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Mirrors > Home > MPE Home > Th. List > reim0d | Structured version Visualization version GIF version |
Description: The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
crred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
reim0d | ⊢ (𝜑 → (ℑ‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | reim0 14579 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (ℑ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6349 ℝcr 10626 0cc0 10627 ℑcim 14559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-2 11791 df-cj 14560 df-re 14561 df-im 14562 |
This theorem is referenced by: eqsqrt2d 14830 zgz 16381 ismbf 24392 iblrelem 24555 itgrevallem1 24559 aaliou2b 25101 tanregt0 25295 logcnlem3 25399 logf1o2 25405 logbrec 25532 ang180lem2 25560 isosctrlem2 25569 zetacvg 25764 basellem3 25832 dstregt0 42397 absimlere 42600 sigarid 43953 sharhght 43960 readdcnnred 44376 resubcnnred 44377 |
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