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Mirrors > Home > ILE Home > Th. List > addclnq0 | Unicode version |
Description: Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Ref | Expression |
---|---|
addclnq0 | Q0 Q0 +Q0 Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 7233 | . . 3 Q0 ~Q0 | |
2 | oveq1 5781 | . . . 4 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 | |
3 | 2 | eleq1d 2208 | . . 3 ~Q0 ~Q0 +Q0 ~Q0 ~Q0 +Q0 ~Q0 ~Q0 |
4 | oveq2 5782 | . . . 4 ~Q0 +Q0 ~Q0 +Q0 | |
5 | 4 | eleq1d 2208 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 +Q0 ~Q0 |
6 | addnnnq0 7257 | . . . 4 ~Q0 +Q0 ~Q0 ~Q0 | |
7 | pinn 7117 | . . . . . . . . 9 | |
8 | nnmcl 6377 | . . . . . . . . 9 | |
9 | 7, 8 | sylan2 284 | . . . . . . . 8 |
10 | pinn 7117 | . . . . . . . . 9 | |
11 | nnmcl 6377 | . . . . . . . . 9 | |
12 | 10, 11 | sylan 281 | . . . . . . . 8 |
13 | nnacl 6376 | . . . . . . . 8 | |
14 | 9, 12, 13 | syl2an 287 | . . . . . . 7 |
15 | 14 | an42s 578 | . . . . . 6 |
16 | mulpiord 7125 | . . . . . . . 8 | |
17 | mulclpi 7136 | . . . . . . . 8 | |
18 | 16, 17 | eqeltrrd 2217 | . . . . . . 7 |
19 | 18 | ad2ant2l 499 | . . . . . 6 |
20 | 15, 19 | jca 304 | . . . . 5 |
21 | opelxpi 4571 | . . . . 5 | |
22 | enq0ex 7247 | . . . . . 6 ~Q0 | |
23 | 22 | ecelqsi 6483 | . . . . 5 ~Q0 ~Q0 |
24 | 20, 21, 23 | 3syl 17 | . . . 4 ~Q0 ~Q0 |
25 | 6, 24 | eqeltrd 2216 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 |
26 | 1, 3, 5, 25 | 2ecoptocl 6517 | . 2 Q0 Q0 +Q0 ~Q0 |
27 | 26, 1 | eleqtrrdi 2233 | 1 Q0 Q0 +Q0 Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 com 4504 cxp 4537 (class class class)co 5774 coa 6310 comu 6311 cec 6427 cqs 6428 cnpi 7080 cmi 7082 ~Q0 ceq0 7094 Q0cnq0 7095 +Q0 cplq0 7097 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-enq0 7232 df-nq0 7233 df-plq0 7235 |
This theorem is referenced by: distnq0r 7271 prarloclemcalc 7310 |
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