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Theorem mul0or 9976
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 461 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
21adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  B  e.  CC )
32mul02d 9567 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( 0  x.  B )  =  0 )
43eqeq2d 2454 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  ( 0  x.  B )  <->  ( A  x.  B )  =  0 ) )
5 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
65adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  A  e.  CC )
7 0cnd 9379 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  0  e.  CC )
8 simpr 461 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  B  =/=  0 )
96, 7, 2, 8mulcan2d 9970 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  ( 0  x.  B )  <->  A  = 
0 ) )
104, 9bitr3d 255 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  0  <->  A  = 
0 ) )
1110biimpd 207 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  0  ->  A  =  0 ) )
1211impancom 440 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( B  =/=  0  ->  A  =  0 ) )
1312necon1bd 2679 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( -.  A  =  0  ->  B  =  0 ) )
1413orrd 378 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( A  =  0  \/  B  =  0 ) )
1514ex 434 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  ->  ( A  =  0  \/  B  =  0 ) ) )
161mul02d 9567 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
17 oveq1 6098 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
1817eqeq1d 2451 . . . 4  |-  ( A  =  0  ->  (
( A  x.  B
)  =  0  <->  (
0  x.  B )  =  0 ) )
1916, 18syl5ibrcom 222 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  0  ->  ( A  x.  B )  =  0 ) )
205mul01d 9568 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  0 )  =  0 )
21 oveq2 6099 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
2221eqeq1d 2451 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  =  0  <->  ( A  x.  0 )  =  0 ) )
2320, 22syl5ibrcom 222 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  =  0  ->  ( A  x.  B )  =  0 ) )
2419, 23jaod 380 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  =  0  \/  B  =  0 )  ->  ( A  x.  B )  =  0 ) )
2515, 24impbid 191 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> 
( A  =  0  \/  B  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606  (class class class)co 6091   CCcc 9280   0cc0 9282    x. cmul 9287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598
This theorem is referenced by:  mulne0b  9977  msq0i  9983  mul0ori  9984  msq0d  9985  mul0ord  9986  coseq1  21984  efrlim  22363
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