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Mirrors > Home > ILE Home > Th. List > mulclnq0 | Unicode version |
Description: Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.) |
Ref | Expression |
---|---|
mulclnq0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 7454 |
. . 3
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2 | oveq1 5903 |
. . . 4
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3 | 2 | eleq1d 2258 |
. . 3
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4 | oveq2 5904 |
. . . 4
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5 | 4 | eleq1d 2258 |
. . 3
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6 | mulnnnq0 7479 |
. . . 4
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7 | nnmcl 6506 |
. . . . . . 7
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8 | mulpiord 7346 |
. . . . . . . 8
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9 | mulclpi 7357 |
. . . . . . . 8
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10 | 8, 9 | eqeltrrd 2267 |
. . . . . . 7
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11 | 7, 10 | anim12i 338 |
. . . . . 6
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12 | 11 | an4s 588 |
. . . . 5
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13 | opelxpi 4676 |
. . . . 5
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14 | enq0ex 7468 |
. . . . . 6
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15 | 14 | ecelqsi 6615 |
. . . . 5
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16 | 12, 13, 15 | 3syl 17 |
. . . 4
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17 | 6, 16 | eqeltrd 2266 |
. . 3
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18 | 1, 3, 5, 17 | 2ecoptocl 6649 |
. 2
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19 | 18, 1 | eleqtrrdi 2283 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-oadd 6445 df-omul 6446 df-er 6559 df-ec 6561 df-qs 6565 df-ni 7333 df-mi 7335 df-enq0 7453 df-nq0 7454 df-mq0 7457 |
This theorem is referenced by: prarloclemcalc 7531 |
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