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Theorem mul0or 9341
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 8943 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  B  =/=  0
)  ->  ( 0  x.  B )  =  0 )
43eqeq2d 2267 . . . . . . . 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 8764 . . . . . . . . . 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 9335 . . . . . . . 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 2487 . . . 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 8943 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
18 oveq1 5764 . . . . 5  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
1918eqeq1d 2264 . . . 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 8944 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  0 )  =  0 )
22 oveq2 5765 . . . . 5  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
2322eqeq1d 2264 . . . 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 1619    e. wcel 1621    =/= wne 2419  (class class class)co 5757   CCcc 8668   0cc0 8670    x. cmul 8675
This theorem is referenced by:  mulne0b  9342  msq0i  9348  mul0ori  9349  msq0d  9350  mul0ord  9351  coseq1  19817  efrlim  20191
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973
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