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Mirrors > Home > ILE Home > Th. List > distrprg | Unicode version |
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
distrprg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrlem1prl 7606 |
. . 3
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2 | distrlem5prl 7610 |
. . 3
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3 | 1, 2 | eqssd 3187 |
. 2
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4 | distrlem1pru 7607 |
. . 3
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5 | distrlem5pru 7611 |
. . 3
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6 | 4, 5 | eqssd 3187 |
. 2
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7 | simp1 999 |
. . . 4
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8 | simp2 1000 |
. . . . 5
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9 | simp3 1001 |
. . . . 5
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10 | addclpr 7561 |
. . . . 5
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11 | 8, 9, 10 | syl2anc 411 |
. . . 4
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12 | mulclpr 7596 |
. . . 4
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13 | 7, 11, 12 | syl2anc 411 |
. . 3
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14 | mulclpr 7596 |
. . . . 5
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15 | 7, 8, 14 | syl2anc 411 |
. . . 4
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16 | mulclpr 7596 |
. . . . 5
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17 | 7, 9, 16 | syl2anc 411 |
. . . 4
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18 | addclpr 7561 |
. . . 4
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19 | 15, 17, 18 | syl2anc 411 |
. . 3
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20 | preqlu 7496 |
. . 3
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21 | 13, 19, 20 | syl2anc 411 |
. 2
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22 | 3, 6, 21 | mpbir2and 946 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
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-eprel 4304 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-irdg 6390 df-1o 6436 df-2o 6437 df-oadd 6440 df-omul 6441 df-er 6554 df-ec 6556 df-qs 6560 df-ni 7328 df-pli 7329 df-mi 7330 df-lti 7331 df-plpq 7368 df-mpq 7369 df-enq 7371 df-nqqs 7372 df-plqqs 7373 df-mqqs 7374 df-1nqqs 7375 df-rq 7376 df-ltnqqs 7377 df-enq0 7448 df-nq0 7449 df-0nq0 7450 df-plq0 7451 df-mq0 7452 df-inp 7490 df-iplp 7492 df-imp 7493 |
This theorem is referenced by: ltmprr 7666 mulcmpblnrlemg 7764 mulasssrg 7782 distrsrg 7783 m1m1sr 7785 1idsr 7792 recexgt0sr 7797 mulgt0sr 7802 mulextsr1lem 7804 recidpirqlemcalc 7881 |
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