Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > neg0 | GIF version |
Description: Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.) |
Ref | Expression |
---|---|
neg0 | ⊢ -0 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 7929 | . 2 ⊢ -0 = (0 − 0) | |
2 | 0cn 7751 | . . 3 ⊢ 0 ∈ ℂ | |
3 | subid 7974 | . . 3 ⊢ (0 ∈ ℂ → (0 − 0) = 0) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (0 − 0) = 0 |
5 | 1, 4 | eqtri 2158 | 1 ⊢ -0 = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 0cc0 7613 − cmin 7926 -cneg 7927 |
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-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-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
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-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-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-sub 7928 df-neg 7929 |
This theorem is referenced by: negeq0 8009 lt0neg1 8223 lt0neg2 8224 le0neg1 8225 le0neg2 8226 negap0 8385 neg1lt0 8821 elznn0 9062 znegcl 9078 xneg0 9607 expnegap0 10294 resqrexlemover 10775 sin0 11425 lcmneg 11744 limcimolemlt 12791 |
Copyright terms: Public domain | W3C validator |