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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 7001 | . . . 4 | |
2 | 1 | brel 4410 | . . 3 |
3 | opelreal 6996 | . . . 4 | |
4 | opelreal 6996 | . . . 4 | |
5 | 3, 4 | anbi12i 447 | . . 3 |
6 | 2, 5 | sylib 120 | . 2 |
7 | ltrelsr 6915 | . . 3 | |
8 | 7 | brel 4410 | . 2 |
9 | eleq1 2141 | . . . . . . . . 9 | |
10 | 9 | anbi1d 452 | . . . . . . . 8 |
11 | eqeq1 2087 | . . . . . . . . . . 11 | |
12 | 11 | anbi1d 452 | . . . . . . . . . 10 |
13 | 12 | anbi1d 452 | . . . . . . . . 9 |
14 | 13 | 2exbidv 1789 | . . . . . . . 8 |
15 | 10, 14 | anbi12d 456 | . . . . . . 7 |
16 | eleq1 2141 | . . . . . . . . 9 | |
17 | 16 | anbi2d 451 | . . . . . . . 8 |
18 | eqeq1 2087 | . . . . . . . . . . 11 | |
19 | 18 | anbi2d 451 | . . . . . . . . . 10 |
20 | 19 | anbi1d 452 | . . . . . . . . 9 |
21 | 20 | 2exbidv 1789 | . . . . . . . 8 |
22 | 17, 21 | anbi12d 456 | . . . . . . 7 |
23 | df-lt 6994 | . . . . . . 7 | |
24 | 15, 22, 23 | brabg 4024 | . . . . . 6 |
25 | 24 | bianabs 575 | . . . . 5 |
26 | vex 2604 | . . . . . . . . . . 11 | |
27 | 26 | eqresr 7004 | . . . . . . . . . 10 |
28 | eqcom 2083 | . . . . . . . . . 10 | |
29 | eqcom 2083 | . . . . . . . . . 10 | |
30 | 27, 28, 29 | 3bitr4i 210 | . . . . . . . . 9 |
31 | vex 2604 | . . . . . . . . . . 11 | |
32 | 31 | eqresr 7004 | . . . . . . . . . 10 |
33 | eqcom 2083 | . . . . . . . . . 10 | |
34 | eqcom 2083 | . . . . . . . . . 10 | |
35 | 32, 33, 34 | 3bitr4i 210 | . . . . . . . . 9 |
36 | 30, 35 | anbi12i 447 | . . . . . . . 8 |
37 | 26, 31 | opth2 3995 | . . . . . . . 8 |
38 | 36, 37 | bitr4i 185 | . . . . . . 7 |
39 | 38 | anbi1i 445 | . . . . . 6 |
40 | 39 | 2exbii 1537 | . . . . 5 |
41 | 25, 40 | syl6bb 194 | . . . 4 |
42 | 3, 4, 41 | syl2anbr 286 | . . 3 |
43 | breq12 3790 | . . . 4 | |
44 | 43 | copsex2g 4001 | . . 3 |
45 | 42, 44 | bitrd 186 | . 2 |
46 | 6, 8, 45 | pm5.21nii 652 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cop 3401 class class class wbr 3785 cnr 6487 c0r 6488 cltr 6493 cr 6980 cltrr 6985 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-inp 6656 df-i1p 6657 df-enr 6903 df-nr 6904 df-ltr 6907 df-0r 6908 df-r 6991 df-lt 6994 |
This theorem is referenced by: ltresr2 7008 pitoregt0 7017 ltrennb 7022 ax0lt1 7042 axprecex 7046 axpre-ltirr 7048 axpre-ltwlin 7049 axpre-lttrn 7050 axpre-apti 7051 axpre-ltadd 7052 axpre-mulgt0 7053 axpre-mulext 7054 axarch 7057 axcaucvglemcau 7064 axcaucvglemres 7065 |
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