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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 6568 | . . . 4 | |
2 | nq0nn 6632 | . . . 4 Q0 ~Q0 | |
3 | 1, 2 | anim12i 331 | . . 3 Q0 ~Q0 |
4 | ee4anv 1850 | . . 3 ~Q0 ~Q0 | |
5 | 3, 4 | sylibr 132 | . 2 Q0 ~Q0 |
6 | oveq12 5541 | . . . . . . 7 ~Q0 +Q0 +Q0 ~Q0 | |
7 | 6 | ad2ant2l 491 | . . . . . 6 ~Q0 +Q0 +Q0 ~Q0 |
8 | nqnq0pi 6628 | . . . . . . . . . 10 ~Q0 | |
9 | 8 | oveq1d 5547 | . . . . . . . . 9 ~Q0 +Q0 ~Q0 +Q0 ~Q0 |
10 | 9 | adantr 270 | . . . . . . . 8 ~Q0 +Q0 ~Q0 +Q0 ~Q0 |
11 | pinn 6499 | . . . . . . . . 9 | |
12 | addnnnq0 6639 | . . . . . . . . 9 ~Q0 +Q0 ~Q0 ~Q0 | |
13 | 11, 12 | sylanl1 394 | . . . . . . . 8 ~Q0 +Q0 ~Q0 ~Q0 |
14 | 10, 13 | eqtr3d 2115 | . . . . . . 7 +Q0 ~Q0 ~Q0 |
15 | 14 | ad2ant2r 492 | . . . . . 6 ~Q0 +Q0 ~Q0 ~Q0 |
16 | 7, 15 | eqtrd 2113 | . . . . 5 ~Q0 +Q0 ~Q0 |
17 | pinn 6499 | . . . . . . . . . . . . . 14 | |
18 | nnmcl 6083 | . . . . . . . . . . . . . 14 | |
19 | 17, 18 | sylan 277 | . . . . . . . . . . . . 13 |
20 | 19 | ad2ant2lr 493 | . . . . . . . . . . . 12 |
21 | mulpiord 6507 | . . . . . . . . . . . . . 14 | |
22 | mulclpi 6518 | . . . . . . . . . . . . . 14 | |
23 | 21, 22 | eqeltrrd 2156 | . . . . . . . . . . . . 13 |
24 | 23 | ad2ant2rl 494 | . . . . . . . . . . . 12 |
25 | pinn 6499 | . . . . . . . . . . . . 13 | |
26 | nnacom 6086 | . . . . . . . . . . . . 13 | |
27 | 25, 26 | sylan2 280 | . . . . . . . . . . . 12 |
28 | 20, 24, 27 | syl2anc 403 | . . . . . . . . . . 11 |
29 | nnppipi 6533 | . . . . . . . . . . . 12 | |
30 | 20, 24, 29 | syl2anc 403 | . . . . . . . . . . 11 |
31 | 28, 30 | eqeltrrd 2156 | . . . . . . . . . 10 |
32 | mulpiord 6507 | . . . . . . . . . . . 12 | |
33 | mulclpi 6518 | . . . . . . . . . . . 12 | |
34 | 32, 33 | eqeltrrd 2156 | . . . . . . . . . . 11 |
35 | 34 | ad2ant2l 491 | . . . . . . . . . 10 |
36 | opelxpi 4394 | . . . . . . . . . 10 | |
37 | 31, 35, 36 | syl2anc 403 | . . . . . . . . 9 |
38 | enqex 6550 | . . . . . . . . . 10 | |
39 | 38 | ecelqsi 6183 | . . . . . . . . 9 |
40 | 37, 39 | syl 14 | . . . . . . . 8 |
41 | df-nqqs 6538 | . . . . . . . 8 | |
42 | 40, 41 | syl6eleqr 2172 | . . . . . . 7 |
43 | nqnq0pi 6628 | . . . . . . . . 9 ~Q0 | |
44 | 43 | eleq1d 2147 | . . . . . . . 8 ~Q0 |
45 | 31, 35, 44 | syl2anc 403 | . . . . . . 7 ~Q0 |
46 | 42, 45 | mpbird 165 | . . . . . 6 ~Q0 |
47 | 46 | ad2ant2r 492 | . . . . 5 ~Q0 ~Q0 |
48 | 16, 47 | eqeltrd 2155 | . . . 4 ~Q0 +Q0 |
49 | 48 | exlimivv 1817 | . . 3 ~Q0 +Q0 |
50 | 49 | exlimivv 1817 | . 2 ~Q0 +Q0 |
51 | 5, 50 | syl 14 | 1 Q0 +Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 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 ceq 6469 cnq 6470 ~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-enq 6537 df-nqqs 6538 df-enq0 6614 df-nq0 6615 df-plq0 6617 |
This theorem is referenced by: prarloclemcalc 6692 |
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