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Mirrors > Home > ILE Home > Th. List > mulnqprl | Unicode version |
Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.) |
Ref | Expression |
---|---|
mulnqprl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmnqg 7324 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | simpr 109 | . . . . . 6 | |
4 | prop 7398 | . . . . . . . . 9 | |
5 | elprnql 7404 | . . . . . . . . 9 | |
6 | 4, 5 | sylan 281 | . . . . . . . 8 |
7 | 6 | ad2antrr 480 | . . . . . . 7 |
8 | prop 7398 | . . . . . . . . 9 | |
9 | elprnql 7404 | . . . . . . . . 9 | |
10 | 8, 9 | sylan 281 | . . . . . . . 8 |
11 | 10 | ad2antlr 481 | . . . . . . 7 |
12 | mulclnq 7299 | . . . . . . 7 | |
13 | 7, 11, 12 | syl2anc 409 | . . . . . 6 |
14 | recclnq 7315 | . . . . . . 7 | |
15 | 11, 14 | syl 14 | . . . . . 6 |
16 | mulcomnqg 7306 | . . . . . . 7 | |
17 | 16 | adantl 275 | . . . . . 6 |
18 | 2, 3, 13, 15, 17 | caovord2d 5993 | . . . . 5 |
19 | mulassnqg 7307 | . . . . . . . 8 | |
20 | 7, 11, 15, 19 | syl3anc 1220 | . . . . . . 7 |
21 | recidnq 7316 | . . . . . . . . 9 | |
22 | 21 | oveq2d 5843 | . . . . . . . 8 |
23 | 11, 22 | syl 14 | . . . . . . 7 |
24 | mulidnq 7312 | . . . . . . . 8 | |
25 | 7, 24 | syl 14 | . . . . . . 7 |
26 | 20, 23, 25 | 3eqtrd 2194 | . . . . . 6 |
27 | 26 | breq2d 3979 | . . . . 5 |
28 | 18, 27 | bitrd 187 | . . . 4 |
29 | prcdnql 7407 | . . . . . 6 | |
30 | 4, 29 | sylan 281 | . . . . 5 |
31 | 30 | ad2antrr 480 | . . . 4 |
32 | 28, 31 | sylbid 149 | . . 3 |
33 | df-imp 7392 | . . . . . . . . 9 | |
34 | mulclnq 7299 | . . . . . . . . 9 | |
35 | 33, 34 | genpprecll 7437 | . . . . . . . 8 |
36 | 35 | exp4b 365 | . . . . . . 7 |
37 | 36 | com34 83 | . . . . . 6 |
38 | 37 | imp32 255 | . . . . 5 |
39 | 38 | adantlr 469 | . . . 4 |
40 | 39 | adantr 274 | . . 3 |
41 | 32, 40 | syld 45 | . 2 |
42 | mulassnqg 7307 | . . . . 5 | |
43 | 3, 15, 11, 42 | syl3anc 1220 | . . . 4 |
44 | mulcomnqg 7306 | . . . . . . 7 | |
45 | 15, 11, 44 | syl2anc 409 | . . . . . 6 |
46 | 11, 21 | syl 14 | . . . . . 6 |
47 | 45, 46 | eqtrd 2190 | . . . . 5 |
48 | 47 | oveq2d 5843 | . . . 4 |
49 | mulidnq 7312 | . . . . 5 | |
50 | 49 | adantl 275 | . . . 4 |
51 | 43, 48, 50 | 3eqtrd 2194 | . . 3 |
52 | 51 | eleq1d 2226 | . 2 |
53 | 41, 52 | sylibd 148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 cop 3564 class class class wbr 3967 cfv 5173 (class class class)co 5827 c1st 6089 c2nd 6090 cnq 7203 c1q 7204 cmq 7206 crq 7207 cltq 7208 cnp 7214 cmp 7217 |
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 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 |
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 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-eprel 4252 df-id 4256 df-iord 4329 df-on 4331 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-irdg 6320 df-1o 6366 df-oadd 6370 df-omul 6371 df-er 6483 df-ec 6485 df-qs 6489 df-ni 7227 df-mi 7229 df-lti 7230 df-mpq 7268 df-enq 7270 df-nqqs 7271 df-mqqs 7273 df-1nqqs 7274 df-rq 7275 df-ltnqqs 7276 df-inp 7389 df-imp 7392 |
This theorem is referenced by: mullocprlem 7493 mulclpr 7495 |
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