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Theorem mul0eqap 8391
Description: If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
Hypotheses
Ref Expression
mul0eqap.a  |-  ( ph  ->  A  e.  CC )
mul0eqap.b  |-  ( ph  ->  B  e.  CC )
mul0eqap.ab  |-  ( ph  ->  A #  B )
mul0eqap.0  |-  ( ph  ->  ( A  x.  B
)  =  0 )
Assertion
Ref Expression
mul0eqap  |-  ( ph  ->  ( A  =  0  \/  B  =  0 ) )

Proof of Theorem mul0eqap
StepHypRef Expression
1 mul0eqap.ab . . . 4  |-  ( ph  ->  A #  B )
2 mul0eqap.a . . . . 5  |-  ( ph  ->  A  e.  CC )
3 mul0eqap.b . . . . 5  |-  ( ph  ->  B  e.  CC )
4 0cnd 7723 . . . . 5  |-  ( ph  ->  0  e.  CC )
5 apcotr 8332 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  0  e.  CC )  ->  ( A #  B  ->  ( A #  0  \/  B #  0 ) ) )
62, 3, 4, 5syl3anc 1199 . . . 4  |-  ( ph  ->  ( A #  B  -> 
( A #  0  \/  B #  0 ) ) )
71, 6mpd 13 . . 3  |-  ( ph  ->  ( A #  0  \/  B #  0 ) )
8 mul0eqap.0 . . . . . . 7  |-  ( ph  ->  ( A  x.  B
)  =  0 )
98adantr 272 . . . . . 6  |-  ( (
ph  /\  A #  0
)  ->  ( A  x.  B )  =  0 )
103adantr 272 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  B  e.  CC )
11 0cnd 7723 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  0  e.  CC )
122, 3mulcld 7750 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  CC )
1312adantr 272 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  ( A  x.  B )  e.  CC )
14 ibar 297 . . . . . . . 8  |-  ( A #  0  ->  ( B #  0 
<->  ( A #  0  /\  B #  0 ) ) )
152, 3mulap0bd 8378 . . . . . . . 8  |-  ( ph  ->  ( ( A #  0  /\  B #  0 )  <-> 
( A  x.  B
) #  0 ) )
1614, 15sylan9bbr 456 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  ( B #  0 
<->  ( A  x.  B
) #  0 ) )
1710, 11, 13, 11, 16apcon4bid 8349 . . . . . 6  |-  ( (
ph  /\  A #  0
)  ->  ( B  =  0  <->  ( A  x.  B )  =  0 ) )
189, 17mpbird 166 . . . . 5  |-  ( (
ph  /\  A #  0
)  ->  B  = 
0 )
1918ex 114 . . . 4  |-  ( ph  ->  ( A #  0  ->  B  =  0 ) )
208adantr 272 . . . . . 6  |-  ( (
ph  /\  B #  0
)  ->  ( A  x.  B )  =  0 )
212adantr 272 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  A  e.  CC )
22 0cnd 7723 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  0  e.  CC )
2312adantr 272 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  ( A  x.  B )  e.  CC )
24 iba 296 . . . . . . . 8  |-  ( B #  0  ->  ( A #  0 
<->  ( A #  0  /\  B #  0 ) ) )
2524, 15sylan9bbr 456 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  ( A #  0 
<->  ( A  x.  B
) #  0 ) )
2621, 22, 23, 22, 25apcon4bid 8349 . . . . . 6  |-  ( (
ph  /\  B #  0
)  ->  ( A  =  0  <->  ( A  x.  B )  =  0 ) )
2720, 26mpbird 166 . . . . 5  |-  ( (
ph  /\  B #  0
)  ->  A  = 
0 )
2827ex 114 . . . 4  |-  ( ph  ->  ( B #  0  ->  A  =  0 ) )
2919, 28orim12d 758 . . 3  |-  ( ph  ->  ( ( A #  0  \/  B #  0 )  ->  ( B  =  0  \/  A  =  0 ) ) )
307, 29mpd 13 . 2  |-  ( ph  ->  ( B  =  0  \/  A  =  0 ) )
3130orcomd 701 1  |-  ( ph  ->  ( A  =  0  \/  B  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   CCcc 7582   0cc0 7584    x. cmul 7589   # cap 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307
This theorem is referenced by: (None)
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