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Mirrors > Home > ILE Home > Th. List > nqpnq0nq | Unicode version |
Description: A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.) |
Ref | Expression |
---|---|
nqpnq0nq | Q0 +Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqpi 6476 | . . . 4 | |
2 | nq0nn 6540 | . . . 4 Q0 ~Q0 | |
3 | 1, 2 | anim12i 321 | . . 3 Q0 ~Q0 |
4 | ee4anv 1809 | . . 3 ~Q0 ~Q0 | |
5 | 3, 4 | sylibr 137 | . 2 Q0 ~Q0 |
6 | oveq12 5521 | . . . . . . 7 ~Q0 +Q0 +Q0 ~Q0 | |
7 | 6 | ad2ant2l 477 | . . . . . 6 ~Q0 +Q0 +Q0 ~Q0 |
8 | nqnq0pi 6536 | . . . . . . . . . 10 ~Q0 | |
9 | 8 | oveq1d 5527 | . . . . . . . . 9 ~Q0 +Q0 ~Q0 +Q0 ~Q0 |
10 | 9 | adantr 261 | . . . . . . . 8 ~Q0 +Q0 ~Q0 +Q0 ~Q0 |
11 | pinn 6407 | . . . . . . . . 9 | |
12 | addnnnq0 6547 | . . . . . . . . 9 ~Q0 +Q0 ~Q0 ~Q0 | |
13 | 11, 12 | sylanl1 382 | . . . . . . . 8 ~Q0 +Q0 ~Q0 ~Q0 |
14 | 10, 13 | eqtr3d 2074 | . . . . . . 7 +Q0 ~Q0 ~Q0 |
15 | 14 | ad2ant2r 478 | . . . . . 6 ~Q0 +Q0 ~Q0 ~Q0 |
16 | 7, 15 | eqtrd 2072 | . . . . 5 ~Q0 +Q0 ~Q0 |
17 | pinn 6407 | . . . . . . . . . . . . . 14 | |
18 | nnmcl 6060 | . . . . . . . . . . . . . 14 | |
19 | 17, 18 | sylan 267 | . . . . . . . . . . . . 13 |
20 | 19 | ad2ant2lr 479 | . . . . . . . . . . . 12 |
21 | mulpiord 6415 | . . . . . . . . . . . . . 14 | |
22 | mulclpi 6426 | . . . . . . . . . . . . . 14 | |
23 | 21, 22 | eqeltrrd 2115 | . . . . . . . . . . . . 13 |
24 | 23 | ad2ant2rl 480 | . . . . . . . . . . . 12 |
25 | pinn 6407 | . . . . . . . . . . . . 13 | |
26 | nnacom 6063 | . . . . . . . . . . . . 13 | |
27 | 25, 26 | sylan2 270 | . . . . . . . . . . . 12 |
28 | 20, 24, 27 | syl2anc 391 | . . . . . . . . . . 11 |
29 | nnppipi 6441 | . . . . . . . . . . . 12 | |
30 | 20, 24, 29 | syl2anc 391 | . . . . . . . . . . 11 |
31 | 28, 30 | eqeltrrd 2115 | . . . . . . . . . 10 |
32 | mulpiord 6415 | . . . . . . . . . . . 12 | |
33 | mulclpi 6426 | . . . . . . . . . . . 12 | |
34 | 32, 33 | eqeltrrd 2115 | . . . . . . . . . . 11 |
35 | 34 | ad2ant2l 477 | . . . . . . . . . 10 |
36 | opelxpi 4376 | . . . . . . . . . 10 | |
37 | 31, 35, 36 | syl2anc 391 | . . . . . . . . 9 |
38 | enqex 6458 | . . . . . . . . . 10 | |
39 | 38 | ecelqsi 6160 | . . . . . . . . 9 |
40 | 37, 39 | syl 14 | . . . . . . . 8 |
41 | df-nqqs 6446 | . . . . . . . 8 | |
42 | 40, 41 | syl6eleqr 2131 | . . . . . . 7 |
43 | nqnq0pi 6536 | . . . . . . . . 9 ~Q0 | |
44 | 43 | eleq1d 2106 | . . . . . . . 8 ~Q0 |
45 | 31, 35, 44 | syl2anc 391 | . . . . . . 7 ~Q0 |
46 | 42, 45 | mpbird 156 | . . . . . 6 ~Q0 |
47 | 46 | ad2ant2r 478 | . . . . 5 ~Q0 ~Q0 |
48 | 16, 47 | eqeltrd 2114 | . . . 4 ~Q0 +Q0 |
49 | 48 | exlimivv 1776 | . . 3 ~Q0 +Q0 |
50 | 49 | exlimivv 1776 | . 2 ~Q0 +Q0 |
51 | 5, 50 | syl 14 | 1 Q0 +Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cop 3378 com 4313 cxp 4343 (class class class)co 5512 coa 5998 comu 5999 cec 6104 cqs 6105 cnpi 6370 cmi 6372 ceq 6377 cnq 6378 ~Q0 ceq0 6384 Q0cnq0 6385 +Q0 cplq0 6387 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-enq 6445 df-nqqs 6446 df-enq0 6522 df-nq0 6523 df-plq0 6525 |
This theorem is referenced by: prarloclemcalc 6600 |
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