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Theorem mulext 8585
Description: Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5897. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
Assertion
Ref Expression
mulext  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  B ) #  ( C  x.  D )  ->  ( A #  C  \/  B #  D ) ) )

Proof of Theorem mulext
StepHypRef Expression
1 simpll 527 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  A  e.  CC )
2 simplr 528 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  B  e.  CC )
31, 2mulcld 7992 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  x.  B
)  e.  CC )
4 simprl 529 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  C  e.  CC )
5 simprr 531 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  D  e.  CC )
64, 5mulcld 7992 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  x.  D
)  e.  CC )
74, 2mulcld 7992 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  x.  B
)  e.  CC )
8 apcotr 8578 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( C  x.  D
)  e.  CC  /\  ( C  x.  B
)  e.  CC )  ->  ( ( A  x.  B ) #  ( C  x.  D )  ->  ( ( A  x.  B ) #  ( C  x.  B )  \/  ( C  x.  D ) #  ( C  x.  B ) ) ) )
93, 6, 7, 8syl3anc 1248 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  B ) #  ( C  x.  D )  ->  (
( A  x.  B
) #  ( C  x.  B )  \/  ( C  x.  D ) #  ( C  x.  B
) ) ) )
10 mulext1 8583 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) #  ( C  x.  B )  ->  A #  C ) )
111, 4, 2, 10syl3anc 1248 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  B ) #  ( C  x.  B )  ->  A #  C ) )
12 mulext2 8584 . . . . 5  |-  ( ( D  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  x.  D
) #  ( C  x.  B )  ->  D #  B ) )
135, 2, 4, 12syl3anc 1248 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  x.  D ) #  ( C  x.  B )  ->  D #  B ) )
14 apsym 8577 . . . . 5  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( D #  B  <->  B #  D
) )
155, 2, 14syl2anc 411 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( D #  B  <->  B #  D
) )
1613, 15sylibd 149 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  x.  D ) #  ( C  x.  B )  ->  B #  D ) )
1711, 16orim12d 787 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( ( A  x.  B ) #  ( C  x.  B )  \/  ( C  x.  D ) #  ( C  x.  B ) )  -> 
( A #  C  \/  B #  D ) ) )
189, 17syld 45 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  B ) #  ( C  x.  D )  ->  ( A #  C  \/  B #  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7823    x. cmul 7830   # cap 8552
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553
This theorem is referenced by:  mulap0r  8586  lt2msq  8857  apexp1  10712  absext  11086
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