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Theorem nnm00 6425
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 108 . . . . . . 7  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  A  =  (/) )
2 simpl 108 . . . . . . 7  |-  ( ( A  =  (/)  /\  (/)  e.  B
)  ->  A  =  (/) )
31, 2jaoi 705 . . . . . 6  |-  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  ->  A  =  (/) )
43orcd 722 . . . . 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 109 . . . . . . 7  |-  ( (
(/)  e.  A  /\  B  =  (/) )  ->  B  =  (/) )
76olcd 723 . . . . . 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 519 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  .o  B )  =  (/) )
10 nnmordi 6412 . . . . . . . . . . . . 13  |-  ( ( ( B  e.  om  /\  A  e.  om )  /\  (/)  e.  A )  ->  ( (/)  e.  B  ->  ( A  .o  (/) )  e.  ( A  .o  B
) ) )
1110expimpd 360 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  om )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  ->  ( A  .o  (/) )  e.  ( A  .o  B ) ) )
1211ancoms 266 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  ->  ( A  .o  (/) )  e.  ( A  .o  B ) ) )
13 nnm0 6371 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
1413adantr 274 . . . . . . . . . . . 12  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  .o  (/) )  =  (/) )
1514eleq1d 2208 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  .o  (/) )  e.  ( A  .o  B )  <->  (/)  e.  ( A  .o  B ) ) )
1612, 15sylibd 148 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( (/)  e.  A  /\  (/)  e.  B )  ->  (/)  e.  ( A  .o  B ) ) )
1716adantr 274 . . . . . . . . 9  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( (/)  e.  A  /\  (/)  e.  B )  ->  (/)  e.  ( A  .o  B ) ) )
1817imp 123 . . . . . . . 8  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  (/)  e.  ( A  .o  B ) )
19 n0i 3368 . . . . . . . 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 609 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) )
2221ex 114 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( (/)  e.  A  /\  (/)  e.  B )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
238, 22jaod 706 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( ( (/)  e.  A  /\  B  =  (/) )  \/  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
24 0elnn 4532 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
25 0elnn 4532 . . . . . . 7  |-  ( B  e.  om  ->  ( B  =  (/)  \/  (/)  e.  B
) )
2624, 25anim12i 336 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  =  (/)  \/  (/)  e.  A )  /\  ( B  =  (/)  \/  (/)  e.  B ) ) )
27 anddi 810 . . . . . 6  |-  ( ( ( A  =  (/)  \/  (/)  e.  A )  /\  ( B  =  (/)  \/  (/)  e.  B
) )  <->  ( (
( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  \/  ( ( (/)  e.  A  /\  B  =  (/) )  \/  ( (/) 
e.  A  /\  (/)  e.  B
) ) ) )
2826, 27sylib 121 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B
) )  \/  (
( (/)  e.  A  /\  B  =  (/) )  \/  ( (/)  e.  A  /\  (/)  e.  B ) ) ) )
2928adantr 274 . . . 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 707 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( A  =  (/)  \/  B  =  (/) ) )
3130ex 114 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  .o  B )  =  (/)  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
32 oveq1 5781 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
33 nnm0r 6375 . . . . . 6  |-  ( B  e.  om  ->  ( (/) 
.o  B )  =  (/) )
3432, 33sylan9eqr 2194 . . . . 5  |-  ( ( B  e.  om  /\  A  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3534ex 114 . . . 4  |-  ( B  e.  om  ->  ( A  =  (/)  ->  ( A  .o  B )  =  (/) ) )
3635adantl 275 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
37 oveq2 5782 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
3837, 13sylan9eqr 2194 . . . . 5  |-  ( ( A  e.  om  /\  B  =  (/) )  -> 
( A  .o  B
)  =  (/) )
3938ex 114 . . . 4  |-  ( A  e.  om  ->  ( B  =  (/)  ->  ( A  .o  B )  =  (/) ) )
4039adantr 274 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  =  (/)  ->  ( A  .o  B
)  =  (/) ) )
4136, 40jaod 706 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  .o  B )  =  (/) ) )
4231, 41impbid 128 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 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   (/)c0 3363   omcom 4504  (class class class)co 5774    .o comu 6311
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318
This theorem is referenced by:  enq0tr  7242  nqnq0pi  7246
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