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Theorem nnm00 6268
Description: The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)
Assertion
Ref Expression
nnm00  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )

Proof of Theorem nnm00
StepHypRef Expression
1 simpl 107 . . . . . . 7  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  (/) )
2 simpl 107 . . . . . . 7  |-  ( ( A  =  (/)  /\  (/)  e.  B
)  ->  A  =  (/) )
31, 2jaoi 671 . . . . . 6  |-  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  ->  A  =  (/) )
43orcd 687 . . . . 5  |-  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  ->  ( A  =  (/)  \/  B  =  (/) ) )
54a1i 9 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
6 simpr 108 . . . . . . 7  |-  ( (
(/)  e.  A  /\  B  =  (/) )  ->  B  =  (/) )
76olcd 688 . . . . . 6  |-  ( (
(/)  e.  A  /\  B  =  (/) )  -> 
( A  =  (/)  \/  B  =  (/) ) )
87a1i 9 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( (/)  e.  A  /\  B  =  (/) )  -> 
( A  =  (/)  \/  B  =  (/) ) ) )
9 simplr 497 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  .o  B )  =  (/) )
10 nnmordi 6255 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  om  /\  A  e.  om )  /\  (/)  e.  A )  ->  ( (/)  e.  B  ->  ( A  .o  (/) )  e.  ( A  .o  B
) ) )
1110expimpd 355 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  om )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  ->  ( A  .o  (/) )  e.  ( A  .o  B ) ) )
1211ancoms 264 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  ->  ( A  .o  (/) )  e.  ( A  .o  B ) ) )
13 nnm0 6218 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
1413adantr 270 . . . . . . . . . . . 12  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  .o  (/) )  =  (/) )
1514eleq1d 2156 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  .o  (/) )  e.  ( A  .o  B )  <->  (/)  e.  ( A  .o  B ) ) )
1612, 15sylibd 147 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  ->  (/)  e.  ( A  .o  B ) ) )
1716adantr 270 . . . . . . . . 9  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( (/)  e.  A  /\  (/)  e.  B )  ->  (/)  e.  ( A  .o  B ) ) )
1817imp 122 . . . . . . . 8  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  (/)  e.  ( A  .o  B ) )
19 n0i 3289 . . . . . . . 8  |-  ( (/)  e.  ( A  .o  B
)  ->  -.  ( A  .o  B )  =  (/) )
2018, 19syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  -.  ( A  .o  B
)  =  (/) )
219, 20pm2.21dd 585 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) )
2221ex 113 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( (/)  e.  A  /\  (/)  e.  B )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
238, 22jaod 672 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( ( (/)  e.  A  /\  B  =  (/) )  \/  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
24 0elnn 4422 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
25 0elnn 4422 . . . . . . 7  |-  ( B  e.  om  ->  ( B  =  (/)  \/  (/)  e.  B
) )
2624, 25anim12i 331 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  =  (/)  \/  (/)  e.  A )  /\  ( B  =  (/)  \/  (/)  e.  B ) ) )
27 anddi 770 . . . . . 6  |-  ( ( ( A  =  (/)  \/  (/)  e.  A )  /\  ( B  =  (/)  \/  (/)  e.  B
) )  <->  ( (
( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  \/  ( ( (/)  e.  A  /\  B  =  (/) )  \/  ( (/) 
e.  A  /\  (/)  e.  B
) ) ) )
2826, 27sylib 120 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B
) )  \/  (
( (/)  e.  A  /\  B  =  (/) )  \/  ( (/)  e.  A  /\  (/)  e.  B ) ) ) )
2928adantr 270 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B
) )  \/  (
( (/)  e.  A  /\  B  =  (/) )  \/  ( (/)  e.  A  /\  (/)  e.  B ) ) ) )
305, 23, 29mpjaod 673 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( A  =  (/)  \/  B  =  (/) ) )
3130ex 113 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  .o  B )  =  (/)  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
32 oveq1 5641 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
33 nnm0r 6222 . . . . . 6  |-  ( B  e.  om  ->  ( (/) 
.o  B )  =  (/) )
3432, 33sylan9eqr 2142 . . . . 5  |-  ( ( B  e.  om  /\  A  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3534ex 113 . . . 4  |-  ( B  e.  om  ->  ( A  =  (/)  ->  ( A  .o  B )  =  (/) ) )
3635adantl 271 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
37 oveq2 5642 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
3837, 13sylan9eqr 2142 . . . . 5  |-  ( ( A  e.  om  /\  B  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3938ex 113 . . . 4  |-  ( A  e.  om  ->  ( B  =  (/)  ->  ( A  .o  B )  =  (/) ) )
4039adantr 270 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
4136, 40jaod 672 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =  (/) ) )
4231, 41impbid 127 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  .o  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   (/)c0 3284   omcom 4395  (class class class)co 5634    .o comu 6161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168
This theorem is referenced by:  enq0tr  6972  nqnq0pi  6976
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