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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 11471 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | 0cn 11230 | . . 3 ⊢ 0 ∈ ℂ | |
3 | subcl 11483 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 689 | . 2 ⊢ (𝐴 ∈ ℂ → (0 − 𝐴) ∈ ℂ) |
5 | 1, 4 | eqeltrid 2833 | 1 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 (class class class)co 7414 ℂcc 11130 0cc0 11132 − cmin 11468 -cneg 11469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470 df-neg 11471 |
This theorem is referenced by: negicn 11485 negcon1 11536 negdi 11541 negdi2 11542 negsubdi2 11543 neg2sub 11544 negcli 11552 negcld 11582 mulneg2 11675 mul2neg 11677 mulsub 11681 divneg 11930 divsubdir 11932 divsubdiv 11954 eqneg 11958 div2neg 11961 divneg2 11962 zeo 12672 sqneg 14106 binom2sub 14208 shftval4 15050 shftcan1 15056 shftcan2 15057 crim 15088 resub 15100 imsub 15108 cjneg 15120 cjsub 15122 absneg 15250 abs2dif2 15306 sqreulem 15332 sqreu 15333 subcn2 15565 risefallfac 15994 fallrisefac 15995 fallfac0 15998 binomrisefac 16012 efcan 16066 efne0 16067 efneg 16068 efsub 16070 sinneg 16116 cosneg 16117 tanneg 16118 efmival 16123 sinhval 16124 coshval 16125 sinsub 16138 cossub 16139 sincossq 16146 cnaddablx 19816 cnaddabl 19817 cnaddinv 19819 cncrng 21309 cncrngOLD 21310 cnfldneg 21316 cnlmod 25060 cnstrcvs 25061 cncvs 25065 plyremlem 26232 reeff1o 26377 sin2pim 26413 cos2pim 26414 cxpsub 26609 cxpsqrt 26630 logrec 26688 asinlem3 26796 asinneg 26811 acosneg 26812 sinasin 26814 asinsin 26817 cosasin 26829 atantan 26848 cnaddabloOLD 30384 hvsubdistr2 30853 spanunsni 31382 ltflcei 37075 dvasin 37171 lcmineqlem1 41494 sqrtcvallem4 43063 sub2times 44648 cosknegpi 45251 etransclem18 45634 etransclem46 45662 addsubeq0 46670 altgsumbcALT 47411 1subrec1sub 47772 sinhpcosh 48165 |
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