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Theorem mulap0r 8571
Description: A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
Assertion
Ref Expression
mulap0r  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A #  0  /\  B #  0 ) )

Proof of Theorem mulap0r
StepHypRef Expression
1 simp3 999 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A  x.  B ) #  0 )
2 simp2 998 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  B  e.  CC )
32mul02d 8348 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  (
0  x.  B )  =  0 )
41, 3breqtrrd 4031 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A  x.  B ) #  ( 0  x.  B
) )
5 simp1 997 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  A  e.  CC )
6 0cnd 7949 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  0  e.  CC )
7 mulext 8570 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( 0  e.  CC  /\  B  e.  CC ) )  -> 
( ( A  x.  B ) #  ( 0  x.  B )  -> 
( A #  0  \/  B #  B ) ) )
85, 2, 6, 2, 7syl22anc 1239 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  (
( A  x.  B
) #  ( 0  x.  B )  ->  ( A #  0  \/  B #  B ) ) )
94, 8mpd 13 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A #  0  \/  B #  B ) )
109orcomd 729 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( B #  B  \/  A #  0 ) )
11 apirr 8561 . . . 4  |-  ( B  e.  CC  ->  -.  B #  B )
12 biorf 744 . . . 4  |-  ( -.  B #  B  ->  ( A #  0  <->  ( B #  B  \/  A #  0 )
) )
132, 11, 123syl 17 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A #  0  <->  ( B #  B  \/  A #  0 )
) )
1410, 13mpbird 167 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  A #  0 )
155mul01d 8349 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A  x.  0 )  =  0 )
161, 15breqtrrd 4031 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A  x.  B ) #  ( A  x.  0
) )
17 mulext 8570 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( A  e.  CC  /\  0  e.  CC ) )  -> 
( ( A  x.  B ) #  ( A  x.  0 )  ->  ( A #  A  \/  B #  0 ) ) )
185, 2, 5, 6, 17syl22anc 1239 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  (
( A  x.  B
) #  ( A  x.  0 )  ->  ( A #  A  \/  B #  0 ) ) )
1916, 18mpd 13 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A #  A  \/  B #  0 ) )
20 apirr 8561 . . . 4  |-  ( A  e.  CC  ->  -.  A #  A )
21 biorf 744 . . . 4  |-  ( -.  A #  A  ->  ( B #  0  <->  ( A #  A  \/  B #  0 )
) )
225, 20, 213syl 17 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( B #  0  <->  ( A #  A  \/  B #  0 )
) )
2319, 22mpbird 167 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  B #  0 )
2414, 23jca 306 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A #  0  /\  B #  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808   0cc0 7810    x. cmul 7815   # cap 8537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-ltxr 7996  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538
This theorem is referenced by:  msqge0  8572  mulge0  8575  mulap0b  8611
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