Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6407 | . . . . 5 | |
2 | pinn 6407 | . . . . 5 | |
3 | elni2 6412 | . . . . . 6 | |
4 | iba 284 | . . . . . . . . 9 | |
5 | nnmord 6090 | . . . . . . . . 9 | |
6 | 4, 5 | sylan9bbr 436 | . . . . . . . 8 |
7 | 6 | 3exp1 1120 | . . . . . . 7 |
8 | 7 | imp4b 332 | . . . . . 6 |
9 | 3, 8 | syl5bi 141 | . . . . 5 |
10 | 1, 2, 9 | syl2an 273 | . . . 4 |
11 | 10 | imp 115 | . . 3 |
12 | ltpiord 6417 | . . . 4 | |
13 | 12 | adantr 261 | . . 3 |
14 | mulclpi 6426 | . . . . . . 7 | |
15 | mulclpi 6426 | . . . . . . 7 | |
16 | ltpiord 6417 | . . . . . . 7 | |
17 | 14, 15, 16 | syl2an 273 | . . . . . 6 |
18 | mulpiord 6415 | . . . . . . . 8 | |
19 | 18 | adantr 261 | . . . . . . 7 |
20 | mulpiord 6415 | . . . . . . . 8 | |
21 | 20 | adantl 262 | . . . . . . 7 |
22 | 19, 21 | eleq12d 2108 | . . . . . 6 |
23 | 17, 22 | bitrd 177 | . . . . 5 |
24 | 23 | anandis 526 | . . . 4 |
25 | 24 | ancoms 255 | . . 3 |
26 | 11, 13, 25 | 3bitr4d 209 | . 2 |
27 | 26 | 3impa 1099 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 c0 3224 class class class wbr 3764 com 4313 (class class class)co 5512 comu 5999 cnpi 6370 cmi 6372 clti 6373 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-ni 6402 df-mi 6404 df-lti 6405 |
This theorem is referenced by: ordpipqqs 6472 ltsonq 6496 ltanqg 6498 ltmnqg 6499 1lt2nq 6504 |
Copyright terms: Public domain | W3C validator |