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Mirrors > Home > ILE Home > Th. List > lediv23 | Unicode version |
Description: Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.) |
Ref | Expression |
---|---|
lediv23 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 944 |
. . . 4
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2 | simp2l 970 |
. . . 4
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3 | simp2r 971 |
. . . . 5
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4 | 2, 3 | gt0ap0d 8168 |
. . . 4
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5 | 1, 2, 4 | redivclapd 8364 |
. . 3
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6 | simp3l 972 |
. . 3
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7 | lemul1 8133 |
. . 3
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8 | 5, 6, 2, 3, 7 | syl112anc 1179 |
. 2
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9 | 1 | recnd 7579 |
. . . 4
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10 | 2 | recnd 7579 |
. . . 4
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11 | 9, 10, 4 | divcanap1d 8321 |
. . 3
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12 | 11 | breq1d 3863 |
. 2
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13 | 6, 2 | remulcld 7581 |
. . . 4
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14 | lediv1 8393 |
. . . 4
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15 | 13, 14 | syld3an2 1222 |
. . 3
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16 | 6 | recnd 7579 |
. . . . 5
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17 | simp3r 973 |
. . . . . 6
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18 | 6, 17 | gt0ap0d 8168 |
. . . . 5
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19 | 10, 16, 18 | divcanap3d 8325 |
. . . 4
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20 | 19 | breq2d 3865 |
. . 3
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21 | 15, 20 | bitrd 187 |
. 2
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22 | 8, 12, 21 | 3bitrd 213 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-id 4131 df-po 4134 df-iso 4135 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 |
This theorem is referenced by: divle1le 9265 ledivge1le 9266 lediv23d 9298 |
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