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Mirrors > Home > ILE Home > Th. List > ltmul1 | Unicode version |
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltmul1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1a 8480 | . . 3 | |
2 | 1 | ex 114 | . 2 |
3 | recexgt0 8469 | . . . 4 | |
4 | 3 | 3ad2ant3 1009 | . . 3 |
5 | simpl1 989 | . . . . . . . . . 10 | |
6 | simpl3l 1041 | . . . . . . . . . 10 | |
7 | 5, 6 | remulcld 7920 | . . . . . . . . 9 |
8 | simpl2 990 | . . . . . . . . . 10 | |
9 | 8, 6 | remulcld 7920 | . . . . . . . . 9 |
10 | simprl 521 | . . . . . . . . . 10 | |
11 | simprrl 529 | . . . . . . . . . 10 | |
12 | 10, 11 | jca 304 | . . . . . . . . 9 |
13 | 7, 9, 12 | 3jca 1166 | . . . . . . . 8 |
14 | ltmul1a 8480 | . . . . . . . 8 | |
15 | 13, 14 | sylan 281 | . . . . . . 7 |
16 | 5 | recnd 7918 | . . . . . . . . 9 |
17 | 16 | adantr 274 | . . . . . . . 8 |
18 | 6 | recnd 7918 | . . . . . . . . 9 |
19 | 18 | adantr 274 | . . . . . . . 8 |
20 | 10 | recnd 7918 | . . . . . . . . 9 |
21 | 20 | adantr 274 | . . . . . . . 8 |
22 | 17, 19, 21 | mulassd 7913 | . . . . . . 7 |
23 | 8 | recnd 7918 | . . . . . . . . 9 |
24 | 23 | adantr 274 | . . . . . . . 8 |
25 | 24, 19, 21 | mulassd 7913 | . . . . . . 7 |
26 | 15, 22, 25 | 3brtr3d 4007 | . . . . . 6 |
27 | simprrr 530 | . . . . . . . 8 | |
28 | 27 | adantr 274 | . . . . . . 7 |
29 | 28 | oveq2d 5852 | . . . . . 6 |
30 | 28 | oveq2d 5852 | . . . . . 6 |
31 | 26, 29, 30 | 3brtr3d 4007 | . . . . 5 |
32 | 17 | mulid1d 7907 | . . . . 5 |
33 | 24 | mulid1d 7907 | . . . . 5 |
34 | 31, 32, 33 | 3brtr3d 4007 | . . . 4 |
35 | 34 | ex 114 | . . 3 |
36 | 4, 35 | rexlimddv 2586 | . 2 |
37 | 2, 36 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 c1 7745 cmul 7749 clt 7924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-sub 8062 df-neg 8063 |
This theorem is referenced by: lemul1 8482 reapmul1lem 8483 ltmul2 8742 ltdiv1 8754 ltdiv23 8778 recp1lt1 8785 ltmul1i 8806 ltmul1d 9665 mertenslemi1 11462 flodddiv4t2lthalf 11859 qnumgt0 12109 tangtx 13306 |
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