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Mirrors > Home > ILE Home > Th. List > ltsrprg | Unicode version |
Description: Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.) |
Ref | Expression |
---|---|
ltsrprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enrex 7545 | . 2 | |
2 | enrer 7543 | . 2 | |
3 | df-nr 7535 | . 2 | |
4 | df-ltr 7538 | . 2 | |
5 | enreceq 7544 | . . . . 5 | |
6 | enreceq 7544 | . . . . . 6 | |
7 | eqcom 2141 | . . . . . 6 | |
8 | 6, 7 | syl6bb 195 | . . . . 5 |
9 | 5, 8 | bi2anan9 595 | . . . 4 |
10 | oveq12 5783 | . . . . . . 7 | |
11 | 10 | adantl 275 | . . . . . 6 |
12 | simprlr 527 | . . . . . . . . . . 11 | |
13 | simplrr 525 | . . . . . . . . . . 11 | |
14 | addcomprg 7386 | . . . . . . . . . . . 12 | |
15 | 14 | oveq1d 5789 | . . . . . . . . . . 11 |
16 | 12, 13, 15 | syl2anc 408 | . . . . . . . . . 10 |
17 | simprrl 528 | . . . . . . . . . . 11 | |
18 | addassprg 7387 | . . . . . . . . . . 11 | |
19 | 12, 13, 17, 18 | syl3anc 1216 | . . . . . . . . . 10 |
20 | addassprg 7387 | . . . . . . . . . . 11 | |
21 | 13, 12, 17, 20 | syl3anc 1216 | . . . . . . . . . 10 |
22 | 16, 19, 21 | 3eqtr3d 2180 | . . . . . . . . 9 |
23 | 22 | oveq2d 5790 | . . . . . . . 8 |
24 | simplll 522 | . . . . . . . . 9 | |
25 | addclpr 7345 | . . . . . . . . . . . . 13 | |
26 | 25 | ad2ant2lr 501 | . . . . . . . . . . . 12 |
27 | addclpr 7345 | . . . . . . . . . . . . 13 | |
28 | 27 | ad2ant2lr 501 | . . . . . . . . . . . 12 |
29 | 26, 28 | anim12ci 337 | . . . . . . . . . . 11 |
30 | 29 | an4s 577 | . . . . . . . . . 10 |
31 | 30 | simpld 111 | . . . . . . . . 9 |
32 | addassprg 7387 | . . . . . . . . 9 | |
33 | 24, 12, 31, 32 | syl3anc 1216 | . . . . . . . 8 |
34 | addclpr 7345 | . . . . . . . . . 10 | |
35 | 12, 17, 34 | syl2anc 408 | . . . . . . . . 9 |
36 | addassprg 7387 | . . . . . . . . 9 | |
37 | 24, 13, 35, 36 | syl3anc 1216 | . . . . . . . 8 |
38 | 23, 33, 37 | 3eqtr4d 2182 | . . . . . . 7 |
39 | 38 | adantr 274 | . . . . . 6 |
40 | simprll 526 | . . . . . . . . . . . 12 | |
41 | simplrl 524 | . . . . . . . . . . . 12 | |
42 | addcomprg 7386 | . . . . . . . . . . . 12 | |
43 | 40, 41, 42 | syl2anc 408 | . . . . . . . . . . 11 |
44 | 43 | oveq1d 5789 | . . . . . . . . . 10 |
45 | simprrr 529 | . . . . . . . . . . 11 | |
46 | addassprg 7387 | . . . . . . . . . . 11 | |
47 | 40, 41, 45, 46 | syl3anc 1216 | . . . . . . . . . 10 |
48 | addassprg 7387 | . . . . . . . . . . 11 | |
49 | 41, 40, 45, 48 | syl3anc 1216 | . . . . . . . . . 10 |
50 | 44, 47, 49 | 3eqtr3d 2180 | . . . . . . . . 9 |
51 | 50 | oveq2d 5790 | . . . . . . . 8 |
52 | simpllr 523 | . . . . . . . . 9 | |
53 | addclpr 7345 | . . . . . . . . . 10 | |
54 | 41, 45, 53 | syl2anc 408 | . . . . . . . . 9 |
55 | addassprg 7387 | . . . . . . . . 9 | |
56 | 52, 40, 54, 55 | syl3anc 1216 | . . . . . . . 8 |
57 | addclpr 7345 | . . . . . . . . . 10 | |
58 | 40, 45, 57 | syl2anc 408 | . . . . . . . . 9 |
59 | addassprg 7387 | . . . . . . . . 9 | |
60 | 52, 41, 58, 59 | syl3anc 1216 | . . . . . . . 8 |
61 | 51, 56, 60 | 3eqtr4d 2182 | . . . . . . 7 |
62 | 61 | adantr 274 | . . . . . 6 |
63 | 11, 39, 62 | 3eqtr4d 2182 | . . . . 5 |
64 | 63 | ex 114 | . . . 4 |
65 | 9, 64 | sylbid 149 | . . 3 |
66 | ltaprg 7427 | . . . . 5 | |
67 | 66 | adantl 275 | . . . 4 |
68 | addclpr 7345 | . . . . 5 | |
69 | 24, 12, 68 | syl2anc 408 | . . . 4 |
70 | 30 | simprd 113 | . . . 4 |
71 | addcomprg 7386 | . . . . 5 | |
72 | 71 | adantl 275 | . . . 4 |
73 | 67, 69, 31, 70, 72, 54 | caovord3d 5941 | . . 3 |
74 | 65, 73 | syld 45 | . 2 |
75 | 1, 2, 3, 4, 74 | brecop 6519 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cop 3530 class class class wbr 3929 (class class class)co 5774 cec 6427 cnp 7099 cpp 7101 cltp 7103 cer 7104 cnr 7105 cltr 7111 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 df-iltp 7278 df-enr 7534 df-nr 7535 df-ltr 7538 |
This theorem is referenced by: gt0srpr 7556 lttrsr 7570 ltposr 7571 ltsosr 7572 0lt1sr 7573 ltasrg 7578 aptisr 7587 mulextsr1 7589 archsr 7590 prsrlt 7595 ltpsrprg 7611 mappsrprg 7612 map2psrprg 7613 pitoregt0 7657 |
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