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Mirrors > Home > ILE Home > Th. List > mulclsr | Unicode version |
Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) |
Ref | Expression |
---|---|
mulclsr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7535 | . . 3 | |
2 | oveq1 5781 | . . . 4 | |
3 | 2 | eleq1d 2208 | . . 3 |
4 | oveq2 5782 | . . . 4 | |
5 | 4 | eleq1d 2208 | . . 3 |
6 | mulsrpr 7554 | . . . 4 | |
7 | mulclpr 7380 | . . . . . . . 8 | |
8 | mulclpr 7380 | . . . . . . . 8 | |
9 | addclpr 7345 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2an 287 | . . . . . . 7 |
11 | 10 | an4s 577 | . . . . . 6 |
12 | mulclpr 7380 | . . . . . . . 8 | |
13 | mulclpr 7380 | . . . . . . . 8 | |
14 | addclpr 7345 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2an 287 | . . . . . . 7 |
16 | 15 | an42s 578 | . . . . . 6 |
17 | 11, 16 | jca 304 | . . . . 5 |
18 | opelxpi 4571 | . . . . 5 | |
19 | enrex 7545 | . . . . . 6 | |
20 | 19 | ecelqsi 6483 | . . . . 5 |
21 | 17, 18, 20 | 3syl 17 | . . . 4 |
22 | 6, 21 | eqeltrd 2216 | . . 3 |
23 | 1, 3, 5, 22 | 2ecoptocl 6517 | . 2 |
24 | 23, 1 | eleqtrrdi 2233 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 cxp 4537 (class class class)co 5774 cec 6427 cqs 6428 cnp 7099 cpp 7101 cmp 7102 cer 7104 cnr 7105 cmr 7110 |
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-imp 7277 df-enr 7534 df-nr 7535 df-mr 7537 |
This theorem is referenced by: pn0sr 7579 negexsr 7580 caucvgsrlemoffval 7604 caucvgsrlemofff 7605 map2psrprg 7613 mulcnsr 7643 mulresr 7646 mulcnsrec 7651 axmulcl 7674 axmulrcl 7675 axmulcom 7679 axmulass 7681 axdistr 7682 axrnegex 7687 |
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