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Mirrors > Home > ILE Home > Th. List > nnm00 | Unicode version |
Description: The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.) |
Ref | Expression |
---|---|
nnm00 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . . . . 7 | |
2 | simpl 102 | . . . . . . 7 | |
3 | 1, 2 | jaoi 636 | . . . . . 6 |
4 | 3 | orcd 652 | . . . . 5 |
5 | 4 | a1i 9 | . . . 4 |
6 | simpr 103 | . . . . . . 7 | |
7 | 6 | olcd 653 | . . . . . 6 |
8 | 7 | a1i 9 | . . . . 5 |
9 | simplr 482 | . . . . . . 7 | |
10 | nnmordi 6089 | . . . . . . . . . . . . 13 | |
11 | 10 | expimpd 345 | . . . . . . . . . . . 12 |
12 | 11 | ancoms 255 | . . . . . . . . . . 11 |
13 | nnm0 6054 | . . . . . . . . . . . . 13 | |
14 | 13 | adantr 261 | . . . . . . . . . . . 12 |
15 | 14 | eleq1d 2106 | . . . . . . . . . . 11 |
16 | 12, 15 | sylibd 138 | . . . . . . . . . 10 |
17 | 16 | adantr 261 | . . . . . . . . 9 |
18 | 17 | imp 115 | . . . . . . . 8 |
19 | n0i 3229 | . . . . . . . 8 | |
20 | 18, 19 | syl 14 | . . . . . . 7 |
21 | 9, 20 | pm2.21dd 550 | . . . . . 6 |
22 | 21 | ex 108 | . . . . 5 |
23 | 8, 22 | jaod 637 | . . . 4 |
24 | 0elnn 4340 | . . . . . . 7 | |
25 | 0elnn 4340 | . . . . . . 7 | |
26 | 24, 25 | anim12i 321 | . . . . . 6 |
27 | anddi 734 | . . . . . 6 | |
28 | 26, 27 | sylib 127 | . . . . 5 |
29 | 28 | adantr 261 | . . . 4 |
30 | 5, 23, 29 | mpjaod 638 | . . 3 |
31 | 30 | ex 108 | . 2 |
32 | oveq1 5519 | . . . . . 6 | |
33 | nnm0r 6058 | . . . . . 6 | |
34 | 32, 33 | sylan9eqr 2094 | . . . . 5 |
35 | 34 | ex 108 | . . . 4 |
36 | 35 | adantl 262 | . . 3 |
37 | oveq2 5520 | . . . . . 6 | |
38 | 37, 13 | sylan9eqr 2094 | . . . . 5 |
39 | 38 | ex 108 | . . . 4 |
40 | 39 | adantr 261 | . . 3 |
41 | 36, 40 | jaod 637 | . 2 |
42 | 31, 41 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 c0 3224 com 4313 (class class class)co 5512 comu 5999 |
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-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 |
This theorem is referenced by: enq0tr 6532 nqnq0pi 6536 |
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