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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 7634 | . . . 4 | |
2 | 1 | brel 4586 | . . 3 |
3 | opelreal 7628 | . . . 4 | |
4 | opelreal 7628 | . . . 4 | |
5 | 3, 4 | anbi12i 455 | . . 3 |
6 | 2, 5 | sylib 121 | . 2 |
7 | ltrelsr 7539 | . . 3 | |
8 | 7 | brel 4586 | . 2 |
9 | eleq1 2200 | . . . . . . . . 9 | |
10 | 9 | anbi1d 460 | . . . . . . . 8 |
11 | eqeq1 2144 | . . . . . . . . . . 11 | |
12 | 11 | anbi1d 460 | . . . . . . . . . 10 |
13 | 12 | anbi1d 460 | . . . . . . . . 9 |
14 | 13 | 2exbidv 1840 | . . . . . . . 8 |
15 | 10, 14 | anbi12d 464 | . . . . . . 7 |
16 | eleq1 2200 | . . . . . . . . 9 | |
17 | 16 | anbi2d 459 | . . . . . . . 8 |
18 | eqeq1 2144 | . . . . . . . . . . 11 | |
19 | 18 | anbi2d 459 | . . . . . . . . . 10 |
20 | 19 | anbi1d 460 | . . . . . . . . 9 |
21 | 20 | 2exbidv 1840 | . . . . . . . 8 |
22 | 17, 21 | anbi12d 464 | . . . . . . 7 |
23 | df-lt 7626 | . . . . . . 7 | |
24 | 15, 22, 23 | brabg 4186 | . . . . . 6 |
25 | 24 | bianabs 600 | . . . . 5 |
26 | vex 2684 | . . . . . . . . . . 11 | |
27 | 26 | eqresr 7637 | . . . . . . . . . 10 |
28 | eqcom 2139 | . . . . . . . . . 10 | |
29 | eqcom 2139 | . . . . . . . . . 10 | |
30 | 27, 28, 29 | 3bitr4i 211 | . . . . . . . . 9 |
31 | vex 2684 | . . . . . . . . . . 11 | |
32 | 31 | eqresr 7637 | . . . . . . . . . 10 |
33 | eqcom 2139 | . . . . . . . . . 10 | |
34 | eqcom 2139 | . . . . . . . . . 10 | |
35 | 32, 33, 34 | 3bitr4i 211 | . . . . . . . . 9 |
36 | 30, 35 | anbi12i 455 | . . . . . . . 8 |
37 | 26, 31 | opth2 4157 | . . . . . . . 8 |
38 | 36, 37 | bitr4i 186 | . . . . . . 7 |
39 | 38 | anbi1i 453 | . . . . . 6 |
40 | 39 | 2exbii 1585 | . . . . 5 |
41 | 25, 40 | syl6bb 195 | . . . 4 |
42 | 3, 4, 41 | syl2anbr 290 | . . 3 |
43 | breq12 3929 | . . . 4 | |
44 | 43 | copsex2g 4163 | . . 3 |
45 | 42, 44 | bitrd 187 | . 2 |
46 | 6, 8, 45 | pm5.21nii 693 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3525 class class class wbr 3924 cnr 7098 c0r 7099 cltr 7104 cr 7612 cltrr 7617 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-inp 7267 df-i1p 7268 df-enr 7527 df-nr 7528 df-ltr 7531 df-0r 7532 df-r 7623 df-lt 7626 |
This theorem is referenced by: ltresr2 7641 pitoregt0 7650 ltrennb 7655 ax0lt1 7677 axprecex 7681 axpre-ltirr 7683 axpre-ltwlin 7684 axpre-lttrn 7685 axpre-apti 7686 axpre-ltadd 7687 axpre-mulgt0 7688 axpre-mulext 7689 axarch 7692 axcaucvglemcau 7699 axcaucvglemres 7700 axpre-suploclemres 7702 |
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