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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 7710 |
. 2
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2 | 0cn 7534 |
. . 3
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3 | subcl 7735 |
. . 3
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4 | 2, 3 | mpan 416 |
. 2
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5 | 1, 4 | syl5eqel 2175 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-resscn 7491 ax-1cn 7492 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7709 df-neg 7710 |
This theorem is referenced by: negicn 7737 negcon1 7788 negdi 7793 negdi2 7794 negsubdi2 7795 neg2sub 7796 negcli 7804 negcld 7834 mulneg2 7928 mul2neg 7930 mulsub 7933 apsub1 8171 subap0 8172 divnegap 8227 divsubdirap 8229 divsubdivap 8249 eqneg 8253 div2negap 8256 divneg2ap 8257 zeo 8905 sqneg 10068 binom2sub 10121 shftval4 10316 shftcan1 10322 shftcan2 10323 crim 10346 resub 10358 imsub 10366 cjneg 10378 cjsub 10380 absneg 10537 abs2dif2 10594 subcn2 10754 efcan 11020 efap0 11021 efne0 11022 efneg 11023 efsub 11025 sinneg 11071 cosneg 11072 tannegap 11073 efmival 11078 sinsub 11085 cossub 11086 sincossq 11093 |
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