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Theorem nnm00 6393
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 690 . . . . . 6  |-  ( ( ( A  =  (/)  /\  B  =  (/) )  \/  ( A  =  (/)  /\  (/)  e.  B ) )  ->  A  =  (/) )
43orcd 707 . . . . 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 708 . . . . . 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 504 . . . . . . 7  |-  ( ( ( ( A  e. 
om  /\  B  e.  om )  /\  ( A  .o  B )  =  (/) )  /\  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  .o  B )  =  (/) )
10 nnmordi 6380 . . . . . . . . . . . . 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 6339 . . . . . . . . . . . . 13  |-  ( A  e.  om  ->  ( A  .o  (/) )  =  (/) )
1413adantr 274 . . . . . . . . . . . 12  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  .o  (/) )  =  (/) )
1514eleq1d 2186 . . . . . . . . . . 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 3338 . . . . . . . 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 594 . . . . . 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 691 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( A  .o  B
)  =  (/) )  -> 
( ( ( (/)  e.  A  /\  B  =  (/) )  \/  ( (/) 
e.  A  /\  (/)  e.  B
) )  ->  ( A  =  (/)  \/  B  =  (/) ) ) )
24 0elnn 4502 . . . . . . 7  |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A
) )
25 0elnn 4502 . . . . . . 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 795 . . . . . 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 692 . . 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 5749 . . . . . 6  |-  ( A  =  (/)  ->  ( A  .o  B )  =  ( (/)  .o  B
) )
33 nnm0r 6343 . . . . . 6  |-  ( B  e.  om  ->  ( (/) 
.o  B )  =  (/) )
3432, 33sylan9eqr 2172 . . . . 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 5750 . . . . . 6  |-  ( B  =  (/)  ->  ( A  .o  B )  =  ( A  .o  (/) ) )
3837, 13sylan9eqr 2172 . . . . 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 691 . 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 682    = wceq 1316    e. wcel 1465   (/)c0 3333   omcom 4474  (class class class)co 5742    .o 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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