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Theorem mul0eqap 8544
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 7871 . . . . 5  |-  ( ph  ->  0  e.  CC )
5 apcotr 8482 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  0  e.  CC )  ->  ( A #  B  ->  ( A #  0  \/  B #  0 ) ) )
62, 3, 4, 5syl3anc 1220 . . . 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 274 . . . . . 6  |-  ( (
ph  /\  A #  0
)  ->  ( A  x.  B )  =  0 )
103adantr 274 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  B  e.  CC )
11 0cnd 7871 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  0  e.  CC )
122, 3mulcld 7898 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  CC )
1312adantr 274 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  ( A  x.  B )  e.  CC )
14 ibar 299 . . . . . . . 8  |-  ( A #  0  ->  ( B #  0 
<->  ( A #  0  /\  B #  0 ) ) )
152, 3mulap0bd 8531 . . . . . . . 8  |-  ( ph  ->  ( ( A #  0  /\  B #  0 )  <-> 
( A  x.  B
) #  0 ) )
1614, 15sylan9bbr 459 . . . . . . 7  |-  ( (
ph  /\  A #  0
)  ->  ( B #  0 
<->  ( A  x.  B
) #  0 ) )
1710, 11, 13, 11, 16apcon4bid 8499 . . . . . 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 274 . . . . . 6  |-  ( (
ph  /\  B #  0
)  ->  ( A  x.  B )  =  0 )
212adantr 274 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  A  e.  CC )
22 0cnd 7871 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  0  e.  CC )
2312adantr 274 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  ( A  x.  B )  e.  CC )
24 iba 298 . . . . . . . 8  |-  ( B #  0  ->  ( A #  0 
<->  ( A #  0  /\  B #  0 ) ) )
2524, 15sylan9bbr 459 . . . . . . 7  |-  ( (
ph  /\  B #  0
)  ->  ( A #  0 
<->  ( A  x.  B
) #  0 ) )
2621, 22, 23, 22, 25apcon4bid 8499 . . . . . 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 776 . . 3  |-  ( ph  ->  ( ( A #  0  \/  B #  0 )  ->  ( B  =  0  \/  A  =  0 ) ) )
307, 29mpd 13 . 2  |-  ( ph  ->  ( B  =  0  \/  A  =  0 ) )
3130orcomd 719 1  |-  ( ph  ->  ( A  =  0  \/  B  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1335    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   CCcc 7730   0cc0 7732    x. cmul 7737   # cap 8456
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457
This theorem is referenced by: (None)
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