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Mirrors > Home > ILE Home > Th. List > mulpipqqs | Unicode version |
Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) |
Ref | Expression |
---|---|
mulpipqqs |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulclpi 7358 |
. . . 4
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2 | mulclpi 7358 |
. . . 4
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3 | opelxpi 4676 |
. . . 4
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4 | 1, 2, 3 | syl2an 289 |
. . 3
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5 | 4 | an4s 588 |
. 2
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6 | mulclpi 7358 |
. . . 4
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7 | mulclpi 7358 |
. . . 4
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8 | opelxpi 4676 |
. . . 4
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9 | 6, 7, 8 | syl2an 289 |
. . 3
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10 | 9 | an4s 588 |
. 2
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11 | mulclpi 7358 |
. . . 4
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12 | mulclpi 7358 |
. . . 4
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13 | opelxpi 4676 |
. . . 4
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14 | 11, 12, 13 | syl2an 289 |
. . 3
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15 | 14 | an4s 588 |
. 2
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16 | enqex 7390 |
. 2
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17 | enqer 7388 |
. 2
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18 | df-enq 7377 |
. 2
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19 | simpll 527 |
. . . 4
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20 | simprr 531 |
. . . 4
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21 | 19, 20 | oveq12d 5915 |
. . 3
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22 | simplr 528 |
. . . 4
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23 | simprl 529 |
. . . 4
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24 | 22, 23 | oveq12d 5915 |
. . 3
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25 | 21, 24 | eqeq12d 2204 |
. 2
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26 | simpll 527 |
. . . 4
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27 | simprr 531 |
. . . 4
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28 | 26, 27 | oveq12d 5915 |
. . 3
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29 | simplr 528 |
. . . 4
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30 | simprl 529 |
. . . 4
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31 | 29, 30 | oveq12d 5915 |
. . 3
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32 | 28, 31 | eqeq12d 2204 |
. 2
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33 | dfmpq2 7385 |
. 2
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34 | simpll 527 |
. . . 4
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35 | simprl 529 |
. . . 4
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36 | 34, 35 | oveq12d 5915 |
. . 3
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37 | simplr 528 |
. . . 4
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38 | simprr 531 |
. . . 4
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39 | 37, 38 | oveq12d 5915 |
. . 3
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40 | 36, 39 | opeq12d 3801 |
. 2
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41 | simpll 527 |
. . . 4
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42 | simprl 529 |
. . . 4
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43 | 41, 42 | oveq12d 5915 |
. . 3
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44 | simplr 528 |
. . . 4
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45 | simprr 531 |
. . . 4
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46 | 44, 45 | oveq12d 5915 |
. . 3
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47 | 43, 46 | opeq12d 3801 |
. 2
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48 | simpll 527 |
. . . 4
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49 | simprl 529 |
. . . 4
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50 | 48, 49 | oveq12d 5915 |
. . 3
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51 | simplr 528 |
. . . 4
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52 | simprr 531 |
. . . 4
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53 | 51, 52 | oveq12d 5915 |
. . 3
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54 | 50, 53 | opeq12d 3801 |
. 2
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55 | df-mqqs 7380 |
. 2
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56 | df-nqqs 7378 |
. 2
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57 | mulcmpblnq 7398 |
. 2
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58 | 5, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57 | oviec 6668 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-mi 7336 df-mpq 7375 df-enq 7377 df-nqqs 7378 df-mqqs 7380 |
This theorem is referenced by: mulclnq 7406 mulcomnqg 7413 mulassnqg 7414 distrnqg 7417 mulidnq 7419 recexnq 7420 ltmnqg 7431 nqnq0m 7485 |
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