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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 11495 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
| 2 | 0cn 11253 | . . 3 ⊢ 0 ∈ ℂ | |
| 3 | subcl 11507 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − 𝐴) ∈ ℂ) | |
| 4 | 2, 3 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℂ → (0 − 𝐴) ∈ ℂ) |
| 5 | 1, 4 | eqeltrid 2845 | 1 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 − cmin 11492 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: negicn 11509 negcon1 11561 negdi 11566 negdi2 11567 negsubdi2 11568 neg2sub 11569 negcli 11577 negcld 11607 mulneg2 11700 mul2neg 11702 mulsub 11706 divneg 11959 divsubdir 11961 divsubdiv 11983 eqneg 11987 div2neg 11990 divneg2 11991 zeo 12704 sqneg 14156 binom2sub 14259 shftval4 15116 shftcan1 15122 shftcan2 15123 crim 15154 resub 15166 imsub 15174 cjneg 15186 cjsub 15188 absneg 15316 abs2dif2 15372 sqreulem 15398 sqreu 15399 subcn2 15631 risefallfac 16060 fallrisefac 16061 fallfac0 16064 binomrisefac 16078 efcan 16132 efne0 16133 efneg 16134 efsub 16136 sinneg 16182 cosneg 16183 tanneg 16184 efmival 16189 sinhval 16190 coshval 16191 sinsub 16204 cossub 16205 sincossq 16212 cnaddablx 19886 cnaddabl 19887 cnaddinv 19889 cncrng 21401 cncrngOLD 21402 cnfldneg 21408 cnlmod 25173 cnstrcvs 25174 cncvs 25178 plyremlem 26346 reeff1o 26491 sin2pim 26527 cos2pim 26528 cxpsub 26724 cxpsqrt 26745 logrec 26806 asinlem3 26914 asinneg 26929 acosneg 26930 sinasin 26932 asinsin 26935 cosasin 26947 atantan 26966 cnaddabloOLD 30600 hvsubdistr2 31069 spanunsni 31598 ltflcei 37615 dvasin 37711 lcmineqlem1 42030 sqrtcvallem4 43652 sub2times 45284 cosknegpi 45884 etransclem18 46267 etransclem46 46295 addsubeq0 47308 altgsumbcALT 48269 1subrec1sub 48626 sinhpcosh 49259 |
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