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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 7202 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | simpr 109 | . . . . . 6 | |
4 | prop 7276 | . . . . . . . . 9 | |
5 | elprnql 7282 | . . . . . . . . 9 | |
6 | 4, 5 | sylan 281 | . . . . . . . 8 |
7 | 6 | ad2antrr 479 | . . . . . . 7 |
8 | prop 7276 | . . . . . . . . 9 | |
9 | elprnql 7282 | . . . . . . . . 9 | |
10 | 8, 9 | sylan 281 | . . . . . . . 8 |
11 | 10 | ad2antlr 480 | . . . . . . 7 |
12 | mulclnq 7177 | . . . . . . 7 | |
13 | 7, 11, 12 | syl2anc 408 | . . . . . 6 |
14 | recclnq 7193 | . . . . . . 7 | |
15 | 11, 14 | syl 14 | . . . . . 6 |
16 | mulcomnqg 7184 | . . . . . . 7 | |
17 | 16 | adantl 275 | . . . . . 6 |
18 | 2, 3, 13, 15, 17 | caovord2d 5933 | . . . . 5 |
19 | mulassnqg 7185 | . . . . . . . 8 | |
20 | 7, 11, 15, 19 | syl3anc 1216 | . . . . . . 7 |
21 | recidnq 7194 | . . . . . . . . 9 | |
22 | 21 | oveq2d 5783 | . . . . . . . 8 |
23 | 11, 22 | syl 14 | . . . . . . 7 |
24 | mulidnq 7190 | . . . . . . . 8 | |
25 | 7, 24 | syl 14 | . . . . . . 7 |
26 | 20, 23, 25 | 3eqtrd 2174 | . . . . . 6 |
27 | 26 | breq2d 3936 | . . . . 5 |
28 | 18, 27 | bitrd 187 | . . . 4 |
29 | prcdnql 7285 | . . . . . 6 | |
30 | 4, 29 | sylan 281 | . . . . 5 |
31 | 30 | ad2antrr 479 | . . . 4 |
32 | 28, 31 | sylbid 149 | . . 3 |
33 | df-imp 7270 | . . . . . . . . 9 | |
34 | mulclnq 7177 | . . . . . . . . 9 | |
35 | 33, 34 | genpprecll 7315 | . . . . . . . 8 |
36 | 35 | exp4b 364 | . . . . . . 7 |
37 | 36 | com34 83 | . . . . . 6 |
38 | 37 | imp32 255 | . . . . 5 |
39 | 38 | adantlr 468 | . . . 4 |
40 | 39 | adantr 274 | . . 3 |
41 | 32, 40 | syld 45 | . 2 |
42 | mulassnqg 7185 | . . . . 5 | |
43 | 3, 15, 11, 42 | syl3anc 1216 | . . . 4 |
44 | mulcomnqg 7184 | . . . . . . 7 | |
45 | 15, 11, 44 | syl2anc 408 | . . . . . 6 |
46 | 11, 21 | syl 14 | . . . . . 6 |
47 | 45, 46 | eqtrd 2170 | . . . . 5 |
48 | 47 | oveq2d 5783 | . . . 4 |
49 | mulidnq 7190 | . . . . 5 | |
50 | 49 | adantl 275 | . . . 4 |
51 | 43, 48, 50 | 3eqtrd 2174 | . . 3 |
52 | 51 | eleq1d 2206 | . 2 |
53 | 41, 52 | sylibd 148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cop 3525 class class class wbr 3924 cfv 5118 (class class class)co 5767 c1st 6029 c2nd 6030 cnq 7081 c1q 7082 cmq 7084 crq 7085 cltq 7086 cnp 7092 cmp 7095 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-mi 7107 df-lti 7108 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-inp 7267 df-imp 7270 |
This theorem is referenced by: mullocprlem 7371 mulclpr 7373 |
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