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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 8476 | . . 3 | |
2 | recexre 8470 | . . . . . 6 #ℝ | |
3 | 2 | adantlr 469 | . . . . 5 #ℝ |
4 | simplll 523 | . . . . . . . . 9 #ℝ | |
5 | 4 | recnd 7921 | . . . . . . . 8 #ℝ |
6 | simprl 521 | . . . . . . . . 9 #ℝ | |
7 | 6 | recnd 7921 | . . . . . . . 8 #ℝ |
8 | ax-icn 7842 | . . . . . . . . 9 | |
9 | mulass 7878 | . . . . . . . . 9 | |
10 | 8, 9 | mp3an1 1313 | . . . . . . . 8 |
11 | 5, 7, 10 | syl2anc 409 | . . . . . . 7 #ℝ |
12 | oveq2 5847 | . . . . . . . . 9 | |
13 | 8 | mulid1i 7895 | . . . . . . . . 9 |
14 | 12, 13 | eqtrdi 2213 | . . . . . . . 8 |
15 | 14 | ad2antll 483 | . . . . . . 7 #ℝ |
16 | 11, 15 | eqtrd 2197 | . . . . . 6 #ℝ |
17 | simpllr 524 | . . . . . . 7 #ℝ | |
18 | 17, 6 | remulcld 7923 | . . . . . 6 #ℝ |
19 | 16, 18 | eqeltrrd 2242 | . . . . 5 #ℝ |
20 | 3, 19 | rexlimddv 2586 | . . . 4 #ℝ |
21 | 20 | ex 114 | . . 3 #ℝ |
22 | 1, 21 | mtoi 654 | . 2 #ℝ |
23 | 0re 7893 | . . . 4 | |
24 | reapti 8471 | . . . 4 #ℝ | |
25 | 23, 24 | mpan2 422 | . . 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 1342 wcel 2135 wrex 2443 class class class wbr 3979 (class class class)co 5839 cc 7745 cr 7746 cc0 7747 c1 7748 ci 7749 cmul 7752 #ℝ creap 8466 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-ltxr 7932 df-sub 8065 df-neg 8066 df-reap 8467 |
This theorem is referenced by: rereim 8478 cru 8494 cju 8850 crre 10793 |
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