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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 8790 | . . . . . . 7 | |
2 | 1re 7765 | . . . . . . 7 | |
3 | ltsub1 8220 | . . . . . . 7 | |
4 | 1, 2, 3 | mp3an13 1306 | . . . . . 6 |
5 | 2m1e1 8838 | . . . . . . 7 | |
6 | 5 | breq1i 3936 | . . . . . 6 |
7 | 4, 6 | syl6bb 195 | . . . . 5 |
8 | ltsub1 8220 | . . . . . . 7 | |
9 | 1, 2, 8 | mp3an13 1306 | . . . . . 6 |
10 | 5 | breq1i 3936 | . . . . . 6 |
11 | 9, 10 | syl6bb 195 | . . . . 5 |
12 | 7, 11 | bi2anan9 595 | . . . 4 |
13 | peano2rem 8029 | . . . . 5 | |
14 | peano2rem 8029 | . . . . 5 | |
15 | mulgt1 8621 | . . . . . 6 | |
16 | 15 | ex 114 | . . . . 5 |
17 | 13, 14, 16 | syl2an 287 | . . . 4 |
18 | 12, 17 | sylbid 149 | . . 3 |
19 | recn 7753 | . . . . . 6 | |
20 | recn 7753 | . . . . . 6 | |
21 | ax-1cn 7713 | . . . . . . 7 | |
22 | mulsub 8163 | . . . . . . . 8 | |
23 | 21, 22 | mpanl2 431 | . . . . . . 7 |
24 | 21, 23 | mpanr2 434 | . . . . . 6 |
25 | 19, 20, 24 | syl2an 287 | . . . . 5 |
26 | 25 | breq2d 3941 | . . . 4 |
27 | 1t1e1 8872 | . . . . . . 7 | |
28 | 27 | oveq2i 5785 | . . . . . 6 |
29 | 28 | breq2i 3937 | . . . . 5 |
30 | remulcl 7748 | . . . . . . . 8 | |
31 | 2, 30 | mpan2 421 | . . . . . . 7 |
32 | remulcl 7748 | . . . . . . . 8 | |
33 | 2, 32 | mpan2 421 | . . . . . . 7 |
34 | readdcl 7746 | . . . . . . 7 | |
35 | 31, 33, 34 | syl2an 287 | . . . . . 6 |
36 | remulcl 7748 | . . . . . . 7 | |
37 | 2, 2 | remulcli 7780 | . . . . . . 7 |
38 | readdcl 7746 | . . . . . . 7 | |
39 | 36, 37, 38 | sylancl 409 | . . . . . 6 |
40 | ltaddsub2 8199 | . . . . . . 7 | |
41 | 2, 40 | mp3an2 1303 | . . . . . 6 |
42 | 35, 39, 41 | syl2anc 408 | . . . . 5 |
43 | 29, 42 | syl5rbbr 194 | . . . 4 |
44 | ltadd1 8191 | . . . . . . 7 | |
45 | 2, 44 | mp3an3 1304 | . . . . . 6 |
46 | 35, 36, 45 | syl2anc 408 | . . . . 5 |
47 | ax-1rid 7727 | . . . . . . 7 | |
48 | ax-1rid 7727 | . . . . . . 7 | |
49 | 47, 48 | oveqan12d 5793 | . . . . . 6 |
50 | 49 | breq1d 3939 | . . . . 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 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 c1 7621 caddc 7623 cmul 7625 clt 7800 cmin 7933 c2 8771 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-2 8779 |
This theorem is referenced by: (None) |
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