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Mirrors > Home > ILE Home > Th. List > mulcmpblnq0 | Unicode version |
Description: Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.) |
Ref | Expression |
---|---|
mulcmpblnq0 | ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 5521 | . 2 | |
2 | nnmcl 6060 | . . . . . . . 8 | |
3 | mulpiord 6415 | . . . . . . . . 9 | |
4 | mulclpi 6426 | . . . . . . . . 9 | |
5 | 3, 4 | eqeltrrd 2115 | . . . . . . . 8 |
6 | 2, 5 | anim12i 321 | . . . . . . 7 |
7 | 6 | an4s 522 | . . . . . 6 |
8 | nnmcl 6060 | . . . . . . . 8 | |
9 | mulpiord 6415 | . . . . . . . . 9 | |
10 | mulclpi 6426 | . . . . . . . . 9 | |
11 | 9, 10 | eqeltrrd 2115 | . . . . . . . 8 |
12 | 8, 11 | anim12i 321 | . . . . . . 7 |
13 | 12 | an4s 522 | . . . . . 6 |
14 | 7, 13 | anim12i 321 | . . . . 5 |
15 | 14 | an4s 522 | . . . 4 |
16 | enq0breq 6534 | . . . 4 ~Q0 | |
17 | 15, 16 | syl 14 | . . 3 ~Q0 |
18 | simplll 485 | . . . . 5 | |
19 | simprll 489 | . . . . 5 | |
20 | simplrr 488 | . . . . . 6 | |
21 | pinn 6407 | . . . . . 6 | |
22 | 20, 21 | syl 14 | . . . . 5 |
23 | nnmcom 6068 | . . . . . 6 | |
24 | 23 | adantl 262 | . . . . 5 |
25 | nnmass 6066 | . . . . . 6 | |
26 | 25 | adantl 262 | . . . . 5 |
27 | simprrr 492 | . . . . . 6 | |
28 | pinn 6407 | . . . . . 6 | |
29 | 27, 28 | syl 14 | . . . . 5 |
30 | nnmcl 6060 | . . . . . 6 | |
31 | 30 | adantl 262 | . . . . 5 |
32 | 18, 19, 22, 24, 26, 29, 31 | caov4d 5685 | . . . 4 |
33 | simpllr 486 | . . . . . 6 | |
34 | pinn 6407 | . . . . . 6 | |
35 | 33, 34 | syl 14 | . . . . 5 |
36 | simprlr 490 | . . . . . 6 | |
37 | pinn 6407 | . . . . . 6 | |
38 | 36, 37 | syl 14 | . . . . 5 |
39 | simplrl 487 | . . . . 5 | |
40 | simprrl 491 | . . . . 5 | |
41 | 35, 38, 39, 24, 26, 40, 31 | caov4d 5685 | . . . 4 |
42 | 32, 41 | eqeq12d 2054 | . . 3 |
43 | 17, 42 | bitrd 177 | . 2 ~Q0 |
44 | 1, 43 | syl5ibr 145 | 1 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 cop 3378 class class class wbr 3764 com 4313 (class class class)co 5512 comu 5999 cnpi 6370 cmi 6372 ~Q0 ceq0 6384 |
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-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-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-enq0 6522 |
This theorem is referenced by: mulnq0mo 6546 |
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