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Mirrors > Home > ILE Home > Th. List > ledivp1 | Unicode version |
Description: Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.) |
Ref | Expression |
---|---|
ledivp1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 499 |
. . 3
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2 | peano2re 7679 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | simpll 497 |
. . . . 5
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5 | 0red 7550 |
. . . . . . 7
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6 | simprr 500 |
. . . . . . 7
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7 | ltp1 8366 |
. . . . . . . 8
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8 | 1, 7 | syl 14 |
. . . . . . 7
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9 | 5, 1, 3, 6, 8 | lelttrd 7669 |
. . . . . 6
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10 | 3, 9 | gt0ap0d 8166 |
. . . . 5
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11 | 4, 3, 10 | redivclapd 8362 |
. . . 4
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12 | simpl 108 |
. . . . 5
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13 | divge0 8395 |
. . . . 5
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14 | 12, 3, 9, 13 | syl12anc 1173 |
. . . 4
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15 | 11, 14 | jca 301 |
. . 3
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16 | 1, 3, 8 | ltled 7663 |
. . 3
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17 | lemul2a 8381 |
. . 3
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18 | 1, 3, 15, 16, 17 | syl31anc 1178 |
. 2
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19 | 4 | recnd 7577 |
. . 3
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20 | 3 | recnd 7577 |
. . 3
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21 | 19, 20, 10 | divcanap1d 8319 |
. 2
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22 | 18, 21 | breqtrd 3875 |
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 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
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 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 |
This theorem is referenced by: (None) |
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