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Mirrors > Home > MPE Home > Th. List > neg0 | Structured version Visualization version GIF version |
Description: Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.) |
Ref | Expression |
---|---|
neg0 | ⊢ -0 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 10873 | . 2 ⊢ -0 = (0 − 0) | |
2 | 0cn 10633 | . . 3 ⊢ 0 ∈ ℂ | |
3 | subid 10905 | . . 3 ⊢ (0 ∈ ℂ → (0 − 0) = 0) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (0 − 0) = 0 |
5 | 1, 4 | eqtri 2844 | 1 ⊢ -0 = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 0cc0 10537 − cmin 10870 -cneg 10871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 |
This theorem is referenced by: negeq0 10940 lt0neg1 11146 lt0neg2 11147 le0neg1 11148 le0neg2 11149 neg1lt0 11755 elznn0 11997 znegcl 12018 xneg0 12606 expneg 13438 sqeqd 14525 sqrmo 14611 0risefac 15392 sin0 15502 m1bits 15789 lcmneg 15947 pcneg 16210 mulgneg 18246 mulgneg2 18261 iblrelem 24391 itgrevallem1 24395 ditg0 24451 ditgneg 24455 logtayl 25243 dcubic2 25422 atan0 25486 atancj 25488 ppiub 25780 lgsneg1 25898 rpvmasum2 26088 ostth3 26214 divnumden2 30534 archirngz 30818 ccfldextdgrr 31057 xrge0iif1 31181 fsum2dsub 31878 bj-pinftyccb 34506 bj-minftyccb 34510 itgaddnclem2 34966 ftc1anclem5 34986 areacirc 35002 monotoddzzfi 39559 acongeq 39600 sqwvfourb 42534 etransclem46 42585 sigariz 43140 sigarcol 43141 sigaradd 43143 |
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