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Theorem nnm00 6168
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 669 . . . . . 6  |-  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  ->  A  =  (/) )
43orcd 685 . . . . 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 686 . . . . . 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 6155 . . . . . . . . . . . . 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 6119 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
1413adantr 270 . . . . . . . . . . . 12  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  .o  (/) )  =  (/) )
1514eleq1d 2148 . . . . . . . . . . 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 3263 . . . . . . . 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 583 . . . . . 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 670 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( ( (/)  e.  A  /\  B  =  (/) )  \/  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
24 0elnn 4366 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
25 0elnn 4366 . . . . . . 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 768 . . . . . 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 671 . . 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 5550 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
33 nnm0r 6123 . . . . . 6  |-  ( B  e.  om  ->  ( (/) 
.o  B )  =  (/) )
3432, 33sylan9eqr 2136 . . . . 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 5551 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
3837, 13sylan9eqr 2136 . . . . 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 670 . 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 662    = wceq 1285    e. wcel 1434   (/)c0 3258   omcom 4339  (class class class)co 5543    .o comu 6063
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070
This theorem is referenced by:  enq0tr  6686  nqnq0pi  6690
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