MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mul0or Unicode version

Theorem mul0or 9403
Description: If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
mul0or  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> 
( A  =  0  \/  B  =  0 ) ) )

Proof of Theorem mul0or
StepHypRef Expression
1 simpr 449 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
21adantr 453 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  B  e.  CC )
32mul02d 9005 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( 0  x.  B )  =  0 )
43eqeq2d 2295 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  ( 0  x.  B )  <->  ( A  x.  B )  =  0 ) )
5 simpl 445 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
65adantr 453 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  A  e.  CC )
7 0cn 8826 . . . . . . . . . 10  |-  0  e.  CC
87a1i 12 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  0  e.  CC )
9 simpr 449 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  B  =/=  0 )
106, 8, 2, 9mulcan2d 9397 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  ( 0  x.  B )  <->  A  = 
0 ) )
114, 10bitr3d 248 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  0  <->  A  = 
0 ) )
1211biimpd 200 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  0  ->  A  =  0 ) )
1312impancom 429 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( B  =/=  0  ->  A  =  0 ) )
1413necon1bd 2515 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( -.  A  =  0  ->  B  =  0 ) )
1514orrd 369 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( A  =  0  \/  B  =  0 ) )
1615ex 425 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  ->  ( A  =  0  \/  B  =  0 ) ) )
171mul02d 9005 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
18 oveq1 5826 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
1918eqeq1d 2292 . . . 4  |-  ( A  =  0  ->  (
( A  x.  B
)  =  0  <->  (
0  x.  B )  =  0 ) )
2017, 19syl5ibrcom 215 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  0  ->  ( A  x.  B )  =  0 ) )
215mul01d 9006 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  0 )  =  0 )
22 oveq2 5827 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
2322eqeq1d 2292 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  =  0  <->  ( A  x.  0 )  =  0 ) )
2421, 23syl5ibrcom 215 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  =  0  ->  ( A  x.  B )  =  0 ) )
2520, 24jaod 371 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  =  0  \/  B  =  0 )  ->  ( A  x.  B )  =  0 ) )
2616, 25impbid 185 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> 
( A  =  0  \/  B  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1628    e. wcel 1688    =/= wne 2447  (class class class)co 5819   CCcc 8730   0cc0 8732    x. cmul 8737
This theorem is referenced by:  mulne0b  9404  msq0i  9410  mul0ori  9411  msq0d  9412  mul0ord  9413  coseq1  19884  efrlim  20258
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035
  Copyright terms: Public domain W3C validator