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Mirrors > Home > ILE Home > Th. List > ltpsrprg | Unicode version |
Description: Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) |
Ref | Expression |
---|---|
ltpsrprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 982 | . . . 4 | |
2 | 1pr 7469 | . . . 4 | |
3 | enrex 7652 | . . . . 5 | |
4 | df-nr 7642 | . . . . 5 | |
5 | 3, 4 | ecopqsi 6532 | . . . 4 |
6 | 1, 2, 5 | sylancl 410 | . . 3 |
7 | simp2 983 | . . . 4 | |
8 | 3, 4 | ecopqsi 6532 | . . . 4 |
9 | 7, 2, 8 | sylancl 410 | . . 3 |
10 | simp3 984 | . . 3 | |
11 | ltasrg 7685 | . . 3 | |
12 | 6, 9, 10, 11 | syl3anc 1220 | . 2 |
13 | addcomprg 7493 | . . . . 5 | |
14 | 1, 2, 13 | sylancl 410 | . . . 4 |
15 | 14 | breq1d 3975 | . . 3 |
16 | 2 | a1i 9 | . . . 4 |
17 | ltsrprg 7662 | . . . 4 | |
18 | 1, 16, 7, 16, 17 | syl22anc 1221 | . . 3 |
19 | ltaprg 7534 | . . . 4 | |
20 | 1, 7, 16, 19 | syl3anc 1220 | . . 3 |
21 | 15, 18, 20 | 3bitr4d 219 | . 2 |
22 | 12, 21 | bitr3d 189 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 963 wceq 1335 wcel 2128 cop 3563 class class class wbr 3965 (class class class)co 5821 cec 6475 cnp 7206 c1p 7207 cpp 7208 cltp 7210 cer 7211 cnr 7212 cplr 7216 cltr 7218 |
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-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 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-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-1o 6360 df-2o 6361 df-oadd 6364 df-omul 6365 df-er 6477 df-ec 6479 df-qs 6483 df-ni 7219 df-pli 7220 df-mi 7221 df-lti 7222 df-plpq 7259 df-mpq 7260 df-enq 7262 df-nqqs 7263 df-plqqs 7264 df-mqqs 7265 df-1nqqs 7266 df-rq 7267 df-ltnqqs 7268 df-enq0 7339 df-nq0 7340 df-0nq0 7341 df-plq0 7342 df-mq0 7343 df-inp 7381 df-i1p 7382 df-iplp 7383 df-iltp 7385 df-enr 7641 df-nr 7642 df-plr 7643 df-ltr 7645 |
This theorem is referenced by: suplocsrlemb 7721 suplocsrlempr 7722 suplocsrlem 7723 |
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