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Mirrors > Home > MPE Home > Th. List > ledivge1le | Structured version Visualization version Unicode version |
Description: If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.) |
Ref | Expression |
---|---|
ledivge1le |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divle1le 11900 | . . . . . . . . 9 | |
2 | 1 | adantr 481 | . . . . . . . 8 |
3 | rerpdivcl 11861 | . . . . . . . . . . 11 | |
4 | 3 | adantr 481 | . . . . . . . . . 10 |
5 | 1red 10055 | . . . . . . . . . 10 | |
6 | rpre 11839 | . . . . . . . . . . 11 | |
7 | 6 | adantl 482 | . . . . . . . . . 10 |
8 | letr 10131 | . . . . . . . . . 10 | |
9 | 4, 5, 7, 8 | syl3anc 1326 | . . . . . . . . 9 |
10 | 9 | expd 452 | . . . . . . . 8 |
11 | 2, 10 | sylbird 250 | . . . . . . 7 |
12 | 11 | com23 86 | . . . . . 6 |
13 | 12 | expimpd 629 | . . . . 5 |
14 | 13 | ex 450 | . . . 4 |
15 | 14 | 3imp1 1280 | . . 3 |
16 | simp1 1061 | . . . . . 6 | |
17 | 6 | adantr 481 | . . . . . . . 8 |
18 | 0lt1 10550 | . . . . . . . . . 10 | |
19 | 0red 10041 | . . . . . . . . . . 11 | |
20 | 1red 10055 | . . . . . . . . . . 11 | |
21 | ltletr 10129 | . . . . . . . . . . 11 | |
22 | 19, 20, 6, 21 | syl3anc 1326 | . . . . . . . . . 10 |
23 | 18, 22 | mpani 712 | . . . . . . . . 9 |
24 | 23 | imp 445 | . . . . . . . 8 |
25 | 17, 24 | jca 554 | . . . . . . 7 |
26 | 25 | 3ad2ant3 1084 | . . . . . 6 |
27 | rpregt0 11846 | . . . . . . 7 | |
28 | 27 | 3ad2ant2 1083 | . . . . . 6 |
29 | 16, 26, 28 | 3jca 1242 | . . . . 5 |
30 | 29 | adantr 481 | . . . 4 |
31 | lediv23 10915 | . . . 4 | |
32 | 30, 31 | syl 17 | . . 3 |
33 | 15, 32 | mpbird 247 | . 2 |
34 | 33 | ex 450 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wcel 1990 class class class wbr 4653 (class class class)co 6650 cr 9935 cc0 9936 c1 9937 clt 10074 cle 10075 cdiv 10684 crp 11832 |
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-rmo 2920 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 df-div 10685 df-rp 11833 |
This theorem is referenced by: gausslemma2dlem1a 25090 |
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