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Mirrors > Home > ILE Home > Th. List > addltmul | Unicode version |
Description: Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.) |
Ref | Expression |
---|---|
addltmul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8918 | . . . . . . 7 | |
2 | 1re 7889 | . . . . . . 7 | |
3 | ltsub1 8347 | . . . . . . 7 | |
4 | 1, 2, 3 | mp3an13 1317 | . . . . . 6 |
5 | 2m1e1 8966 | . . . . . . 7 | |
6 | 5 | breq1i 3983 | . . . . . 6 |
7 | 4, 6 | bitrdi 195 | . . . . 5 |
8 | ltsub1 8347 | . . . . . . 7 | |
9 | 1, 2, 8 | mp3an13 1317 | . . . . . 6 |
10 | 5 | breq1i 3983 | . . . . . 6 |
11 | 9, 10 | bitrdi 195 | . . . . 5 |
12 | 7, 11 | bi2anan9 596 | . . . 4 |
13 | peano2rem 8156 | . . . . 5 | |
14 | peano2rem 8156 | . . . . 5 | |
15 | mulgt1 8749 | . . . . . 6 | |
16 | 15 | ex 114 | . . . . 5 |
17 | 13, 14, 16 | syl2an 287 | . . . 4 |
18 | 12, 17 | sylbid 149 | . . 3 |
19 | recn 7877 | . . . . . 6 | |
20 | recn 7877 | . . . . . 6 | |
21 | ax-1cn 7837 | . . . . . . 7 | |
22 | mulsub 8290 | . . . . . . . 8 | |
23 | 21, 22 | mpanl2 432 | . . . . . . 7 |
24 | 21, 23 | mpanr2 435 | . . . . . 6 |
25 | 19, 20, 24 | syl2an 287 | . . . . 5 |
26 | 25 | breq2d 3988 | . . . 4 |
27 | remulcl 7872 | . . . . . . . 8 | |
28 | 2, 27 | mpan2 422 | . . . . . . 7 |
29 | remulcl 7872 | . . . . . . . 8 | |
30 | 2, 29 | mpan2 422 | . . . . . . 7 |
31 | readdcl 7870 | . . . . . . 7 | |
32 | 28, 30, 31 | syl2an 287 | . . . . . 6 |
33 | remulcl 7872 | . . . . . . 7 | |
34 | 2, 2 | remulcli 7904 | . . . . . . 7 |
35 | readdcl 7870 | . . . . . . 7 | |
36 | 33, 34, 35 | sylancl 410 | . . . . . 6 |
37 | ltaddsub2 8326 | . . . . . . 7 | |
38 | 2, 37 | mp3an2 1314 | . . . . . 6 |
39 | 32, 36, 38 | syl2anc 409 | . . . . 5 |
40 | 1t1e1 9000 | . . . . . . 7 | |
41 | 40 | oveq2i 5847 | . . . . . 6 |
42 | 41 | breq2i 3984 | . . . . 5 |
43 | 39, 42 | bitr3di 194 | . . . 4 |
44 | ltadd1 8318 | . . . . . . 7 | |
45 | 2, 44 | mp3an3 1315 | . . . . . 6 |
46 | 32, 33, 45 | syl2anc 409 | . . . . 5 |
47 | ax-1rid 7851 | . . . . . . 7 | |
48 | ax-1rid 7851 | . . . . . . 7 | |
49 | 47, 48 | oveqan12d 5855 | . . . . . 6 |
50 | 49 | breq1d 3986 | . . . . 5 |
51 | 46, 50 | bitr3d 189 | . . . 4 |
52 | 26, 43, 51 | 3bitrd 213 | . . 3 |
53 | 18, 52 | sylibd 148 | . 2 |
54 | 53 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 class class class wbr 3976 (class class class)co 5836 cc 7742 cr 7743 c1 7745 caddc 7747 cmul 7749 clt 7924 cmin 8060 c2 8899 |
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-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-lttrn 7858 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 df-2 8907 |
This theorem is referenced by: (None) |
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