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Mirrors > Home > ILE Home > Th. List > ltmul2 | Unicode version |
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) |
Ref | Expression |
---|---|
ltmul2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1 8547 |
. 2
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2 | recn 7943 |
. . . 4
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3 | recn 7943 |
. . . . . . 7
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4 | mulcom 7939 |
. . . . . . 7
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5 | 3, 4 | sylan 283 |
. . . . . 6
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6 | 5 | 3adant2 1016 |
. . . . 5
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7 | recn 7943 |
. . . . . . 7
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8 | mulcom 7939 |
. . . . . . 7
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9 | 7, 8 | sylan 283 |
. . . . . 6
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10 | 9 | 3adant1 1015 |
. . . . 5
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11 | 6, 10 | breq12d 4016 |
. . . 4
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12 | 2, 11 | syl3an3 1273 |
. . 3
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13 | 12 | 3adant3r 1235 |
. 2
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14 | 1, 13 | bitrd 188 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-sub 8128 df-neg 8129 |
This theorem is referenced by: ltmul12a 8815 mulgt1 8818 ltmulgt11 8819 lt2msq1 8840 ltdiv2 8842 ltmul2i 8878 ltmul2d 9737 ef01bndlem 11759 cos01gt0 11765 sin4lt0 11769 pockthg 12349 tangtx 14152 lgsdilem 14321 |
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