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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 108 | . . . . . . 7 | |
2 | simpl 108 | . . . . . . 7 | |
3 | 1, 2 | jaoi 690 | . . . . . 6 |
4 | 3 | orcd 707 | . . . . 5 |
5 | 4 | a1i 9 | . . . 4 |
6 | simpr 109 | . . . . . . 7 | |
7 | 6 | olcd 708 | . . . . . 6 |
8 | 7 | a1i 9 | . . . . 5 |
9 | simplr 504 | . . . . . . 7 | |
10 | nnmordi 6380 | . . . . . . . . . . . . 13 | |
11 | 10 | expimpd 360 | . . . . . . . . . . . 12 |
12 | 11 | ancoms 266 | . . . . . . . . . . 11 |
13 | nnm0 6339 | . . . . . . . . . . . . 13 | |
14 | 13 | adantr 274 | . . . . . . . . . . . 12 |
15 | 14 | eleq1d 2186 | . . . . . . . . . . 11 |
16 | 12, 15 | sylibd 148 | . . . . . . . . . 10 |
17 | 16 | adantr 274 | . . . . . . . . 9 |
18 | 17 | imp 123 | . . . . . . . 8 |
19 | n0i 3338 | . . . . . . . 8 | |
20 | 18, 19 | syl 14 | . . . . . . 7 |
21 | 9, 20 | pm2.21dd 594 | . . . . . 6 |
22 | 21 | ex 114 | . . . . 5 |
23 | 8, 22 | jaod 691 | . . . 4 |
24 | 0elnn 4502 | . . . . . . 7 | |
25 | 0elnn 4502 | . . . . . . 7 | |
26 | 24, 25 | anim12i 336 | . . . . . 6 |
27 | anddi 795 | . . . . . 6 | |
28 | 26, 27 | sylib 121 | . . . . 5 |
29 | 28 | adantr 274 | . . . 4 |
30 | 5, 23, 29 | mpjaod 692 | . . 3 |
31 | 30 | ex 114 | . 2 |
32 | oveq1 5749 | . . . . . 6 | |
33 | nnm0r 6343 | . . . . . 6 | |
34 | 32, 33 | sylan9eqr 2172 | . . . . 5 |
35 | 34 | ex 114 | . . . 4 |
36 | 35 | adantl 275 | . . 3 |
37 | oveq2 5750 | . . . . . 6 | |
38 | 37, 13 | sylan9eqr 2172 | . . . . 5 |
39 | 38 | ex 114 | . . . 4 |
40 | 39 | adantr 274 | . . 3 |
41 | 36, 40 | jaod 691 | . 2 |
42 | 31, 41 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wceq 1316 wcel 1465 c0 3333 com 4474 (class class class)co 5742 comu 6279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 |
This theorem is referenced by: enq0tr 7210 nqnq0pi 7214 |
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