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Mirrors > Home > ILE Home > Th. List > prmuloc | Unicode version |
Description: Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Ref | Expression |
---|---|
prmuloc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexnqi 7185 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | prml 7253 | . . . 4 | |
4 | 3 | ad2antrr 479 | . . 3 |
5 | simprl 505 | . . . . . 6 | |
6 | simplrl 509 | . . . . . 6 | |
7 | mulclnq 7152 | . . . . . 6 | |
8 | 5, 6, 7 | syl2anc 408 | . . . . 5 |
9 | ltrelnq 7141 | . . . . . . . 8 | |
10 | 9 | brel 4561 | . . . . . . 7 |
11 | 10 | simprd 113 | . . . . . 6 |
12 | 11 | ad3antlr 484 | . . . . 5 |
13 | appdiv0nq 7340 | . . . . 5 | |
14 | 8, 12, 13 | syl2anc 408 | . . . 4 |
15 | prarloc 7279 | . . . . . . . . . 10 | |
16 | 15 | adantlr 468 | . . . . . . . . 9 |
17 | 16 | adantlr 468 | . . . . . . . 8 |
18 | 17 | ad2ant2r 500 | . . . . . . 7 |
19 | r2ex 2432 | . . . . . . 7 | |
20 | 18, 19 | sylib 121 | . . . . . 6 |
21 | elprnql 7257 | . . . . . . . . . . . . . 14 | |
22 | 21 | adantlr 468 | . . . . . . . . . . . . 13 |
23 | 22 | adantlr 468 | . . . . . . . . . . . 12 |
24 | 23 | adantlr 468 | . . . . . . . . . . 11 |
25 | 24 | ad2ant2r 500 | . . . . . . . . . 10 |
26 | 25 | adantrr 470 | . . . . . . . . 9 |
27 | simplll 507 | . . . . . . . . . . 11 | |
28 | 27 | ad2antrr 479 | . . . . . . . . . 10 |
29 | simprl 505 | . . . . . . . . . . 11 | |
30 | 29 | simprd 113 | . . . . . . . . . 10 |
31 | elprnqu 7258 | . . . . . . . . . 10 | |
32 | 28, 30, 31 | syl2anc 408 | . . . . . . . . 9 |
33 | prltlu 7263 | . . . . . . . . . . . . . . . . 17 | |
34 | 33 | 3adant1r 1194 | . . . . . . . . . . . . . . . 16 |
35 | 34 | 3adant2l 1195 | . . . . . . . . . . . . . . 15 |
36 | 35 | 3adant3l 1197 | . . . . . . . . . . . . . 14 |
37 | 36 | 3adant1r 1194 | . . . . . . . . . . . . 13 |
38 | 37 | 3expa 1166 | . . . . . . . . . . . 12 |
39 | 38 | ad2ant2r 500 | . . . . . . . . . . 11 |
40 | simprr 506 | . . . . . . . . . . 11 | |
41 | simplrr 510 | . . . . . . . . . . . 12 | |
42 | 41 | ad2antrr 479 | . . . . . . . . . . 11 |
43 | simplrr 510 | . . . . . . . . . . 11 | |
44 | 10 | simpld 111 | . . . . . . . . . . . . 13 |
45 | 44 | ad3antlr 484 | . . . . . . . . . . . 12 |
46 | 45 | ad2antrr 479 | . . . . . . . . . . 11 |
47 | 12 | ad2antrr 479 | . . . . . . . . . . 11 |
48 | simplrl 509 | . . . . . . . . . . 11 | |
49 | 6 | ad2antrr 479 | . . . . . . . . . . 11 |
50 | 39, 40, 42, 43, 46, 47, 26, 48, 49 | prmuloclemcalc 7341 | . . . . . . . . . 10 |
51 | df-3an 949 | . . . . . . . . . 10 | |
52 | 29, 50, 51 | sylanbrc 413 | . . . . . . . . 9 |
53 | 26, 32, 52 | jca31 307 | . . . . . . . 8 |
54 | 53 | ex 114 | . . . . . . 7 |
55 | 54 | 2eximdv 1838 | . . . . . 6 |
56 | 20, 55 | mpd 13 | . . . . 5 |
57 | r2ex 2432 | . . . . 5 | |
58 | 56, 57 | sylibr 133 | . . . 4 |
59 | 14, 58 | rexlimddv 2531 | . . 3 |
60 | 4, 59 | rexlimddv 2531 | . 2 |
61 | 2, 60 | rexlimddv 2531 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wex 1453 wcel 1465 wrex 2394 cop 3500 class class class wbr 3899 (class class class)co 5742 cnq 7056 cplq 7058 cmq 7059 cltq 7061 cnp 7067 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 |
This theorem is referenced by: prmuloc2 7343 mullocpr 7347 |
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