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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 6670 |
. . 3
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2 | oveq1 5571 |
. . . 4
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3 | 2 | eleq1d 2151 |
. . 3
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4 | oveq2 5572 |
. . . 4
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5 | 4 | eleq1d 2151 |
. . 3
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6 | mulpipqqs 6695 |
. . . 4
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7 | mulclpi 6650 |
. . . . . . 7
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8 | mulclpi 6650 |
. . . . . . 7
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9 | 7, 8 | anim12i 331 |
. . . . . 6
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10 | 9 | an4s 553 |
. . . . 5
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11 | opelxpi 4422 |
. . . . 5
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12 | enqex 6682 |
. . . . . 6
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13 | 12 | ecelqsi 6248 |
. . . . 5
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14 | 10, 11, 13 | 3syl 17 |
. . . 4
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15 | 6, 14 | eqeltrd 2159 |
. . 3
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16 | 1, 3, 5, 15 | 2ecoptocl 6282 |
. 2
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17 | 16, 1 | syl6eleqr 2176 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-iord 4149 df-on 4151 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-1st 5819 df-2nd 5820 df-recs 5975 df-irdg 6040 df-oadd 6090 df-omul 6091 df-er 6194 df-ec 6196 df-qs 6200 df-ni 6626 df-mi 6628 df-mpq 6667 df-enq 6669 df-nqqs 6670 df-mqqs 6672 |
This theorem is referenced by: halfnqq 6732 prarloclemarch 6740 prarloclemarch2 6741 ltrnqg 6742 prarloclemlt 6815 prarloclemlo 6816 prarloclemcalc 6824 addnqprllem 6849 addnqprulem 6850 addnqprl 6851 addnqpru 6852 mpvlu 6861 dmmp 6863 appdivnq 6885 prmuloclemcalc 6887 prmuloc 6888 mulnqprl 6890 mulnqpru 6891 mullocprlem 6892 mullocpr 6893 mulclpr 6894 mulnqprlemrl 6895 mulnqprlemru 6896 mulnqprlemfl 6897 mulnqprlemfu 6898 mulnqpr 6899 mulassprg 6903 distrlem1prl 6904 distrlem1pru 6905 distrlem4prl 6906 distrlem4pru 6907 distrlem5prl 6908 distrlem5pru 6909 1idprl 6912 1idpru 6913 recexprlem1ssl 6955 recexprlem1ssu 6956 recexprlemss1l 6957 recexprlemss1u 6958 |
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