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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 7960 |
. 2
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2 | 0cn 7782 |
. . 3
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3 | subcl 7985 |
. . 3
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4 | 2, 3 | mpan 421 |
. 2
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5 | 1, 4 | eqeltrid 2227 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-neg 7960 |
This theorem is referenced by: negicn 7987 negcon1 8038 negdi 8043 negdi2 8044 negsubdi2 8045 neg2sub 8046 negcli 8054 negcld 8084 mulneg2 8182 mul2neg 8184 mulsub 8187 apsub1 8428 subap0 8429 divnegap 8490 divsubdirap 8492 divsubdivap 8512 eqneg 8516 div2negap 8519 divneg2ap 8520 zeo 9180 sqneg 10383 binom2sub 10436 shftval4 10632 shftcan1 10638 shftcan2 10639 crim 10662 resub 10674 imsub 10682 cjneg 10694 cjsub 10696 absneg 10854 abs2dif2 10911 subcn2 11112 efcan 11419 efap0 11420 efne0 11421 efneg 11422 efsub 11424 sinneg 11469 cosneg 11470 tannegap 11471 efmival 11476 sinsub 11483 cossub 11484 sincossq 11491 sin2pim 12942 cos2pim 12943 rpcxpsub 13037 |
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