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Mirrors > Home > ILE Home > Th. List > nq0a0 | Unicode version |
Description: Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Ref | Expression |
---|---|
nq0a0 | Q0 +Q0 0Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nq0nn 6632 | . 2 Q0 ~Q0 | |
2 | df-0nq0 6616 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5541 | . . . . . 6 ~Q0 0Q0 ~Q0 +Q0 0Q0 ~Q0 +Q0 ~Q0 | |
4 | 2, 3 | mpan2 415 | . . . . 5 ~Q0 +Q0 0Q0 ~Q0 +Q0 ~Q0 |
5 | peano1 4335 | . . . . . 6 | |
6 | 1pi 6505 | . . . . . 6 | |
7 | addnnnq0 6639 | . . . . . 6 ~Q0 +Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanr12 429 | . . . . 5 ~Q0 +Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2135 | . . . 4 ~Q0 +Q0 0Q0 ~Q0 |
10 | pinn 6499 | . . . . . . . . . 10 | |
11 | nnm0 6077 | . . . . . . . . . . 11 | |
12 | 11 | oveq2d 5548 | . . . . . . . . . 10 |
13 | 10, 12 | syl 14 | . . . . . . . . 9 |
14 | nnm1 6120 | . . . . . . . . . . 11 | |
15 | 14 | oveq1d 5547 | . . . . . . . . . 10 |
16 | nna0 6076 | . . . . . . . . . 10 | |
17 | 15, 16 | eqtrd 2113 | . . . . . . . . 9 |
18 | 13, 17 | sylan9eqr 2135 | . . . . . . . 8 |
19 | nnm1 6120 | . . . . . . . . . 10 | |
20 | 10, 19 | syl 14 | . . . . . . . . 9 |
21 | 20 | adantl 271 | . . . . . . . 8 |
22 | 18, 21 | opeq12d 3578 | . . . . . . 7 |
23 | 22 | eceq1d 6165 | . . . . . 6 ~Q0 ~Q0 |
24 | 23 | eqeq2d 2092 | . . . . 5 ~Q0 ~Q0 |
25 | 24 | biimpar 291 | . . . 4 ~Q0 ~Q0 |
26 | 9, 25 | eqtr4d 2116 | . . 3 ~Q0 +Q0 0Q0 |
27 | 26 | exlimivv 1817 | . 2 ~Q0 +Q0 0Q0 |
28 | 1, 27 | syl 14 | 1 Q0 +Q0 0Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 c0 3251 cop 3401 com 4331 (class class class)co 5532 c1o 6017 coa 6021 comu 6022 cec 6127 cnpi 6462 ~Q0 ceq0 6476 Q0cnq0 6477 0Q0c0q0 6478 +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-1o 6024 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-0nq0 6616 df-plq0 6617 |
This theorem is referenced by: prarloclem5 6690 |
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