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Mirrors > Home > ILE Home > Th. List > mulclnq | Unicode version |
Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) |
Ref | Expression |
---|---|
mulclnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 6446 | . . 3 | |
2 | oveq1 5519 | . . . 4 | |
3 | 2 | eleq1d 2106 | . . 3 |
4 | oveq2 5520 | . . . 4 | |
5 | 4 | eleq1d 2106 | . . 3 |
6 | mulpipqqs 6471 | . . . 4 | |
7 | mulclpi 6426 | . . . . . . 7 | |
8 | mulclpi 6426 | . . . . . . 7 | |
9 | 7, 8 | anim12i 321 | . . . . . 6 |
10 | 9 | an4s 522 | . . . . 5 |
11 | opelxpi 4376 | . . . . 5 | |
12 | enqex 6458 | . . . . . 6 | |
13 | 12 | ecelqsi 6160 | . . . . 5 |
14 | 10, 11, 13 | 3syl 17 | . . . 4 |
15 | 6, 14 | eqeltrd 2114 | . . 3 |
16 | 1, 3, 5, 15 | 2ecoptocl 6194 | . 2 |
17 | 16, 1 | syl6eleqr 2131 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cop 3378 cxp 4343 (class class class)co 5512 cec 6104 cqs 6105 cnpi 6370 cmi 6372 ceq 6377 cnq 6378 cmq 6381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 |
This theorem is referenced by: halfnqq 6508 prarloclemarch 6516 prarloclemarch2 6517 ltrnqg 6518 prarloclemlt 6591 prarloclemlo 6592 prarloclemcalc 6600 addnqprllem 6625 addnqprulem 6626 addnqprl 6627 addnqpru 6628 mpvlu 6637 dmmp 6639 appdivnq 6661 prmuloclemcalc 6663 prmuloc 6664 mulnqprl 6666 mulnqpru 6667 mullocprlem 6668 mullocpr 6669 mulclpr 6670 mulnqprlemrl 6671 mulnqprlemru 6672 mulnqprlemfl 6673 mulnqprlemfu 6674 mulnqpr 6675 mulassprg 6679 distrlem1prl 6680 distrlem1pru 6681 distrlem4prl 6682 distrlem4pru 6683 distrlem5prl 6684 distrlem5pru 6685 1idprl 6688 1idpru 6689 recexprlem1ssl 6731 recexprlem1ssu 6732 recexprlemss1l 6733 recexprlemss1u 6734 |
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