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Mirrors > Home > ILE Home > Th. List > addclnq0 | Unicode version |
Description: Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Ref | Expression |
---|---|
addclnq0 | Q0 Q0 +Q0 Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 6615 | . . 3 Q0 ~Q0 | |
2 | oveq1 5539 | . . . 4 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 | |
3 | 2 | eleq1d 2147 | . . 3 ~Q0 ~Q0 +Q0 ~Q0 ~Q0 +Q0 ~Q0 ~Q0 |
4 | oveq2 5540 | . . . 4 ~Q0 +Q0 ~Q0 +Q0 | |
5 | 4 | eleq1d 2147 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 +Q0 ~Q0 |
6 | addnnnq0 6639 | . . . 4 ~Q0 +Q0 ~Q0 ~Q0 | |
7 | pinn 6499 | . . . . . . . . 9 | |
8 | nnmcl 6083 | . . . . . . . . 9 | |
9 | 7, 8 | sylan2 280 | . . . . . . . 8 |
10 | pinn 6499 | . . . . . . . . 9 | |
11 | nnmcl 6083 | . . . . . . . . 9 | |
12 | 10, 11 | sylan 277 | . . . . . . . 8 |
13 | nnacl 6082 | . . . . . . . 8 | |
14 | 9, 12, 13 | syl2an 283 | . . . . . . 7 |
15 | 14 | an42s 553 | . . . . . 6 |
16 | mulpiord 6507 | . . . . . . . 8 | |
17 | mulclpi 6518 | . . . . . . . 8 | |
18 | 16, 17 | eqeltrrd 2156 | . . . . . . 7 |
19 | 18 | ad2ant2l 491 | . . . . . 6 |
20 | 15, 19 | jca 300 | . . . . 5 |
21 | opelxpi 4394 | . . . . 5 | |
22 | enq0ex 6629 | . . . . . 6 ~Q0 | |
23 | 22 | ecelqsi 6183 | . . . . 5 ~Q0 ~Q0 |
24 | 20, 21, 23 | 3syl 17 | . . . 4 ~Q0 ~Q0 |
25 | 6, 24 | eqeltrd 2155 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 |
26 | 1, 3, 5, 25 | 2ecoptocl 6217 | . 2 Q0 Q0 +Q0 ~Q0 |
27 | 26, 1 | syl6eleqr 2172 | 1 Q0 Q0 +Q0 Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cop 3401 com 4331 cxp 4361 (class class class)co 5532 coa 6021 comu 6022 cec 6127 cqs 6128 cnpi 6462 cmi 6464 ~Q0 ceq0 6476 Q0cnq0 6477 +Q0 cplq0 6479 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-enq0 6614 df-nq0 6615 df-plq0 6617 |
This theorem is referenced by: distnq0r 6653 prarloclemcalc 6692 |
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