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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 7414 |
. . 3
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2 | distrlem5prl 7418 |
. . 3
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3 | 1, 2 | eqssd 3119 |
. 2
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4 | distrlem1pru 7415 |
. . 3
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5 | distrlem5pru 7419 |
. . 3
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6 | 4, 5 | eqssd 3119 |
. 2
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7 | simp1 982 |
. . . 4
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8 | simp2 983 |
. . . . 5
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9 | simp3 984 |
. . . . 5
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10 | addclpr 7369 |
. . . . 5
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11 | 8, 9, 10 | syl2anc 409 |
. . . 4
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12 | mulclpr 7404 |
. . . 4
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13 | 7, 11, 12 | syl2anc 409 |
. . 3
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14 | mulclpr 7404 |
. . . . 5
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15 | 7, 8, 14 | syl2anc 409 |
. . . 4
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16 | mulclpr 7404 |
. . . . 5
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17 | 7, 9, 16 | syl2anc 409 |
. . . 4
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18 | addclpr 7369 |
. . . 4
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19 | 15, 17, 18 | syl2anc 409 |
. . 3
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20 | preqlu 7304 |
. . 3
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21 | 13, 19, 20 | syl2anc 409 |
. 2
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22 | 3, 6, 21 | mpbir2and 929 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-iplp 7300 df-imp 7301 |
This theorem is referenced by: ltmprr 7474 mulcmpblnrlemg 7572 mulasssrg 7590 distrsrg 7591 m1m1sr 7593 1idsr 7600 recexgt0sr 7605 mulgt0sr 7610 mulextsr1lem 7612 recidpirqlemcalc 7689 |
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