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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-m1r 7668 | . . 3 | |
2 | df-1r 7667 | . . 3 | |
3 | 1, 2 | oveq12i 5851 | . 2 |
4 | df-0r 7666 | . . 3 | |
5 | 1pr 7489 | . . . . 5 | |
6 | addclpr 7472 | . . . . . 6 | |
7 | 5, 5, 6 | mp2an 423 | . . . . 5 |
8 | addsrpr 7680 | . . . . 5 | |
9 | 5, 7, 7, 5, 8 | mp4an 424 | . . . 4 |
10 | addassprg 7514 | . . . . . . 7 | |
11 | 5, 5, 5, 10 | mp3an 1326 | . . . . . 6 |
12 | 11 | oveq2i 5850 | . . . . 5 |
13 | addclpr 7472 | . . . . . . 7 | |
14 | 5, 7, 13 | mp2an 423 | . . . . . 6 |
15 | addclpr 7472 | . . . . . . 7 | |
16 | 7, 5, 15 | mp2an 423 | . . . . . 6 |
17 | enreceq 7671 | . . . . . 6 | |
18 | 5, 5, 14, 16, 17 | mp4an 424 | . . . . 5 |
19 | 12, 18 | mpbir 145 | . . . 4 |
20 | 9, 19 | eqtr4i 2188 | . . 3 |
21 | 4, 20 | eqtr4i 2188 | . 2 |
22 | 3, 21 | eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1342 wcel 2135 cop 3576 (class class class)co 5839 cec 6493 cnp 7226 c1p 7227 cpp 7228 cer 7231 c0r 7233 c1r 7234 cm1r 7235 cplr 7236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-eprel 4264 df-id 4268 df-po 4271 df-iso 4272 df-iord 4341 df-on 4343 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-1o 6378 df-2o 6379 df-oadd 6382 df-omul 6383 df-er 6495 df-ec 6497 df-qs 6501 df-ni 7239 df-pli 7240 df-mi 7241 df-lti 7242 df-plpq 7279 df-mpq 7280 df-enq 7282 df-nqqs 7283 df-plqqs 7284 df-mqqs 7285 df-1nqqs 7286 df-rq 7287 df-ltnqqs 7288 df-enq0 7359 df-nq0 7360 df-0nq0 7361 df-plq0 7362 df-mq0 7363 df-inp 7401 df-i1p 7402 df-iplp 7403 df-enr 7661 df-nr 7662 df-plr 7663 df-0r 7666 df-1r 7667 df-m1r 7668 |
This theorem is referenced by: pn0sr 7706 ltm1sr 7712 caucvgsrlemoffres 7735 caucvgsr 7737 suplocsrlempr 7742 axi2m1 7810 |
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