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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 8145 | . 2 ⊢ -𝐴 = (0 − 𝐴) | |
2 | 0cn 7963 | . . 3 ⊢ 0 ∈ ℂ | |
3 | subcl 8170 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 424 | . 2 ⊢ (𝐴 ∈ ℂ → (0 − 𝐴) ∈ ℂ) |
5 | 1, 4 | eqeltrid 2274 | 1 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 (class class class)co 5888 ℂcc 7823 0cc0 7825 − cmin 8142 -cneg 8143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 df-neg 8145 |
This theorem is referenced by: negicn 8172 negcon1 8223 negdi 8228 negdi2 8229 negsubdi2 8230 neg2sub 8231 negcli 8239 negcld 8269 mulneg2 8367 mul2neg 8369 mulsub 8372 apsub1 8613 subap0 8614 divnegap 8677 divsubdirap 8679 divsubdivap 8699 eqneg 8703 div2negap 8706 divneg2ap 8707 zeo 9372 sqneg 10593 binom2sub 10648 shftval4 10851 shftcan1 10857 shftcan2 10858 crim 10881 resub 10893 imsub 10901 cjneg 10913 cjsub 10915 absneg 11073 abs2dif2 11130 subcn2 11333 efcan 11698 efap0 11699 efne0 11700 efneg 11701 efsub 11703 sinneg 11748 cosneg 11749 tannegap 11750 efmival 11755 sinsub 11762 cossub 11763 sincossq 11770 cncrng 13745 cnfldneg 13749 sin2pim 14530 cos2pim 14531 rpcxpsub 14625 |
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