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Mirrors > Home > MPE Home > Th. List > ltmul2 | Structured version 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 10283 |
. 2
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2 | recn 9476 |
. . . 4
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3 | recn 9476 |
. . . . . . 7
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4 | mulcom 9472 |
. . . . . . 7
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5 | 3, 4 | sylan 471 |
. . . . . 6
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6 | 5 | 3adant2 1007 |
. . . . 5
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7 | recn 9476 |
. . . . . . 7
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8 | mulcom 9472 |
. . . . . . 7
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9 | 7, 8 | sylan 471 |
. . . . . 6
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10 | 9 | 3adant1 1006 |
. . . . 5
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11 | 6, 10 | breq12d 4406 |
. . . 4
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12 | 2, 11 | syl3an3 1254 |
. . 3
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13 | 12 | 3adant3r 1216 |
. 2
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14 | 1, 13 | bitrd 253 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-po 4742 df-so 4743 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-er 7204 df-en 7414 df-dom 7415 df-sdom 7416 df-pnf 9524 df-mnf 9525 df-ltxr 9527 df-sub 9701 df-neg 9702 |
This theorem is referenced by: ltmul12a 10289 mulgt1 10292 ltmulgt11 10293 lt2msq1 10319 ltdiv2 10321 ltmul2i 10358 ltmul2d 11169 ef01bndlem 13579 cos01gt0 13586 sin4lt0 13590 iserodd 14013 pockthg 14078 prmreclem1 14088 prmreclem5 14092 blcvx 20500 dvcvx 21618 itgulm 21999 tangtx 22093 chtub 22677 bposlem1 22749 bposlem2 22750 bposlem7 22755 lgsdilem 22787 lgsquadlem1 22819 lgsquadlem2 22820 chebbnd1lem3 22846 chto1ub 22851 pntlemb 22972 irrapxlem1 29304 irrapxlem2 29305 irrapxlem5 29308 pellexlem2 29312 rmspecsqrnq 29388 stoweidlem11 29947 stoweidlem26 29962 |
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