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Mirrors > Home > ILE Home > Th. List > inelr | Unicode version |
Description: The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.) |
Ref | Expression |
---|---|
inelr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ine0 8283 | . . 3 | |
2 | 1 | neii 2336 | . 2 |
3 | 0lt1 8016 | . . . . . 6 | |
4 | 0re 7890 | . . . . . . 7 | |
5 | 1re 7889 | . . . . . . 7 | |
6 | 4, 5 | ltnsymi 7989 | . . . . . 6 |
7 | 3, 6 | ax-mp 5 | . . . . 5 |
8 | ixi 8472 | . . . . . . . 8 | |
9 | 5 | renegcli 8151 | . . . . . . . 8 |
10 | 8, 9 | eqeltri 2237 | . . . . . . 7 |
11 | 4, 10, 5 | ltadd1i 8391 | . . . . . 6 |
12 | ax-1cn 7837 | . . . . . . . 8 | |
13 | 12 | addid2i 8032 | . . . . . . 7 |
14 | ax-i2m1 7849 | . . . . . . 7 | |
15 | 13, 14 | breq12i 3985 | . . . . . 6 |
16 | 11, 15 | bitri 183 | . . . . 5 |
17 | 7, 16 | mtbir 661 | . . . 4 |
18 | mullt0 8369 | . . . . . 6 | |
19 | 18 | anidms 395 | . . . . 5 |
20 | 19 | ex 114 | . . . 4 |
21 | 17, 20 | mtoi 654 | . . 3 |
22 | mulgt0 7964 | . . . . . 6 | |
23 | 22 | anidms 395 | . . . . 5 |
24 | 23 | ex 114 | . . . 4 |
25 | 17, 24 | mtoi 654 | . . 3 |
26 | lttri3 7969 | . . . 4 | |
27 | 4, 26 | mpan2 422 | . . 3 |
28 | 21, 25, 27 | mpbir2and 933 | . 2 |
29 | 2, 28 | mto 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wceq 1342 wcel 2135 class class class wbr 3976 (class class class)co 5836 cr 7743 cc0 7744 c1 7745 ci 7746 caddc 7747 cmul 7749 clt 7924 cneg 8061 |
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 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
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 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-sub 8062 df-neg 8063 |
This theorem is referenced by: rimul 8474 |
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