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Mirrors > Home > ILE Home > Th. List > addpipqqs | Unicode version |
Description: Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) |
Ref | Expression |
---|---|
addpipqqs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addpipqqslem 7289 | . 2 | |
2 | addpipqqslem 7289 | . 2 | |
3 | addpipqqslem 7289 | . 2 | |
4 | enqex 7280 | . 2 | |
5 | enqer 7278 | . 2 | |
6 | df-enq 7267 | . 2 | |
7 | oveq12 5833 | . . . 4 | |
8 | oveq12 5833 | . . . 4 | |
9 | 7, 8 | eqeqan12d 2173 | . . 3 |
10 | 9 | an42s 579 | . 2 |
11 | oveq12 5833 | . . . 4 | |
12 | oveq12 5833 | . . . 4 | |
13 | 11, 12 | eqeqan12d 2173 | . . 3 |
14 | 13 | an42s 579 | . 2 |
15 | dfplpq2 7274 | . 2 | |
16 | oveq12 5833 | . . . . 5 | |
17 | oveq12 5833 | . . . . 5 | |
18 | 16, 17 | oveqan12d 5843 | . . . 4 |
19 | 18 | an42s 579 | . . 3 |
20 | oveq12 5833 | . . . 4 | |
21 | 20 | ad2ant2l 500 | . . 3 |
22 | 19, 21 | opeq12d 3749 | . 2 |
23 | oveq12 5833 | . . . . 5 | |
24 | oveq12 5833 | . . . . 5 | |
25 | 23, 24 | oveqan12d 5843 | . . . 4 |
26 | 25 | an42s 579 | . . 3 |
27 | oveq12 5833 | . . . 4 | |
28 | 27 | ad2ant2l 500 | . . 3 |
29 | 26, 28 | opeq12d 3749 | . 2 |
30 | oveq12 5833 | . . . . 5 | |
31 | oveq12 5833 | . . . . 5 | |
32 | 30, 31 | oveqan12d 5843 | . . . 4 |
33 | 32 | an42s 579 | . . 3 |
34 | oveq12 5833 | . . . 4 | |
35 | 34 | ad2ant2l 500 | . . 3 |
36 | 33, 35 | opeq12d 3749 | . 2 |
37 | df-plqqs 7269 | . 2 | |
38 | df-nqqs 7268 | . 2 | |
39 | addcmpblnq 7287 | . 2 | |
40 | 1, 2, 3, 4, 5, 6, 10, 14, 15, 22, 29, 36, 37, 38, 39 | oviec 6586 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cop 3563 (class class class)co 5824 cec 6478 cnpi 7192 cpli 7193 cmi 7194 cplpq 7196 ceq 7199 cnq 7200 cplq 7202 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-plpq 7264 df-enq 7267 df-nqqs 7268 df-plqqs 7269 |
This theorem is referenced by: addclnq 7295 addcomnqg 7301 addassnqg 7302 distrnqg 7307 ltanqg 7320 1lt2nq 7326 ltexnqq 7328 nqnq0a 7374 addpinq1 7384 |
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