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Mirrors > Home > ILE Home > Th. List > addnqprl | Unicode version |
Description: Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Ref | Expression |
---|---|
addnqprl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 6665 | . . . . . 6 | |
2 | addnqprllem 6717 | . . . . . 6 | |
3 | 1, 2 | sylanl1 394 | . . . . 5 |
4 | 3 | adantlr 460 | . . . 4 |
5 | prop 6665 | . . . . . 6 | |
6 | addnqprllem 6717 | . . . . . 6 | |
7 | 5, 6 | sylanl1 394 | . . . . 5 |
8 | 7 | adantll 459 | . . . 4 |
9 | 4, 8 | jcad 301 | . . 3 |
10 | simpl 107 | . . . 4 | |
11 | simpl 107 | . . . . 5 | |
12 | simpl 107 | . . . . 5 | |
13 | 11, 12 | anim12i 331 | . . . 4 |
14 | df-iplp 6658 | . . . . 5 | |
15 | addclnq 6565 | . . . . 5 | |
16 | 14, 15 | genpprecll 6704 | . . . 4 |
17 | 10, 13, 16 | 3syl 17 | . . 3 |
18 | 9, 17 | syld 44 | . 2 |
19 | simpr 108 | . . . . 5 | |
20 | elprnql 6671 | . . . . . . . . 9 | |
21 | 1, 20 | sylan 277 | . . . . . . . 8 |
22 | 21 | ad2antrr 471 | . . . . . . 7 |
23 | elprnql 6671 | . . . . . . . . 9 | |
24 | 5, 23 | sylan 277 | . . . . . . . 8 |
25 | 24 | ad2antlr 472 | . . . . . . 7 |
26 | addclnq 6565 | . . . . . . 7 | |
27 | 22, 25, 26 | syl2anc 403 | . . . . . 6 |
28 | recclnq 6582 | . . . . . 6 | |
29 | 27, 28 | syl 14 | . . . . 5 |
30 | mulassnqg 6574 | . . . . 5 | |
31 | 19, 29, 27, 30 | syl3anc 1169 | . . . 4 |
32 | mulclnq 6566 | . . . . . 6 | |
33 | 19, 29, 32 | syl2anc 403 | . . . . 5 |
34 | distrnqg 6577 | . . . . 5 | |
35 | 33, 22, 25, 34 | syl3anc 1169 | . . . 4 |
36 | mulcomnqg 6573 | . . . . . . . 8 | |
37 | 29, 27, 36 | syl2anc 403 | . . . . . . 7 |
38 | recidnq 6583 | . . . . . . . 8 | |
39 | 27, 38 | syl 14 | . . . . . . 7 |
40 | 37, 39 | eqtrd 2113 | . . . . . 6 |
41 | 40 | oveq2d 5548 | . . . . 5 |
42 | mulidnq 6579 | . . . . . 6 | |
43 | 42 | adantl 271 | . . . . 5 |
44 | 41, 43 | eqtrd 2113 | . . . 4 |
45 | 31, 35, 44 | 3eqtr3d 2121 | . . 3 |
46 | 45 | eleq1d 2147 | . 2 |
47 | 18, 46 | sylibd 147 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cop 3401 class class class wbr 3785 cfv 4922 (class class class)co 5532 c1st 5785 c2nd 5786 cnq 6470 c1q 6471 cplq 6472 cmq 6473 crq 6474 cltq 6475 cnp 6481 cpp 6483 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-inp 6656 df-iplp 6658 |
This theorem is referenced by: addlocprlemlt 6721 addclpr 6727 |
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