Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > add20 | Unicode version |
Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
add20 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 486 | . . . . . . . . 9 | |
2 | simplrl 487 | . . . . . . . . . 10 | |
3 | simplll 485 | . . . . . . . . . 10 | |
4 | addge02 7468 | . . . . . . . . . 10 | |
5 | 2, 3, 4 | syl2anc 391 | . . . . . . . . 9 |
6 | 1, 5 | mpbid 135 | . . . . . . . 8 |
7 | simpr 103 | . . . . . . . 8 | |
8 | 6, 7 | breqtrd 3788 | . . . . . . 7 |
9 | simplrr 488 | . . . . . . 7 | |
10 | 0red 7028 | . . . . . . . 8 | |
11 | 2, 10 | letri3d 7133 | . . . . . . 7 |
12 | 8, 9, 11 | mpbir2and 851 | . . . . . 6 |
13 | 12 | oveq2d 5528 | . . . . 5 |
14 | 3 | recnd 7054 | . . . . . 6 |
15 | 14 | addid1d 7162 | . . . . 5 |
16 | 13, 7, 15 | 3eqtr3rd 2081 | . . . 4 |
17 | 16, 12 | jca 290 | . . 3 |
18 | 17 | ex 108 | . 2 |
19 | oveq12 5521 | . . 3 | |
20 | 00id 7154 | . . 3 | |
21 | 19, 20 | syl6eq 2088 | . 2 |
22 | 18, 21 | impbid1 130 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 cc0 6889 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-1re 6978 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-ltirr 6996 ax-pre-apti 6999 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: add20i 7484 sumsqeq0 9332 |
Copyright terms: Public domain | W3C validator |