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Mirrors > Home > MPE Home > Th. List > negcl | Structured version Visualization version 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 10861 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | 0cn 10621 | . . 3 ⊢ 0 ∈ ℂ | |
3 | subcl 10873 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 686 | . 2 ⊢ (𝐴 ∈ ℂ → (0 − 𝐴) ∈ ℂ) |
5 | 1, 4 | eqeltrid 2914 | 1 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 0cc0 10525 − cmin 10858 -cneg 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 |
This theorem is referenced by: negicn 10875 negcon1 10926 negdi 10931 negdi2 10932 negsubdi2 10933 neg2sub 10934 negcli 10942 negcld 10972 mulneg2 11065 mul2neg 11067 mulsub 11071 divneg 11320 divsubdir 11322 divsubdiv 11344 eqneg 11348 div2neg 11351 divneg2 11352 zeo 12056 sqneg 13470 binom2sub 13569 shftval4 14424 shftcan1 14430 shftcan2 14431 crim 14462 resub 14474 imsub 14482 cjneg 14494 cjsub 14496 absneg 14625 abs2dif2 14681 sqreulem 14707 sqreu 14708 subcn2 14939 risefallfac 15366 fallrisefac 15367 fallfac0 15370 binomrisefac 15384 efcan 15437 efne0 15438 efneg 15439 efsub 15441 sinneg 15487 cosneg 15488 tanneg 15489 efmival 15494 sinhval 15495 coshval 15496 sinsub 15509 cossub 15510 sincossq 15517 cnaddablx 18917 cnaddabl 18918 cnaddinv 18920 cncrng 20494 cnfldneg 20499 cnlmod 23671 cnstrcvs 23672 cncvs 23676 plyremlem 24820 reeff1o 24962 sin2pim 24998 cos2pim 24999 cxpsub 25192 cxpsqrt 25213 logrec 25268 asinlem3 25376 asinneg 25391 acosneg 25392 sinasin 25394 asinsin 25397 cosasin 25409 atantan 25428 cnaddabloOLD 28285 hvsubdistr2 28754 spanunsni 29283 ltflcei 34761 dvasin 34859 sub2times 41416 cosknegpi 42026 etransclem18 42414 etransclem46 42442 addsubeq0 43373 altgsumbcALT 44329 1subrec1sub 44620 sinhpcosh 44767 |
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