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Mirrors > Home > ILE Home > Th. List > nqnq0a | Unicode version |
Description: Addition of positive fractions is equal with or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Ref | Expression |
---|---|
nqnq0a | +Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqpi 7154 | . . . 4 | |
2 | nqpi 7154 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | ee4anv 1886 | . . 3 | |
5 | 3, 4 | sylibr 133 | . 2 |
6 | oveq12 5751 | . . . . . . 7 | |
7 | mulclpi 7104 | . . . . . . . . . . . . 13 | |
8 | 7 | ad2ant2rl 502 | . . . . . . . . . . . 12 |
9 | mulclpi 7104 | . . . . . . . . . . . . 13 | |
10 | 9 | ad2ant2lr 501 | . . . . . . . . . . . 12 |
11 | addpiord 7092 | . . . . . . . . . . . 12 | |
12 | 8, 10, 11 | syl2anc 408 | . . . . . . . . . . 11 |
13 | mulpiord 7093 | . . . . . . . . . . . . 13 | |
14 | 13 | ad2ant2rl 502 | . . . . . . . . . . . 12 |
15 | mulpiord 7093 | . . . . . . . . . . . . 13 | |
16 | 15 | ad2ant2lr 501 | . . . . . . . . . . . 12 |
17 | 14, 16 | oveq12d 5760 | . . . . . . . . . . 11 |
18 | 12, 17 | eqtrd 2150 | . . . . . . . . . 10 |
19 | mulpiord 7093 | . . . . . . . . . . 11 | |
20 | 19 | ad2ant2l 499 | . . . . . . . . . 10 |
21 | 18, 20 | opeq12d 3683 | . . . . . . . . 9 |
22 | 21 | eceq1d 6433 | . . . . . . . 8 ~Q0 ~Q0 |
23 | addpipqqs 7146 | . . . . . . . . 9 | |
24 | addclpi 7103 | . . . . . . . . . . 11 | |
25 | 8, 10, 24 | syl2anc 408 | . . . . . . . . . 10 |
26 | mulclpi 7104 | . . . . . . . . . . 11 | |
27 | 26 | ad2ant2l 499 | . . . . . . . . . 10 |
28 | nqnq0pi 7214 | . . . . . . . . . 10 ~Q0 | |
29 | 25, 27, 28 | syl2anc 408 | . . . . . . . . 9 ~Q0 |
30 | 23, 29 | eqtr4d 2153 | . . . . . . . 8 ~Q0 |
31 | pinn 7085 | . . . . . . . . . 10 | |
32 | 31 | anim1i 338 | . . . . . . . . 9 |
33 | pinn 7085 | . . . . . . . . . 10 | |
34 | 33 | anim1i 338 | . . . . . . . . 9 |
35 | addnnnq0 7225 | . . . . . . . . 9 ~Q0 +Q0 ~Q0 ~Q0 | |
36 | 32, 34, 35 | syl2an 287 | . . . . . . . 8 ~Q0 +Q0 ~Q0 ~Q0 |
37 | 22, 30, 36 | 3eqtr4d 2160 | . . . . . . 7 ~Q0 +Q0 ~Q0 |
38 | 6, 37 | sylan9eqr 2172 | . . . . . 6 ~Q0 +Q0 ~Q0 |
39 | nqnq0pi 7214 | . . . . . . . . . . 11 ~Q0 | |
40 | 39 | adantr 274 | . . . . . . . . . 10 ~Q0 |
41 | 40 | eqeq2d 2129 | . . . . . . . . 9 ~Q0 |
42 | nqnq0pi 7214 | . . . . . . . . . . 11 ~Q0 | |
43 | 42 | adantl 275 | . . . . . . . . . 10 ~Q0 |
44 | 43 | eqeq2d 2129 | . . . . . . . . 9 ~Q0 |
45 | 41, 44 | anbi12d 464 | . . . . . . . 8 ~Q0 ~Q0 |
46 | 45 | pm5.32i 449 | . . . . . . 7 ~Q0 ~Q0 |
47 | oveq12 5751 | . . . . . . . 8 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 | |
48 | 47 | adantl 275 | . . . . . . 7 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 |
49 | 46, 48 | sylbir 134 | . . . . . 6 +Q0 ~Q0 +Q0 ~Q0 |
50 | 38, 49 | eqtr4d 2153 | . . . . 5 +Q0 |
51 | 50 | an4s 562 | . . . 4 +Q0 |
52 | 51 | exlimivv 1852 | . . 3 +Q0 |
53 | 52 | exlimivv 1852 | . 2 +Q0 |
54 | 5, 53 | syl 14 | 1 +Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 cop 3500 com 4474 (class class class)co 5742 coa 6278 comu 6279 cec 6395 cnpi 7048 cpli 7049 cmi 7050 ceq 7055 cnq 7056 cplq 7058 ~Q0 ceq0 7062 +Q0 cplq0 7065 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-plpq 7120 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-enq0 7200 df-nq0 7201 df-plq0 7203 |
This theorem is referenced by: prarloclemlo 7270 prarloclemcalc 7278 |
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