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Mirrors > Home > MPE Home > Th. List > recl | Structured version Visualization version GIF version |
Description: The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
recl | ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reval 15142 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | cjth 15139 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) | |
3 | 2 | simpld 494 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℝ) |
4 | 3 | rehalfcld 12511 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) / 2) ∈ ℝ) |
5 | 1, 4 | eqeltrd 2839 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 ici 11155 + caddc 11156 · cmul 11158 − cmin 11490 / cdiv 11918 2c2 12319 ∗ccj 15132 ℜcre 15133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 |
This theorem is referenced by: imcl 15147 ref 15148 crre 15150 remim 15153 reim0b 15155 rereb 15156 mulre 15157 cjreb 15159 recj 15160 reneg 15161 readd 15162 resub 15163 remullem 15164 remul2 15166 rediv 15167 imcj 15168 imneg 15169 imadd 15170 immul2 15173 cjadd 15177 ipcnval 15179 cjmulval 15181 cjmulge0 15182 cjneg 15183 imval2 15187 cnrecnv 15201 sqeqd 15202 recli 15203 recld 15230 cnpart 15276 absrele 15344 releabs 15357 efeul 16195 absef 16230 absefib 16231 efieq1re 16232 cnsubrg 21463 mbfconst 25682 itgconst 25869 tanregt0 26596 argregt0 26667 tanarg 26676 logf1o2 26707 abscxp 26749 isosctrlem1 26876 asinsin 26950 acoscos 26951 atancj 26968 atantan 26981 cxploglim2 27037 zetacvg 27073 cncph 30848 ccfldextdgrr 33697 sqrtcvallem2 43627 sqrtcvallem3 43628 sqrtcvallem4 43629 sqrtcvallem5 43630 sqrtcval 43631 |
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