Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ledivge1le | 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 8802 | . . . . . . . . 9 | |
2 | 1 | adantr 270 | . . . . . . . 8 |
3 | rerpdivcl 8764 | . . . . . . . . . . 11 | |
4 | 3 | adantr 270 | . . . . . . . . . 10 |
5 | 1red 7134 | . . . . . . . . . 10 | |
6 | rpre 8740 | . . . . . . . . . . 11 | |
7 | 6 | adantl 271 | . . . . . . . . . 10 |
8 | letr 7194 | . . . . . . . . . 10 | |
9 | 4, 5, 7, 8 | syl3anc 1169 | . . . . . . . . 9 |
10 | 9 | expd 254 | . . . . . . . 8 |
11 | 2, 10 | sylbird 168 | . . . . . . 7 |
12 | 11 | com23 77 | . . . . . 6 |
13 | 12 | expimpd 355 | . . . . 5 |
14 | 13 | ex 113 | . . . 4 |
15 | 14 | 3imp1 1151 | . . 3 |
16 | simp1 938 | . . . . . 6 | |
17 | 6 | adantr 270 | . . . . . . . 8 |
18 | 0lt1 7236 | . . . . . . . . . 10 | |
19 | 0red 7120 | . . . . . . . . . . 11 | |
20 | 1red 7134 | . . . . . . . . . . 11 | |
21 | ltletr 7200 | . . . . . . . . . . 11 | |
22 | 19, 20, 6, 21 | syl3anc 1169 | . . . . . . . . . 10 |
23 | 18, 22 | mpani 420 | . . . . . . . . 9 |
24 | 23 | imp 122 | . . . . . . . 8 |
25 | 17, 24 | jca 300 | . . . . . . 7 |
26 | 25 | 3ad2ant3 961 | . . . . . 6 |
27 | rpregt0 8747 | . . . . . . 7 | |
28 | 27 | 3ad2ant2 960 | . . . . . 6 |
29 | 16, 26, 28 | 3jca 1118 | . . . . 5 |
30 | 29 | adantr 270 | . . . 4 |
31 | lediv23 7971 | . . . 4 | |
32 | 30, 31 | syl 14 | . . 3 |
33 | 15, 32 | mpbird 165 | . 2 |
34 | 33 | ex 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 cc0 6981 c1 6982 clt 7153 cle 7154 cdiv 7760 crp 8734 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-rp 8735 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |