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Mirrors > Home > ILE Home > Th. List > nq0a0 | Unicode version |
Description: Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Ref | Expression |
---|---|
nq0a0 | Q0 +Q0 0Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nq0nn 7218 | . 2 Q0 ~Q0 | |
2 | df-0nq0 7202 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5751 | . . . . . 6 ~Q0 0Q0 ~Q0 +Q0 0Q0 ~Q0 +Q0 ~Q0 | |
4 | 2, 3 | mpan2 421 | . . . . 5 ~Q0 +Q0 0Q0 ~Q0 +Q0 ~Q0 |
5 | peano1 4478 | . . . . . 6 | |
6 | 1pi 7091 | . . . . . 6 | |
7 | addnnnq0 7225 | . . . . . 6 ~Q0 +Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanr12 435 | . . . . 5 ~Q0 +Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2172 | . . . 4 ~Q0 +Q0 0Q0 ~Q0 |
10 | pinn 7085 | . . . . . . . . . 10 | |
11 | nnm0 6339 | . . . . . . . . . . 11 | |
12 | 11 | oveq2d 5758 | . . . . . . . . . 10 |
13 | 10, 12 | syl 14 | . . . . . . . . 9 |
14 | nnm1 6388 | . . . . . . . . . . 11 | |
15 | 14 | oveq1d 5757 | . . . . . . . . . 10 |
16 | nna0 6338 | . . . . . . . . . 10 | |
17 | 15, 16 | eqtrd 2150 | . . . . . . . . 9 |
18 | 13, 17 | sylan9eqr 2172 | . . . . . . . 8 |
19 | nnm1 6388 | . . . . . . . . . 10 | |
20 | 10, 19 | syl 14 | . . . . . . . . 9 |
21 | 20 | adantl 275 | . . . . . . . 8 |
22 | 18, 21 | opeq12d 3683 | . . . . . . 7 |
23 | 22 | eceq1d 6433 | . . . . . 6 ~Q0 ~Q0 |
24 | 23 | eqeq2d 2129 | . . . . 5 ~Q0 ~Q0 |
25 | 24 | biimpar 295 | . . . 4 ~Q0 ~Q0 |
26 | 9, 25 | eqtr4d 2153 | . . 3 ~Q0 +Q0 0Q0 |
27 | 26 | exlimivv 1852 | . 2 ~Q0 +Q0 0Q0 |
28 | 1, 27 | syl 14 | 1 Q0 +Q0 0Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 c0 3333 cop 3500 com 4474 (class class class)co 5742 c1o 6274 coa 6278 comu 6279 cec 6395 cnpi 7048 ~Q0 ceq0 7062 Q0cnq0 7063 0Q0c0q0 7064 +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-1o 6281 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-mi 7082 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 |
This theorem is referenced by: prarloclem5 7276 |
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