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Mirrors > Home > ILE Home > Th. List > m1p1sr | Unicode version |
Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Ref | Expression |
---|---|
m1p1sr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-m1r 7342 |
. . 3
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2 | df-1r 7341 |
. . 3
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3 | 1, 2 | oveq12i 5680 |
. 2
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4 | df-0r 7340 |
. . 3
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5 | 1pr 7176 |
. . . . 5
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6 | addclpr 7159 |
. . . . . 6
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7 | 5, 5, 6 | mp2an 418 |
. . . . 5
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8 | addsrpr 7354 |
. . . . 5
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9 | 5, 7, 7, 5, 8 | mp4an 419 |
. . . 4
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10 | addassprg 7201 |
. . . . . . 7
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11 | 5, 5, 5, 10 | mp3an 1274 |
. . . . . 6
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12 | 11 | oveq2i 5679 |
. . . . 5
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13 | addclpr 7159 |
. . . . . . 7
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14 | 5, 7, 13 | mp2an 418 |
. . . . . 6
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15 | addclpr 7159 |
. . . . . . 7
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16 | 7, 5, 15 | mp2an 418 |
. . . . . 6
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17 | enreceq 7345 |
. . . . . 6
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18 | 5, 5, 14, 16, 17 | mp4an 419 |
. . . . 5
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19 | 12, 18 | mpbir 145 |
. . . 4
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20 | 9, 19 | eqtr4i 2112 |
. . 3
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21 | 4, 20 | eqtr4i 2112 |
. 2
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22 | 3, 21 | eqtr4i 2112 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-eprel 4127 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-irdg 6151 df-1o 6197 df-2o 6198 df-oadd 6201 df-omul 6202 df-er 6308 df-ec 6310 df-qs 6314 df-ni 6926 df-pli 6927 df-mi 6928 df-lti 6929 df-plpq 6966 df-mpq 6967 df-enq 6969 df-nqqs 6970 df-plqqs 6971 df-mqqs 6972 df-1nqqs 6973 df-rq 6974 df-ltnqqs 6975 df-enq0 7046 df-nq0 7047 df-0nq0 7048 df-plq0 7049 df-mq0 7050 df-inp 7088 df-i1p 7089 df-iplp 7090 df-enr 7335 df-nr 7336 df-plr 7337 df-0r 7340 df-1r 7341 df-m1r 7342 |
This theorem is referenced by: pn0sr 7380 caucvgsrlemoffres 7408 caucvgsr 7410 axi2m1 7473 |
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