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Theorem mul0or 9651
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 448 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
21adantr 452 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  B  e.  CC )
32mul02d 9253 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( 0  x.  B )  =  0 )
43eqeq2d 2446 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  ( 0  x.  B )  <->  ( A  x.  B )  =  0 ) )
5 simpl 444 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
65adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  A  e.  CC )
7 0cn 9073 . . . . . . . . . 10  |-  0  e.  CC
87a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  0  e.  CC )
9 simpr 448 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  B  =/=  0 )
106, 8, 2, 9mulcan2d 9645 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  ( 0  x.  B )  <->  A  = 
0 ) )
114, 10bitr3d 247 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  0  <->  A  = 
0 ) )
1211biimpd 199 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( ( A  x.  B )  =  0  ->  A  =  0 ) )
1312impancom 428 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( B  =/=  0  ->  A  =  0 ) )
1413necon1bd 2666 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( -.  A  =  0  ->  B  =  0 ) )
1514orrd 368 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  x.  B )  =  0 )  ->  ( A  =  0  \/  B  =  0 ) )
1615ex 424 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  ->  ( A  =  0  \/  B  =  0 ) ) )
171mul02d 9253 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
18 oveq1 6079 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
1918eqeq1d 2443 . . . 4  |-  ( A  =  0  ->  (
( A  x.  B
)  =  0  <->  (
0  x.  B )  =  0 ) )
2017, 19syl5ibrcom 214 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  0  ->  ( A  x.  B )  =  0 ) )
215mul01d 9254 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  0 )  =  0 )
22 oveq2 6080 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
2322eqeq1d 2443 . . . 4  |-  ( B  =  0  ->  (
( A  x.  B
)  =  0  <->  ( A  x.  0 )  =  0 ) )
2421, 23syl5ibrcom 214 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  =  0  ->  ( A  x.  B )  =  0 ) )
2520, 24jaod 370 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  =  0  \/  B  =  0 )  ->  ( A  x.  B )  =  0 ) )
2616, 25impbid 184 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  0  <-> 
( A  =  0  \/  B  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6072   CCcc 8977   0cc0 8979    x. cmul 8984
This theorem is referenced by:  mulne0b  9652  msq0i  9658  mul0ori  9659  msq0d  9660  mul0ord  9661  coseq1  20418  efrlim  20796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283
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