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Mirrors > Home > ILE Home > Th. List > pn0sr | Unicode version |
Description: A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Ref | Expression |
---|---|
pn0sr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m1r 7780 |
. . . 4
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2 | 1sr 7779 |
. . . 4
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3 | distrsrg 7787 |
. . . 4
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4 | 1, 2, 3 | mp3an23 1340 |
. . 3
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5 | m1p1sr 7788 |
. . . . 5
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6 | 5 | oveq2i 5906 |
. . . 4
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7 | 6 | a1i 9 |
. . 3
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8 | mulclsr 7782 |
. . . . 5
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9 | 1, 8 | mpan2 425 |
. . . 4
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10 | mulclsr 7782 |
. . . . 5
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11 | 2, 10 | mpan2 425 |
. . . 4
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12 | addcomsrg 7783 |
. . . 4
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13 | 9, 11, 12 | syl2anc 411 |
. . 3
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14 | 4, 7, 13 | 3eqtr3d 2230 |
. 2
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15 | 00sr 7797 |
. 2
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16 | 1idsr 7796 |
. . 3
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17 | 16 | oveq1d 5910 |
. 2
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18 | 14, 15, 17 | 3eqtr3rd 2231 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-1o 6440 df-2o 6441 df-oadd 6444 df-omul 6445 df-er 6558 df-ec 6560 df-qs 6564 df-ni 7332 df-pli 7333 df-mi 7334 df-lti 7335 df-plpq 7372 df-mpq 7373 df-enq 7375 df-nqqs 7376 df-plqqs 7377 df-mqqs 7378 df-1nqqs 7379 df-rq 7380 df-ltnqqs 7381 df-enq0 7452 df-nq0 7453 df-0nq0 7454 df-plq0 7455 df-mq0 7456 df-inp 7494 df-i1p 7495 df-iplp 7496 df-imp 7497 df-enr 7754 df-nr 7755 df-plr 7756 df-mr 7757 df-0r 7759 df-1r 7760 df-m1r 7761 |
This theorem is referenced by: negexsr 7800 caucvgsrlemoffval 7824 map2psrprg 7833 axrnegex 7907 |
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