Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7416 | . 2 Q0 ~Q0 | |
2 | df-0nq0 7400 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5874 | . . . . . 6 ~Q0 0Q0 ~Q0 +Q0 0Q0 ~Q0 +Q0 ~Q0 | |
4 | 2, 3 | mpan2 425 | . . . . 5 ~Q0 +Q0 0Q0 ~Q0 +Q0 ~Q0 |
5 | peano1 4587 | . . . . . 6 | |
6 | 1pi 7289 | . . . . . 6 | |
7 | addnnnq0 7423 | . . . . . 6 ~Q0 +Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanr12 439 | . . . . 5 ~Q0 +Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2230 | . . . 4 ~Q0 +Q0 0Q0 ~Q0 |
10 | pinn 7283 | . . . . . . . . . 10 | |
11 | nnm0 6466 | . . . . . . . . . . 11 | |
12 | 11 | oveq2d 5881 | . . . . . . . . . 10 |
13 | 10, 12 | syl 14 | . . . . . . . . 9 |
14 | nnm1 6516 | . . . . . . . . . . 11 | |
15 | 14 | oveq1d 5880 | . . . . . . . . . 10 |
16 | nna0 6465 | . . . . . . . . . 10 | |
17 | 15, 16 | eqtrd 2208 | . . . . . . . . 9 |
18 | 13, 17 | sylan9eqr 2230 | . . . . . . . 8 |
19 | nnm1 6516 | . . . . . . . . . 10 | |
20 | 10, 19 | syl 14 | . . . . . . . . 9 |
21 | 20 | adantl 277 | . . . . . . . 8 |
22 | 18, 21 | opeq12d 3782 | . . . . . . 7 |
23 | 22 | eceq1d 6561 | . . . . . 6 ~Q0 ~Q0 |
24 | 23 | eqeq2d 2187 | . . . . 5 ~Q0 ~Q0 |
25 | 24 | biimpar 297 | . . . 4 ~Q0 ~Q0 |
26 | 9, 25 | eqtr4d 2211 | . . 3 ~Q0 +Q0 0Q0 |
27 | 26 | exlimivv 1894 | . 2 ~Q0 +Q0 0Q0 |
28 | 1, 27 | syl 14 | 1 Q0 +Q0 0Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wceq 1353 wex 1490 wcel 2146 c0 3420 cop 3592 com 4583 (class class class)co 5865 c1o 6400 coa 6404 comu 6405 cec 6523 cnpi 7246 ~Q0 ceq0 7260 Q0cnq0 7261 0Q0c0q0 7262 +Q0 cplq0 7263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-1o 6407 df-oadd 6411 df-omul 6412 df-er 6525 df-ec 6527 df-qs 6531 df-ni 7278 df-mi 7280 df-enq0 7398 df-nq0 7399 df-0nq0 7400 df-plq0 7401 |
This theorem is referenced by: prarloclem5 7474 |
Copyright terms: Public domain | W3C validator |