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Mirrors > Home > ILE Home > Th. List > negcl | Unicode version |
Description: Closure law for negative. (Contributed by NM, 6-Aug-2003.) |
Ref | Expression |
---|---|
negcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 7904 | . 2 | |
2 | 0cn 7726 | . . 3 | |
3 | subcl 7929 | . . 3 | |
4 | 2, 3 | mpan 420 | . 2 |
5 | 1, 4 | eqeltrid 2204 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1465 (class class class)co 5742 cc 7586 cc0 7588 cmin 7901 cneg 7902 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 |
This theorem is referenced by: negicn 7931 negcon1 7982 negdi 7987 negdi2 7988 negsubdi2 7989 neg2sub 7990 negcli 7998 negcld 8028 mulneg2 8126 mul2neg 8128 mulsub 8131 apsub1 8372 subap0 8373 divnegap 8434 divsubdirap 8436 divsubdivap 8456 eqneg 8460 div2negap 8463 divneg2ap 8464 zeo 9124 sqneg 10320 binom2sub 10373 shftval4 10568 shftcan1 10574 shftcan2 10575 crim 10598 resub 10610 imsub 10618 cjneg 10630 cjsub 10632 absneg 10790 abs2dif2 10847 subcn2 11048 efcan 11309 efap0 11310 efne0 11311 efneg 11312 efsub 11314 sinneg 11360 cosneg 11361 tannegap 11362 efmival 11367 sinsub 11374 cossub 11375 sincossq 11382 sin2pim 12821 cos2pim 12822 |
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