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Mirrors > Home > ILE Home > Th. List > ltmul1a | Unicode version |
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltmul1a |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 1001 |
. . . . 5
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2 | simpl1 1000 |
. . . . 5
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3 | 1, 2 | resubcld 8336 |
. . . 4
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4 | simpl3l 1052 |
. . . 4
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5 | simpr 110 |
. . . . 5
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6 | 2, 1 | posdifd 8487 |
. . . . 5
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7 | 5, 6 | mpbid 147 |
. . . 4
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8 | simpl3r 1053 |
. . . 4
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9 | 3, 4, 7, 8 | mulgt0d 8078 |
. . 3
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10 | 1 | recnd 7984 |
. . . 4
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11 | 2 | recnd 7984 |
. . . 4
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12 | 4 | recnd 7984 |
. . . 4
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13 | 10, 11, 12 | subdird 8370 |
. . 3
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14 | 9, 13 | breqtrd 4029 |
. 2
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15 | 2, 4 | remulcld 7986 |
. . 3
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16 | 1, 4 | remulcld 7986 |
. . 3
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17 | 15, 16 | posdifd 8487 |
. 2
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18 | 14, 17 | mpbird 167 |
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-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 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: ltmul1 8547 |
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