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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 7340 | . . . 4 | |
2 | nqpi 7340 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | ee4anv 1927 | . . 3 | |
5 | 3, 4 | sylibr 133 | . 2 |
6 | oveq12 5862 | . . . . . . 7 | |
7 | mulclpi 7290 | . . . . . . . . . . . . 13 | |
8 | 7 | ad2ant2rl 508 | . . . . . . . . . . . 12 |
9 | mulclpi 7290 | . . . . . . . . . . . . 13 | |
10 | 9 | ad2ant2lr 507 | . . . . . . . . . . . 12 |
11 | addpiord 7278 | . . . . . . . . . . . 12 | |
12 | 8, 10, 11 | syl2anc 409 | . . . . . . . . . . 11 |
13 | mulpiord 7279 | . . . . . . . . . . . . 13 | |
14 | 13 | ad2ant2rl 508 | . . . . . . . . . . . 12 |
15 | mulpiord 7279 | . . . . . . . . . . . . 13 | |
16 | 15 | ad2ant2lr 507 | . . . . . . . . . . . 12 |
17 | 14, 16 | oveq12d 5871 | . . . . . . . . . . 11 |
18 | 12, 17 | eqtrd 2203 | . . . . . . . . . 10 |
19 | mulpiord 7279 | . . . . . . . . . . 11 | |
20 | 19 | ad2ant2l 505 | . . . . . . . . . 10 |
21 | 18, 20 | opeq12d 3773 | . . . . . . . . 9 |
22 | 21 | eceq1d 6549 | . . . . . . . 8 ~Q0 ~Q0 |
23 | addpipqqs 7332 | . . . . . . . . 9 | |
24 | addclpi 7289 | . . . . . . . . . . 11 | |
25 | 8, 10, 24 | syl2anc 409 | . . . . . . . . . 10 |
26 | mulclpi 7290 | . . . . . . . . . . 11 | |
27 | 26 | ad2ant2l 505 | . . . . . . . . . 10 |
28 | nqnq0pi 7400 | . . . . . . . . . 10 ~Q0 | |
29 | 25, 27, 28 | syl2anc 409 | . . . . . . . . 9 ~Q0 |
30 | 23, 29 | eqtr4d 2206 | . . . . . . . 8 ~Q0 |
31 | pinn 7271 | . . . . . . . . . 10 | |
32 | 31 | anim1i 338 | . . . . . . . . 9 |
33 | pinn 7271 | . . . . . . . . . 10 | |
34 | 33 | anim1i 338 | . . . . . . . . 9 |
35 | addnnnq0 7411 | . . . . . . . . 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 7400 | . . . . . . . . . . 11 ~Q0 | |
40 | 39 | adantr 274 | . . . . . . . . . 10 ~Q0 |
41 | 40 | eqeq2d 2182 | . . . . . . . . 9 ~Q0 |
42 | nqnq0pi 7400 | . . . . . . . . . . 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 5862 | . . . . . . . 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 3586 com 4574 (class class class)co 5853 coa 6392 comu 6393 cec 6511 cnpi 7234 cpli 7235 cmi 7236 ceq 7241 cnq 7242 cplq 7244 ~Q0 ceq0 7248 +Q0 cplq0 7251 |
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 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-plpq 7306 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-enq0 7386 df-nq0 7387 df-plq0 7389 |
This theorem is referenced by: prarloclemlo 7456 prarloclemcalc 7464 |
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