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Mirrors > Home > ILE Home > Th. List > nq0m0r | Unicode version |
Description: Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.) |
Ref | Expression |
---|---|
nq0m0r | Q0 0Q0 ·Q0 0Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nq0nn 6540 | . 2 Q0 ~Q0 | |
2 | df-0nq0 6524 | . . . . . 6 0Q0 ~Q0 | |
3 | oveq12 5521 | . . . . . 6 0Q0 ~Q0 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 | |
4 | 2, 3 | mpan 400 | . . . . 5 ~Q0 0Q0 ·Q0 ~Q0 ·Q0 ~Q0 |
5 | peano1 4317 | . . . . . 6 | |
6 | 1pi 6413 | . . . . . 6 | |
7 | mulnnnq0 6548 | . . . . . 6 ~Q0 ·Q0 ~Q0 ~Q0 | |
8 | 5, 6, 7 | mpanl12 412 | . . . . 5 ~Q0 ·Q0 ~Q0 ~Q0 |
9 | 4, 8 | sylan9eqr 2094 | . . . 4 ~Q0 0Q0 ·Q0 ~Q0 |
10 | nnm0r 6058 | . . . . . . . . . . 11 | |
11 | 10 | oveq1d 5527 | . . . . . . . . . 10 |
12 | 1onn 6093 | . . . . . . . . . . 11 | |
13 | nnm0r 6058 | . . . . . . . . . . 11 | |
14 | 12, 13 | ax-mp 7 | . . . . . . . . . 10 |
15 | 11, 14 | syl6eq 2088 | . . . . . . . . 9 |
16 | 15 | adantr 261 | . . . . . . . 8 |
17 | mulpiord 6415 | . . . . . . . . . . . 12 | |
18 | mulclpi 6426 | . . . . . . . . . . . 12 | |
19 | 17, 18 | eqeltrrd 2115 | . . . . . . . . . . 11 |
20 | 6, 19 | mpan 400 | . . . . . . . . . 10 |
21 | pinn 6407 | . . . . . . . . . 10 | |
22 | nnm0 6054 | . . . . . . . . . 10 | |
23 | 20, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 23 | adantl 262 | . . . . . . . 8 |
25 | 16, 24 | eqtr4d 2075 | . . . . . . 7 |
26 | 10, 5 | syl6eqel 2128 | . . . . . . . 8 |
27 | enq0eceq 6535 | . . . . . . . . 9 ~Q0 ~Q0 | |
28 | 5, 6, 27 | mpanr12 415 | . . . . . . . 8 ~Q0 ~Q0 |
29 | 26, 20, 28 | syl2an 273 | . . . . . . 7 ~Q0 ~Q0 |
30 | 25, 29 | mpbird 156 | . . . . . 6 ~Q0 ~Q0 |
31 | 30, 2 | syl6eqr 2090 | . . . . 5 ~Q0 0Q0 |
32 | 31 | adantr 261 | . . . 4 ~Q0 ~Q0 0Q0 |
33 | 9, 32 | eqtrd 2072 | . . 3 ~Q0 0Q0 ·Q0 0Q0 |
34 | 33 | exlimivv 1776 | . 2 ~Q0 0Q0 ·Q0 0Q0 |
35 | 1, 34 | syl 14 | 1 Q0 0Q0 ·Q0 0Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 c0 3224 cop 3378 com 4313 (class class class)co 5512 c1o 5994 comu 5999 cec 6104 cnpi 6370 cmi 6372 ~Q0 ceq0 6384 Q0cnq0 6385 0Q0c0q0 6386 ·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-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-mq0 6526 |
This theorem is referenced by: prarloclem5 6598 |
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