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Mirrors > Home > ILE Home > Th. List > nqnq0a | Unicode version |
Description: Addition of positive fractions is equal with or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Ref | Expression |
---|---|
nqnq0a | +Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqpi 7333 | . . . 4 | |
2 | nqpi 7333 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | ee4anv 1927 | . . 3 | |
5 | 3, 4 | sylibr 133 | . 2 |
6 | oveq12 5860 | . . . . . . 7 | |
7 | mulclpi 7283 | . . . . . . . . . . . . 13 | |
8 | 7 | ad2ant2rl 508 | . . . . . . . . . . . 12 |
9 | mulclpi 7283 | . . . . . . . . . . . . 13 | |
10 | 9 | ad2ant2lr 507 | . . . . . . . . . . . 12 |
11 | addpiord 7271 | . . . . . . . . . . . 12 | |
12 | 8, 10, 11 | syl2anc 409 | . . . . . . . . . . 11 |
13 | mulpiord 7272 | . . . . . . . . . . . . 13 | |
14 | 13 | ad2ant2rl 508 | . . . . . . . . . . . 12 |
15 | mulpiord 7272 | . . . . . . . . . . . . 13 | |
16 | 15 | ad2ant2lr 507 | . . . . . . . . . . . 12 |
17 | 14, 16 | oveq12d 5869 | . . . . . . . . . . 11 |
18 | 12, 17 | eqtrd 2203 | . . . . . . . . . 10 |
19 | mulpiord 7272 | . . . . . . . . . . 11 | |
20 | 19 | ad2ant2l 505 | . . . . . . . . . 10 |
21 | 18, 20 | opeq12d 3771 | . . . . . . . . 9 |
22 | 21 | eceq1d 6547 | . . . . . . . 8 ~Q0 ~Q0 |
23 | addpipqqs 7325 | . . . . . . . . 9 | |
24 | addclpi 7282 | . . . . . . . . . . 11 | |
25 | 8, 10, 24 | syl2anc 409 | . . . . . . . . . 10 |
26 | mulclpi 7283 | . . . . . . . . . . 11 | |
27 | 26 | ad2ant2l 505 | . . . . . . . . . 10 |
28 | nqnq0pi 7393 | . . . . . . . . . 10 ~Q0 | |
29 | 25, 27, 28 | syl2anc 409 | . . . . . . . . 9 ~Q0 |
30 | 23, 29 | eqtr4d 2206 | . . . . . . . 8 ~Q0 |
31 | pinn 7264 | . . . . . . . . . 10 | |
32 | 31 | anim1i 338 | . . . . . . . . 9 |
33 | pinn 7264 | . . . . . . . . . 10 | |
34 | 33 | anim1i 338 | . . . . . . . . 9 |
35 | addnnnq0 7404 | . . . . . . . . 9 ~Q0 +Q0 ~Q0 ~Q0 | |
36 | 32, 34, 35 | syl2an 287 | . . . . . . . 8 ~Q0 +Q0 ~Q0 ~Q0 |
37 | 22, 30, 36 | 3eqtr4d 2213 | . . . . . . 7 ~Q0 +Q0 ~Q0 |
38 | 6, 37 | sylan9eqr 2225 | . . . . . 6 ~Q0 +Q0 ~Q0 |
39 | nqnq0pi 7393 | . . . . . . . . . . 11 ~Q0 | |
40 | 39 | adantr 274 | . . . . . . . . . 10 ~Q0 |
41 | 40 | eqeq2d 2182 | . . . . . . . . 9 ~Q0 |
42 | nqnq0pi 7393 | . . . . . . . . . . 11 ~Q0 | |
43 | 42 | adantl 275 | . . . . . . . . . 10 ~Q0 |
44 | 43 | eqeq2d 2182 | . . . . . . . . 9 ~Q0 |
45 | 41, 44 | anbi12d 470 | . . . . . . . 8 ~Q0 ~Q0 |
46 | 45 | pm5.32i 451 | . . . . . . 7 ~Q0 ~Q0 |
47 | oveq12 5860 | . . . . . . . 8 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 | |
48 | 47 | adantl 275 | . . . . . . 7 ~Q0 ~Q0 +Q0 ~Q0 +Q0 ~Q0 |
49 | 46, 48 | sylbir 134 | . . . . . 6 +Q0 ~Q0 +Q0 ~Q0 |
50 | 38, 49 | eqtr4d 2206 | . . . . 5 +Q0 |
51 | 50 | an4s 583 | . . . 4 +Q0 |
52 | 51 | exlimivv 1889 | . . 3 +Q0 |
53 | 52 | exlimivv 1889 | . 2 +Q0 |
54 | 5, 53 | syl 14 | 1 +Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 cop 3584 com 4572 (class class class)co 5851 coa 6390 comu 6391 cec 6509 cnpi 7227 cpli 7228 cmi 7229 ceq 7234 cnq 7235 cplq 7237 ~Q0 ceq0 7241 +Q0 cplq0 7244 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-oadd 6397 df-omul 6398 df-er 6511 df-ec 6513 df-qs 6517 df-ni 7259 df-pli 7260 df-mi 7261 df-plpq 7299 df-enq 7302 df-nqqs 7303 df-plqqs 7304 df-enq0 7379 df-nq0 7380 df-plq0 7382 |
This theorem is referenced by: prarloclemlo 7449 prarloclemcalc 7457 |
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