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Mirrors > Home > ILE Home > Th. List > addclnq0 | Unicode version |
Description: Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.) |
Ref | Expression |
---|---|
addclnq0 | Q0 Q0 +Q0 Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nq0 7358 | . . 3 Q0 ~Q0 | |
2 | oveq1 5844 | . . . 4 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 | |
3 | 2 | eleq1d 2233 | . . 3 ~Q0 ~Q0 +Q0 ~Q0 ~Q0 +Q0 ~Q0 ~Q0 |
4 | oveq2 5845 | . . . 4 ~Q0 +Q0 ~Q0 +Q0 | |
5 | 4 | eleq1d 2233 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 +Q0 ~Q0 |
6 | addnnnq0 7382 | . . . 4 ~Q0 +Q0 ~Q0 ~Q0 | |
7 | pinn 7242 | . . . . . . . . 9 | |
8 | nnmcl 6441 | . . . . . . . . 9 | |
9 | 7, 8 | sylan2 284 | . . . . . . . 8 |
10 | pinn 7242 | . . . . . . . . 9 | |
11 | nnmcl 6441 | . . . . . . . . 9 | |
12 | 10, 11 | sylan 281 | . . . . . . . 8 |
13 | nnacl 6440 | . . . . . . . 8 | |
14 | 9, 12, 13 | syl2an 287 | . . . . . . 7 |
15 | 14 | an42s 579 | . . . . . 6 |
16 | mulpiord 7250 | . . . . . . . 8 | |
17 | mulclpi 7261 | . . . . . . . 8 | |
18 | 16, 17 | eqeltrrd 2242 | . . . . . . 7 |
19 | 18 | ad2ant2l 500 | . . . . . 6 |
20 | 15, 19 | jca 304 | . . . . 5 |
21 | opelxpi 4631 | . . . . 5 | |
22 | enq0ex 7372 | . . . . . 6 ~Q0 | |
23 | 22 | ecelqsi 6547 | . . . . 5 ~Q0 ~Q0 |
24 | 20, 21, 23 | 3syl 17 | . . . 4 ~Q0 ~Q0 |
25 | 6, 24 | eqeltrd 2241 | . . 3 ~Q0 +Q0 ~Q0 ~Q0 |
26 | 1, 3, 5, 25 | 2ecoptocl 6581 | . 2 Q0 Q0 +Q0 ~Q0 |
27 | 26, 1 | eleqtrrdi 2258 | 1 Q0 Q0 +Q0 Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cop 3574 com 4562 cxp 4597 (class class class)co 5837 coa 6373 comu 6374 cec 6491 cqs 6492 cnpi 7205 cmi 7207 ~Q0 ceq0 7219 Q0cnq0 7220 +Q0 cplq0 7222 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-oadd 6380 df-omul 6381 df-er 6493 df-ec 6495 df-qs 6499 df-ni 7237 df-mi 7239 df-enq0 7357 df-nq0 7358 df-plq0 7360 |
This theorem is referenced by: distnq0r 7396 prarloclemcalc 7435 |
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