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Mirrors > Home > ILE Home > Th. List > lemul2a | Unicode version |
Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) |
Ref | Expression |
---|---|
lemul2a |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lemul1a 8319 |
. 2
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2 | recn 7475 |
. . . . . 6
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3 | recn 7475 |
. . . . . 6
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4 | mulcom 7471 |
. . . . . 6
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5 | 2, 3, 4 | syl2an 283 |
. . . . 5
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6 | 5 | adantrr 463 |
. . . 4
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7 | 6 | 3adant2 962 |
. . 3
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8 | 7 | adantr 270 |
. 2
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9 | recn 7475 |
. . . . . 6
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10 | mulcom 7471 |
. . . . . 6
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11 | 9, 3, 10 | syl2an 283 |
. . . . 5
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12 | 11 | adantrr 463 |
. . . 4
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13 | 12 | 3adant1 961 |
. . 3
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14 | 13 | adantr 270 |
. 2
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15 | 1, 8, 14 | 3brtr3d 3874 |
1
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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-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulrcl 7444 ax-addcom 7445 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-1rid 7452 ax-0id 7453 ax-rnegex 7454 ax-precex 7455 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-apti 7460 ax-pre-ltadd 7461 ax-pre-mulgt0 7462 ax-pre-mulext 7463 |
This theorem depends on definitions: df-bi 115 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-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-po 4123 df-iso 4124 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-sub 7655 df-neg 7656 df-reap 8052 df-ap 8059 |
This theorem is referenced by: lemul12b 8322 ledivp1 8364 lemul2ad 8401 facavg 10154 mulcn2 10701 cvgratnnlemnexp 10918 cvgratnnlemmn 10919 mertenslemi1 10929 |
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