Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nqnq0m | Unicode version |
Description: Multiplication of positive fractions is equal with or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.) |
Ref | Expression |
---|---|
nqnq0m | ·Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqpi 6476 | . . . 4 | |
2 | nqpi 6476 | . . . 4 | |
3 | 1, 2 | anim12i 321 | . . 3 |
4 | ee4anv 1809 | . . 3 | |
5 | 3, 4 | sylibr 137 | . 2 |
6 | oveq12 5521 | . . . . . . 7 | |
7 | mulpiord 6415 | . . . . . . . . . . 11 | |
8 | 7 | ad2ant2r 478 | . . . . . . . . . 10 |
9 | mulpiord 6415 | . . . . . . . . . . 11 | |
10 | 9 | ad2ant2l 477 | . . . . . . . . . 10 |
11 | 8, 10 | opeq12d 3557 | . . . . . . . . 9 |
12 | 11 | eceq1d 6142 | . . . . . . . 8 ~Q0 ~Q0 |
13 | mulpipqqs 6471 | . . . . . . . . 9 | |
14 | mulclpi 6426 | . . . . . . . . . . 11 | |
15 | 14 | ad2ant2r 478 | . . . . . . . . . 10 |
16 | mulclpi 6426 | . . . . . . . . . . 11 | |
17 | 16 | ad2ant2l 477 | . . . . . . . . . 10 |
18 | nqnq0pi 6536 | . . . . . . . . . 10 ~Q0 | |
19 | 15, 17, 18 | syl2anc 391 | . . . . . . . . 9 ~Q0 |
20 | 13, 19 | eqtr4d 2075 | . . . . . . . 8 ~Q0 |
21 | pinn 6407 | . . . . . . . . . 10 | |
22 | 21 | anim1i 323 | . . . . . . . . 9 |
23 | pinn 6407 | . . . . . . . . . 10 | |
24 | 23 | anim1i 323 | . . . . . . . . 9 |
25 | mulnnnq0 6548 | . . . . . . . . 9 ~Q0 ·Q0 ~Q0 ~Q0 | |
26 | 22, 24, 25 | syl2an 273 | . . . . . . . 8 ~Q0 ·Q0 ~Q0 ~Q0 |
27 | 12, 20, 26 | 3eqtr4d 2082 | . . . . . . 7 ~Q0 ·Q0 ~Q0 |
28 | 6, 27 | sylan9eqr 2094 | . . . . . 6 ~Q0 ·Q0 ~Q0 |
29 | nqnq0pi 6536 | . . . . . . . . . . 11 ~Q0 | |
30 | 29 | adantr 261 | . . . . . . . . . 10 ~Q0 |
31 | 30 | eqeq2d 2051 | . . . . . . . . 9 ~Q0 |
32 | nqnq0pi 6536 | . . . . . . . . . . 11 ~Q0 | |
33 | 32 | adantl 262 | . . . . . . . . . 10 ~Q0 |
34 | 33 | eqeq2d 2051 | . . . . . . . . 9 ~Q0 |
35 | 31, 34 | anbi12d 442 | . . . . . . . 8 ~Q0 ~Q0 |
36 | 35 | pm5.32i 427 | . . . . . . 7 ~Q0 ~Q0 |
37 | oveq12 5521 | . . . . . . . 8 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
38 | 37 | adantl 262 | . . . . . . 7 ~Q0 ~Q0 ·Q0 ~Q0 ·Q0 ~Q0 |
39 | 36, 38 | sylbir 125 | . . . . . 6 ·Q0 ~Q0 ·Q0 ~Q0 |
40 | 28, 39 | eqtr4d 2075 | . . . . 5 ·Q0 |
41 | 40 | an4s 522 | . . . 4 ·Q0 |
42 | 41 | exlimivv 1776 | . . 3 ·Q0 |
43 | 42 | exlimivv 1776 | . 2 ·Q0 |
44 | 5, 43 | syl 14 | 1 ·Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cop 3378 com 4313 (class class class)co 5512 comu 5999 cec 6104 cnpi 6370 cmi 6372 ceq 6377 cnq 6378 cmq 6381 ~Q0 ceq0 6384 ·Q0 cmq0 6388 |
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-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-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 df-enq0 6522 df-nq0 6523 df-mq0 6526 |
This theorem is referenced by: prarloclemlo 6592 prarloclemcalc 6600 |
Copyright terms: Public domain | W3C validator |