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Mirrors > Home > ILE Home > Th. List > ltresr | Unicode version |
Description: Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) |
Ref | Expression |
---|---|
ltresr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelre 6909 | . . . 4 | |
2 | 1 | brel 4392 | . . 3 |
3 | opelreal 6904 | . . . 4 | |
4 | opelreal 6904 | . . . 4 | |
5 | 3, 4 | anbi12i 433 | . . 3 |
6 | 2, 5 | sylib 127 | . 2 |
7 | ltrelsr 6823 | . . 3 | |
8 | 7 | brel 4392 | . 2 |
9 | eleq1 2100 | . . . . . . . . 9 | |
10 | 9 | anbi1d 438 | . . . . . . . 8 |
11 | eqeq1 2046 | . . . . . . . . . . 11 | |
12 | 11 | anbi1d 438 | . . . . . . . . . 10 |
13 | 12 | anbi1d 438 | . . . . . . . . 9 |
14 | 13 | 2exbidv 1748 | . . . . . . . 8 |
15 | 10, 14 | anbi12d 442 | . . . . . . 7 |
16 | eleq1 2100 | . . . . . . . . 9 | |
17 | 16 | anbi2d 437 | . . . . . . . 8 |
18 | eqeq1 2046 | . . . . . . . . . . 11 | |
19 | 18 | anbi2d 437 | . . . . . . . . . 10 |
20 | 19 | anbi1d 438 | . . . . . . . . 9 |
21 | 20 | 2exbidv 1748 | . . . . . . . 8 |
22 | 17, 21 | anbi12d 442 | . . . . . . 7 |
23 | df-lt 6902 | . . . . . . 7 | |
24 | 15, 22, 23 | brabg 4006 | . . . . . 6 |
25 | 24 | bianabs 543 | . . . . 5 |
26 | vex 2560 | . . . . . . . . . . 11 | |
27 | 26 | eqresr 6912 | . . . . . . . . . 10 |
28 | eqcom 2042 | . . . . . . . . . 10 | |
29 | eqcom 2042 | . . . . . . . . . 10 | |
30 | 27, 28, 29 | 3bitr4i 201 | . . . . . . . . 9 |
31 | vex 2560 | . . . . . . . . . . 11 | |
32 | 31 | eqresr 6912 | . . . . . . . . . 10 |
33 | eqcom 2042 | . . . . . . . . . 10 | |
34 | eqcom 2042 | . . . . . . . . . 10 | |
35 | 32, 33, 34 | 3bitr4i 201 | . . . . . . . . 9 |
36 | 30, 35 | anbi12i 433 | . . . . . . . 8 |
37 | 26, 31 | opth2 3977 | . . . . . . . 8 |
38 | 36, 37 | bitr4i 176 | . . . . . . 7 |
39 | 38 | anbi1i 431 | . . . . . 6 |
40 | 39 | 2exbii 1497 | . . . . 5 |
41 | 25, 40 | syl6bb 185 | . . . 4 |
42 | 3, 4, 41 | syl2anbr 276 | . . 3 |
43 | breq12 3769 | . . . 4 | |
44 | 43 | copsex2g 3983 | . . 3 |
45 | 42, 44 | bitrd 177 | . 2 |
46 | 6, 8, 45 | pm5.21nii 620 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cop 3378 class class class wbr 3764 cnr 6395 c0r 6396 cltr 6401 cr 6888 cltrr 6893 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-i1p 6565 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-r 6899 df-lt 6902 |
This theorem is referenced by: ltresr2 6916 pitoregt0 6925 ltrennb 6930 ax0lt1 6950 axprecex 6954 axpre-ltirr 6956 axpre-ltwlin 6957 axpre-lttrn 6958 axpre-apti 6959 axpre-ltadd 6960 axpre-mulgt0 6961 axpre-mulext 6962 axarch 6965 axcaucvglemcau 6972 axcaucvglemres 6973 ax-0lt1 6990 |
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