Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > le2add | Unicode version |
Description: Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
le2add |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 481 | . . . 4 | |
2 | simprl 483 | . . . 4 | |
3 | simplr 482 | . . . 4 | |
4 | leadd1 7425 | . . . 4 | |
5 | 1, 2, 3, 4 | syl3anc 1135 | . . 3 |
6 | simprr 484 | . . . 4 | |
7 | leadd2 7426 | . . . 4 | |
8 | 3, 6, 2, 7 | syl3anc 1135 | . . 3 |
9 | 5, 8 | anbi12d 442 | . 2 |
10 | 1, 3 | readdcld 7055 | . . 3 |
11 | 2, 3 | readdcld 7055 | . . 3 |
12 | 2, 6 | readdcld 7055 | . . 3 |
13 | letr 7101 | . . 3 | |
14 | 10, 11, 12, 13 | syl3anc 1135 | . 2 |
15 | 9, 14 | sylbid 139 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 caddc 6892 cle 7061 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-pre-ltwlin 6997 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-iota 4867 df-fv 4910 df-ov 5515 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 |
This theorem is referenced by: addge0 7446 le2addi 7503 le2addd 7554 |
Copyright terms: Public domain | W3C validator |