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Mirrors > Home > ILE Home > Th. List > negcl | GIF version |
Description: Closure law for negative. (Contributed by NM, 6-Aug-2003.) |
Ref | Expression |
---|---|
negcl | ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 8080 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | 0cn 7899 | . . 3 ⊢ 0 ∈ ℂ | |
3 | subcl 8105 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 422 | . 2 ⊢ (𝐴 ∈ ℂ → (0 − 𝐴) ∈ ℂ) |
5 | 1, 4 | eqeltrid 2257 | 1 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 (class class class)co 5850 ℂcc 7759 0cc0 7761 − cmin 8077 -cneg 8078 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7853 ax-1cn 7854 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079 df-neg 8080 |
This theorem is referenced by: negicn 8107 negcon1 8158 negdi 8163 negdi2 8164 negsubdi2 8165 neg2sub 8166 negcli 8174 negcld 8204 mulneg2 8302 mul2neg 8304 mulsub 8307 apsub1 8548 subap0 8549 divnegap 8610 divsubdirap 8612 divsubdivap 8632 eqneg 8636 div2negap 8639 divneg2ap 8640 zeo 9304 sqneg 10522 binom2sub 10576 shftval4 10779 shftcan1 10785 shftcan2 10786 crim 10809 resub 10821 imsub 10829 cjneg 10841 cjsub 10843 absneg 11001 abs2dif2 11058 subcn2 11261 efcan 11626 efap0 11627 efne0 11628 efneg 11629 efsub 11631 sinneg 11676 cosneg 11677 tannegap 11678 efmival 11683 sinsub 11690 cossub 11691 sincossq 11698 sin2pim 13449 cos2pim 13450 rpcxpsub 13544 |
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