Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ltsub2 | Unicode version |
Description: Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltsub2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd2 8308 | . . 3 | |
2 | simp3 988 | . . . . 5 | |
3 | simp1 986 | . . . . 5 | |
4 | 2, 3 | readdcld 7919 | . . . 4 |
5 | simp2 987 | . . . 4 | |
6 | ltsubadd 8321 | . . . 4 | |
7 | 4, 5, 2, 6 | syl3anc 1227 | . . 3 |
8 | 2 | recnd 7918 | . . . . 5 |
9 | 3 | recnd 7918 | . . . . 5 |
10 | 5 | recnd 7918 | . . . . 5 |
11 | 8, 9, 10 | addsubd 8221 | . . . 4 |
12 | 11 | breq1d 3986 | . . 3 |
13 | 1, 7, 12 | 3bitr2d 215 | . 2 |
14 | 2, 5 | resubcld 8270 | . . 3 |
15 | ltaddsub 8325 | . . 3 | |
16 | 14, 3, 2, 15 | syl3anc 1227 | . 2 |
17 | 13, 16 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 967 wcel 2135 class class class wbr 3976 (class class class)co 5836 cr 7743 caddc 7747 clt 7924 cmin 8060 |
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-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltadd 7860 |
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 |
This theorem is referenced by: lt2sub 8349 ltneg 8351 ltsub2d 8444 ltm1 8732 |
Copyright terms: Public domain | W3C validator |