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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 15060 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | cjth 15057 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) | |
3 | 2 | simpld 494 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℝ) |
4 | 3 | rehalfcld 12466 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) / 2) ∈ ℝ) |
5 | 1, 4 | eqeltrd 2832 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 ici 11118 + caddc 11119 · cmul 11121 − cmin 11451 / cdiv 11878 2c2 12274 ∗ccj 15050 ℜcre 15051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-cj 15053 df-re 15054 |
This theorem is referenced by: imcl 15065 ref 15066 crre 15068 remim 15071 reim0b 15073 rereb 15074 mulre 15075 cjreb 15077 recj 15078 reneg 15079 readd 15080 resub 15081 remullem 15082 remul2 15084 rediv 15085 imcj 15086 imneg 15087 imadd 15088 immul2 15091 cjadd 15095 ipcnval 15097 cjmulval 15099 cjmulge0 15100 cjneg 15101 imval2 15105 cnrecnv 15119 sqeqd 15120 recli 15121 recld 15148 cnpart 15194 absrele 15262 releabs 15275 efeul 16112 absef 16147 absefib 16148 efieq1re 16149 cnsubrg 21209 mbfconst 25395 itgconst 25581 tanregt0 26299 argregt0 26369 tanarg 26378 logf1o2 26409 abscxp 26451 isosctrlem1 26574 asinsin 26648 acoscos 26649 atancj 26666 atantan 26679 cxploglim2 26734 zetacvg 26770 cncph 30354 ccfldextdgrr 33050 sqrtcvallem2 42703 sqrtcvallem3 42704 sqrtcvallem4 42705 sqrtcvallem5 42706 sqrtcval 42707 |
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