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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 6887 |
. . . 4
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2 | mulclpi 6887 |
. . . 4
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3 | opelxpi 4469 |
. . . 4
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4 | 1, 2, 3 | syl2an 283 |
. . 3
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5 | 4 | an4s 555 |
. 2
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6 | mulclpi 6887 |
. . . 4
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7 | mulclpi 6887 |
. . . 4
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8 | opelxpi 4469 |
. . . 4
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9 | 6, 7, 8 | syl2an 283 |
. . 3
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10 | 9 | an4s 555 |
. 2
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11 | mulclpi 6887 |
. . . 4
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12 | mulclpi 6887 |
. . . 4
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13 | opelxpi 4469 |
. . . 4
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14 | 11, 12, 13 | syl2an 283 |
. . 3
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15 | 14 | an4s 555 |
. 2
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16 | enqex 6919 |
. 2
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17 | enqer 6917 |
. 2
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18 | df-enq 6906 |
. 2
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19 | simpll 496 |
. . . 4
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20 | simprr 499 |
. . . 4
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21 | 19, 20 | oveq12d 5670 |
. . 3
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22 | simplr 497 |
. . . 4
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23 | simprl 498 |
. . . 4
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24 | 22, 23 | oveq12d 5670 |
. . 3
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25 | 21, 24 | eqeq12d 2102 |
. 2
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26 | simpll 496 |
. . . 4
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27 | simprr 499 |
. . . 4
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28 | 26, 27 | oveq12d 5670 |
. . 3
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29 | simplr 497 |
. . . 4
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30 | simprl 498 |
. . . 4
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31 | 29, 30 | oveq12d 5670 |
. . 3
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32 | 28, 31 | eqeq12d 2102 |
. 2
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33 | dfmpq2 6914 |
. 2
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34 | simpll 496 |
. . . 4
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35 | simprl 498 |
. . . 4
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36 | 34, 35 | oveq12d 5670 |
. . 3
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37 | simplr 497 |
. . . 4
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38 | simprr 499 |
. . . 4
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39 | 37, 38 | oveq12d 5670 |
. . 3
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40 | 36, 39 | opeq12d 3630 |
. 2
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41 | simpll 496 |
. . . 4
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42 | simprl 498 |
. . . 4
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43 | 41, 42 | oveq12d 5670 |
. . 3
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44 | simplr 497 |
. . . 4
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45 | simprr 499 |
. . . 4
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46 | 44, 45 | oveq12d 5670 |
. . 3
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47 | 43, 46 | opeq12d 3630 |
. 2
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48 | simpll 496 |
. . . 4
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49 | simprl 498 |
. . . 4
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50 | 48, 49 | oveq12d 5670 |
. . 3
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51 | simplr 497 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
52 | simprr 499 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
53 | 51, 52 | oveq12d 5670 |
. . 3
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54 | 50, 53 | opeq12d 3630 |
. 2
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55 | df-mqqs 6909 |
. 2
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56 | df-nqqs 6907 |
. 2
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57 | mulcmpblnq 6927 |
. 2
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58 | 5, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57 | oviec 6398 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-oadd 6185 df-omul 6186 df-er 6292 df-ec 6294 df-qs 6298 df-ni 6863 df-mi 6865 df-mpq 6904 df-enq 6906 df-nqqs 6907 df-mqqs 6909 |
This theorem is referenced by: mulclnq 6935 mulcomnqg 6942 mulassnqg 6943 distrnqg 6946 mulidnq 6948 recexnq 6949 ltmnqg 6960 nqnq0m 7014 |
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