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Mirrors > Home > ILE Home > Th. List > rimul | Unicode version |
Description: A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
rimul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inelr 8339 | . . 3 | |
2 | recexre 8333 | . . . . . 6 #ℝ | |
3 | 2 | adantlr 468 | . . . . 5 #ℝ |
4 | simplll 522 | . . . . . . . . 9 #ℝ | |
5 | 4 | recnd 7787 | . . . . . . . 8 #ℝ |
6 | simprl 520 | . . . . . . . . 9 #ℝ | |
7 | 6 | recnd 7787 | . . . . . . . 8 #ℝ |
8 | ax-icn 7708 | . . . . . . . . 9 | |
9 | mulass 7744 | . . . . . . . . 9 | |
10 | 8, 9 | mp3an1 1302 | . . . . . . . 8 |
11 | 5, 7, 10 | syl2anc 408 | . . . . . . 7 #ℝ |
12 | oveq2 5775 | . . . . . . . . 9 | |
13 | 8 | mulid1i 7761 | . . . . . . . . 9 |
14 | 12, 13 | syl6eq 2186 | . . . . . . . 8 |
15 | 14 | ad2antll 482 | . . . . . . 7 #ℝ |
16 | 11, 15 | eqtrd 2170 | . . . . . 6 #ℝ |
17 | simpllr 523 | . . . . . . 7 #ℝ | |
18 | 17, 6 | remulcld 7789 | . . . . . 6 #ℝ |
19 | 16, 18 | eqeltrrd 2215 | . . . . 5 #ℝ |
20 | 3, 19 | rexlimddv 2552 | . . . 4 #ℝ |
21 | 20 | ex 114 | . . 3 #ℝ |
22 | 1, 21 | mtoi 653 | . 2 #ℝ |
23 | 0re 7759 | . . . 4 | |
24 | reapti 8334 | . . . 4 #ℝ | |
25 | 23, 24 | mpan2 421 | . . 3 #ℝ |
26 | 25 | adantr 274 | . 2 #ℝ |
27 | 22, 26 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 c1 7614 ci 7615 cmul 7618 #ℝ creap 8329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 |
This theorem is referenced by: rereim 8341 cru 8357 cju 8712 crre 10622 |
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