36358 itgabsnc 36359 ftc1anclem2 36364 ftc1anclem5 36367 ftc1anclem6 36368 ftc1anclem8 36370 cntotbnd 36467 sqrtcvallem1 42151 sqrtcvallem4 42159 isosctrlem1ALT 43464 iblsplit 44453 |
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