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Mirrors > Home > ILE Home > Th. List > nnanq0 | Unicode version |
Description: Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.) |
Ref | Expression |
---|---|
nnanq0 | ~Q0 ~Q0 +Q0 ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addnnnq0 6639 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 | |
2 | 1 | 3impdir 1225 | . 2 ~Q0 +Q0 ~Q0 ~Q0 |
3 | pinn 6499 | . . . . . . . 8 | |
4 | nnmcom 6091 | . . . . . . . 8 | |
5 | 3, 4 | sylan2 280 | . . . . . . 7 |
6 | 5 | 3adant2 957 | . . . . . 6 |
7 | 6 | oveq1d 5547 | . . . . 5 |
8 | nndi 6088 | . . . . . . 7 | |
9 | 8 | 3coml 1145 | . . . . . 6 |
10 | 3, 9 | syl3an3 1204 | . . . . 5 |
11 | 7, 10 | eqtr4d 2116 | . . . 4 |
12 | 11 | opeq1d 3576 | . . 3 |
13 | 12 | eceq1d 6165 | . 2 ~Q0 ~Q0 |
14 | simp3 940 | . . 3 | |
15 | nnacl 6082 | . . . 4 | |
16 | 15 | 3adant3 958 | . . 3 |
17 | mulcanenq0ec 6635 | . . 3 ~Q0 ~Q0 | |
18 | 14, 16, 14, 17 | syl3anc 1169 | . 2 ~Q0 ~Q0 |
19 | 2, 13, 18 | 3eqtrrd 2118 | 1 ~Q0 ~Q0 +Q0 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 919 wceq 1284 wcel 1433 cop 3401 com 4331 (class class class)co 5532 coa 6021 comu 6022 cec 6127 cnpi 6462 ~Q0 ceq0 6476 +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-enq0 6614 df-nq0 6615 df-plq0 6617 |
This theorem is referenced by: nq02m 6655 prarloclemcalc 6692 |
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