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Mirrors > Home > ILE Home > Th. List > iap0 | Unicode version |
Description: The imaginary unit is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Ref | Expression |
---|---|
iap0 | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ap0 8469 | . . . 4 # | |
2 | 1 | olci 722 | . . 3 # # |
3 | 0re 7880 | . . . 4 | |
4 | 1re 7879 | . . . 4 | |
5 | apreim 8482 | . . . 4 # # # | |
6 | 3, 4, 3, 3, 5 | mp4an 424 | . . 3 # # # |
7 | 2, 6 | mpbir 145 | . 2 # |
8 | ax-icn 7829 | . . . . 5 | |
9 | 8 | mulid1i 7882 | . . . 4 |
10 | 9 | oveq2i 5837 | . . 3 |
11 | 8 | addid2i 8022 | . . 3 |
12 | 10, 11 | eqtri 2178 | . 2 |
13 | it0e0 9059 | . . . 4 | |
14 | 13 | oveq2i 5837 | . . 3 |
15 | 00id 8020 | . . 3 | |
16 | 14, 15 | eqtri 2178 | . 2 |
17 | 7, 12, 16 | 3brtr3i 3995 | 1 # |
Colors of variables: wff set class |
Syntax hints: wb 104 wo 698 wcel 2128 class class class wbr 3967 (class class class)co 5826 cr 7733 cc0 7734 c1 7735 ci 7736 caddc 7737 cmul 7739 # cap 8460 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-ltxr 7919 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 |
This theorem is referenced by: 2muliap0 9062 irec 10527 iexpcyc 10532 imval 10761 imre 10762 reim 10763 crim 10769 cjreb 10777 tanval2ap 11621 tanval3ap 11622 efival 11640 |
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