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Mirrors > Home > ILE Home > Th. List > mulclpr | Unicode version |
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) |
Ref | Expression |
---|---|
mulclpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-imp 7497 |
. . . 4
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2 | 1 | genpelxp 7539 |
. . 3
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3 | mulclnq 7404 |
. . . 4
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4 | 1, 3 | genpml 7545 |
. . 3
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5 | 1, 3 | genpmu 7546 |
. . 3
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6 | 2, 4, 5 | jca32 310 |
. 2
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7 | ltmnqg 7429 |
. . . . 5
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8 | mulcomnqg 7411 |
. . . . 5
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9 | mulnqprl 7596 |
. . . . 5
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10 | 1, 3, 7, 8, 9 | genprndl 7549 |
. . . 4
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11 | mulnqpru 7597 |
. . . . 5
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12 | 1, 3, 7, 8, 11 | genprndu 7550 |
. . . 4
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13 | 10, 12 | jca 306 |
. . 3
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14 | 1, 3, 7, 8 | genpdisj 7551 |
. . 3
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15 | mullocpr 7599 |
. . 3
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16 | 13, 14, 15 | 3jca 1179 |
. 2
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17 | elnp1st2nd 7504 |
. 2
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18 | 6, 16, 17 | sylanbrc 417 |
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-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 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 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-1o 6440 df-2o 6441 df-oadd 6444 df-omul 6445 df-er 6558 df-ec 6560 df-qs 6564 df-ni 7332 df-pli 7333 df-mi 7334 df-lti 7335 df-plpq 7372 df-mpq 7373 df-enq 7375 df-nqqs 7376 df-plqqs 7377 df-mqqs 7378 df-1nqqs 7379 df-rq 7380 df-ltnqqs 7381 df-enq0 7452 df-nq0 7453 df-0nq0 7454 df-plq0 7455 df-mq0 7456 df-inp 7494 df-imp 7497 |
This theorem is referenced by: mulnqprlemfl 7603 mulnqprlemfu 7604 mulnqpr 7605 mulassprg 7609 distrlem1prl 7610 distrlem1pru 7611 distrlem4prl 7612 distrlem4pru 7613 distrlem5prl 7614 distrlem5pru 7615 distrprg 7616 1idpr 7620 recexprlemex 7665 ltmprr 7670 mulcmpblnrlemg 7768 mulcmpblnr 7769 mulclsr 7782 mulcomsrg 7785 mulasssrg 7786 distrsrg 7787 m1m1sr 7789 1idsr 7796 00sr 7797 recexgt0sr 7801 mulgt0sr 7806 mulextsr1lem 7808 mulextsr1 7809 recidpirq 7886 |
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