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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 6538 | . . 3 | |
2 | oveq1 5539 | . . . 4 | |
3 | 2 | eleq1d 2147 | . . 3 |
4 | oveq2 5540 | . . . 4 | |
5 | 4 | eleq1d 2147 | . . 3 |
6 | mulpipqqs 6563 | . . . 4 | |
7 | mulclpi 6518 | . . . . . . 7 | |
8 | mulclpi 6518 | . . . . . . 7 | |
9 | 7, 8 | anim12i 331 | . . . . . 6 |
10 | 9 | an4s 552 | . . . . 5 |
11 | opelxpi 4394 | . . . . 5 | |
12 | enqex 6550 | . . . . . 6 | |
13 | 12 | ecelqsi 6183 | . . . . 5 |
14 | 10, 11, 13 | 3syl 17 | . . . 4 |
15 | 6, 14 | eqeltrd 2155 | . . 3 |
16 | 1, 3, 5, 15 | 2ecoptocl 6217 | . 2 |
17 | 16, 1 | syl6eleqr 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cop 3401 cxp 4361 (class class class)co 5532 cec 6127 cqs 6128 cnpi 6462 cmi 6464 ceq 6469 cnq 6470 cmq 6473 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-mqqs 6540 |
This theorem is referenced by: halfnqq 6600 prarloclemarch 6608 prarloclemarch2 6609 ltrnqg 6610 prarloclemlt 6683 prarloclemlo 6684 prarloclemcalc 6692 addnqprllem 6717 addnqprulem 6718 addnqprl 6719 addnqpru 6720 mpvlu 6729 dmmp 6731 appdivnq 6753 prmuloclemcalc 6755 prmuloc 6756 mulnqprl 6758 mulnqpru 6759 mullocprlem 6760 mullocpr 6761 mulclpr 6762 mulnqprlemrl 6763 mulnqprlemru 6764 mulnqprlemfl 6765 mulnqprlemfu 6766 mulnqpr 6767 mulassprg 6771 distrlem1prl 6772 distrlem1pru 6773 distrlem4prl 6774 distrlem4pru 6775 distrlem5prl 6776 distrlem5pru 6777 1idprl 6780 1idpru 6781 recexprlem1ssl 6823 recexprlem1ssu 6824 recexprlemss1l 6825 recexprlemss1u 6826 |
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