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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 529 |
. . 3
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2 | peano2re 8110 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | simpll 527 |
. . . . 5
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5 | 0red 7975 |
. . . . . . 7
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6 | simprr 531 |
. . . . . . 7
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7 | ltp1 8818 |
. . . . . . . 8
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8 | 1, 7 | syl 14 |
. . . . . . 7
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9 | 5, 1, 3, 6, 8 | lelttrd 8099 |
. . . . . 6
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10 | 3, 9 | gt0ap0d 8603 |
. . . . 5
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11 | 4, 3, 10 | redivclapd 8809 |
. . . 4
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12 | simpl 109 |
. . . . 5
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13 | divge0 8847 |
. . . . 5
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14 | 12, 3, 9, 13 | syl12anc 1246 |
. . . 4
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15 | 11, 14 | jca 306 |
. . 3
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16 | 1, 3, 8 | ltled 8093 |
. . 3
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17 | lemul2a 8833 |
. . 3
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18 | 1, 3, 15, 16, 17 | syl31anc 1251 |
. 2
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19 | 4 | recnd 8003 |
. . 3
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20 | 3 | recnd 8003 |
. . 3
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21 | 19, 20, 10 | divcanap1d 8765 |
. 2
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22 | 18, 21 | breqtrd 4043 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-mulrcl 7927 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-precex 7938 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 ax-pre-mulgt0 7945 ax-pre-mulext 7946 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-br 4018 df-opab 4079 df-id 4307 df-po 4310 df-iso 4311 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-reap 8549 df-ap 8556 df-div 8647 |
This theorem is referenced by: (None) |
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