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Mirrors > Home > ILE Home > Th. List > eqneg | Unicode version |
Description: A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
eqneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1p1times 7889 | . . 3 | |
2 | negid 8002 | . . . 4 | |
3 | ax-1cn 7706 | . . . . . 6 | |
4 | 3, 3 | addcli 7763 | . . . . 5 |
5 | 4 | mul01i 8146 | . . . 4 |
6 | 2, 5 | syl6reqr 2189 | . . 3 |
7 | 1, 6 | eqeq12d 2152 | . 2 |
8 | id 19 | . . 3 | |
9 | 0cnd 7752 | . . 3 | |
10 | 4 | a1i 9 | . . 3 |
11 | 1re 7758 | . . . . . 6 | |
12 | 11, 11 | readdcli 7772 | . . . . 5 |
13 | 0lt1 7882 | . . . . . 6 | |
14 | 11, 11, 13, 13 | addgt0ii 8246 | . . . . 5 |
15 | 12, 14 | gt0ap0ii 8383 | . . . 4 # |
16 | 15 | a1i 9 | . . 3 # |
17 | 8, 9, 10, 16 | mulcanapd 8415 | . 2 |
18 | negcl 7955 | . . 3 | |
19 | 8, 8, 18 | addcand 7939 | . 2 |
20 | 7, 17, 19 | 3bitr3rd 218 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wcel 1480 class class class wbr 3924 (class class class)co 5767 cc 7611 cc0 7613 c1 7614 caddc 7616 cmul 7618 cneg 7927 # cap 8336 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
This theorem is referenced by: eqnegd 8486 eqnegi 8494 |
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