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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 7104 |
. . 3
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2 | oveq1 5735 |
. . . 4
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3 | 2 | eleq1d 2183 |
. . 3
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4 | oveq2 5736 |
. . . 4
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5 | 4 | eleq1d 2183 |
. . 3
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6 | mulpipqqs 7129 |
. . . 4
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7 | mulclpi 7084 |
. . . . . . 7
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8 | mulclpi 7084 |
. . . . . . 7
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9 | 7, 8 | anim12i 334 |
. . . . . 6
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10 | 9 | an4s 560 |
. . . . 5
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11 | opelxpi 4531 |
. . . . 5
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12 | enqex 7116 |
. . . . . 6
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13 | 12 | ecelqsi 6437 |
. . . . 5
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14 | 10, 11, 13 | 3syl 17 |
. . . 4
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15 | 6, 14 | eqeltrd 2191 |
. . 3
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16 | 1, 3, 5, 15 | 2ecoptocl 6471 |
. 2
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17 | 16, 1 | syl6eleqr 2208 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-id 4175 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-recs 6156 df-irdg 6221 df-oadd 6271 df-omul 6272 df-er 6383 df-ec 6385 df-qs 6389 df-ni 7060 df-mi 7062 df-mpq 7101 df-enq 7103 df-nqqs 7104 df-mqqs 7106 |
This theorem is referenced by: halfnqq 7166 prarloclemarch 7174 prarloclemarch2 7175 ltrnqg 7176 prarloclemlt 7249 prarloclemlo 7250 prarloclemcalc 7258 addnqprllem 7283 addnqprulem 7284 addnqprl 7285 addnqpru 7286 mpvlu 7295 dmmp 7297 appdivnq 7319 prmuloclemcalc 7321 prmuloc 7322 mulnqprl 7324 mulnqpru 7325 mullocprlem 7326 mullocpr 7327 mulclpr 7328 mulnqprlemrl 7329 mulnqprlemru 7330 mulnqprlemfl 7331 mulnqprlemfu 7332 mulnqpr 7333 mulassprg 7337 distrlem1prl 7338 distrlem1pru 7339 distrlem4prl 7340 distrlem4pru 7341 distrlem5prl 7342 distrlem5pru 7343 1idprl 7346 1idpru 7347 recexprlem1ssl 7389 recexprlem1ssu 7390 recexprlemss1l 7391 recexprlemss1u 7392 |
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