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Theorem mul0or 10210
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 9795 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( 0  x.  B )  =  0 )
43eqeq2d 2471 . . . . . . . 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 9606 . . . . . . . . 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 10204 . . . . . . . 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 2675 . . . 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 9795 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
17 oveq1 6303 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
1817eqeq1d 2459 . . . 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 9796 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  0 )  =  0 )
21 oveq2 6304 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
2221eqeq1d 2459 . . . 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 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   CCcc 9507   0cc0 9509    x. cmul 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827
This theorem is referenced by:  mulne0b  10211  msq0i  10217  mul0ori  10218  msq0d  10219  mul0ord  10220  coseq1  23040  efrlim  23424
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