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Mirrors > Home > ILE Home > Th. List > ltmpig | Unicode version |
Description: Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
ltmpig |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7117 | . . . . 5 | |
2 | pinn 7117 | . . . . 5 | |
3 | elni2 7122 | . . . . . 6 | |
4 | iba 298 | . . . . . . . . 9 | |
5 | nnmord 6413 | . . . . . . . . 9 | |
6 | 4, 5 | sylan9bbr 458 | . . . . . . . 8 |
7 | 6 | 3exp1 1201 | . . . . . . 7 |
8 | 7 | imp4b 347 | . . . . . 6 |
9 | 3, 8 | syl5bi 151 | . . . . 5 |
10 | 1, 2, 9 | syl2an 287 | . . . 4 |
11 | 10 | imp 123 | . . 3 |
12 | ltpiord 7127 | . . . 4 | |
13 | 12 | adantr 274 | . . 3 |
14 | mulclpi 7136 | . . . . . . 7 | |
15 | mulclpi 7136 | . . . . . . 7 | |
16 | ltpiord 7127 | . . . . . . 7 | |
17 | 14, 15, 16 | syl2an 287 | . . . . . 6 |
18 | mulpiord 7125 | . . . . . . . 8 | |
19 | 18 | adantr 274 | . . . . . . 7 |
20 | mulpiord 7125 | . . . . . . . 8 | |
21 | 20 | adantl 275 | . . . . . . 7 |
22 | 19, 21 | eleq12d 2210 | . . . . . 6 |
23 | 17, 22 | bitrd 187 | . . . . 5 |
24 | 23 | anandis 581 | . . . 4 |
25 | 24 | ancoms 266 | . . 3 |
26 | 11, 13, 25 | 3bitr4d 219 | . 2 |
27 | 26 | 3impa 1176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 c0 3363 class class class wbr 3929 com 4504 (class class class)co 5774 comu 6311 cnpi 7080 cmi 7082 clti 7083 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-ni 7112 df-mi 7114 df-lti 7115 |
This theorem is referenced by: ordpipqqs 7182 ltsonq 7206 ltanqg 7208 ltmnqg 7209 1lt2nq 7214 |
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