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Mirrors > Home > MPE Home > Th. List > ltmul12a | Structured version Visualization version Unicode version |
Description: Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
Ref | Expression |
---|---|
ltmul12a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 798 | . . . 4 | |
2 | simpllr 799 | . . . 4 | |
3 | simpll 790 | . . . . . 6 | |
4 | simprl 794 | . . . . . 6 | |
5 | 3, 4 | jca 554 | . . . . 5 |
6 | 5 | ad2ant2l 782 | . . . 4 |
7 | ltle 10126 | . . . . . . 7 | |
8 | 7 | imp 445 | . . . . . 6 |
9 | 8 | adantrl 752 | . . . . 5 |
10 | 9 | ad2ant2r 783 | . . . 4 |
11 | lemul1a 10877 | . . . 4 | |
12 | 1, 2, 6, 10, 11 | syl31anc 1329 | . . 3 |
13 | simplrl 800 | . . . . . . 7 | |
14 | simplrr 801 | . . . . . . 7 | |
15 | simpllr 799 | . . . . . . 7 | |
16 | 0re 10040 | . . . . . . . . . 10 | |
17 | lelttr 10128 | . . . . . . . . . 10 | |
18 | 16, 17 | mp3an1 1411 | . . . . . . . . 9 |
19 | 18 | imp 445 | . . . . . . . 8 |
20 | 19 | adantlr 751 | . . . . . . 7 |
21 | ltmul2 10874 | . . . . . . 7 | |
22 | 13, 14, 15, 20, 21 | syl112anc 1330 | . . . . . 6 |
23 | 22 | biimpa 501 | . . . . 5 |
24 | 23 | anasss 679 | . . . 4 |
25 | 24 | adantrrl 760 | . . 3 |
26 | remulcl 10021 | . . . . . 6 | |
27 | 26 | ad2ant2r 783 | . . . . 5 |
28 | remulcl 10021 | . . . . . 6 | |
29 | 28 | ad2ant2lr 784 | . . . . 5 |
30 | remulcl 10021 | . . . . . 6 | |
31 | 30 | ad2ant2l 782 | . . . . 5 |
32 | lelttr 10128 | . . . . 5 | |
33 | 27, 29, 31, 32 | syl3anc 1326 | . . . 4 |
34 | 33 | adantr 481 | . . 3 |
35 | 12, 25, 34 | mp2and 715 | . 2 |
36 | 35 | an4s 869 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 class class class wbr 4653 (class class class)co 6650 cr 9935 cc0 9936 cmul 9941 clt 10074 cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 |
This theorem is referenced by: ltmul12ad 10965 hgt750lem2 30730 expmordi 37512 tgblthelfgott 41703 tgoldbach 41705 tgblthelfgottOLD 41709 tgoldbachOLD 41712 |
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