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Theorem mulext 8526
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 5860. 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 524 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  A  e.  CC )
2 simplr 525 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  B  e.  CC )
31, 2mulcld 7933 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  x.  B
)  e.  CC )
4 simprl 526 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  C  e.  CC )
5 simprr 527 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  D  e.  CC )
64, 5mulcld 7933 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  x.  D
)  e.  CC )
74, 2mulcld 7933 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  x.  B
)  e.  CC )
8 apcotr 8519 . . 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 1233 . 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 8524 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) #  ( C  x.  B )  ->  A #  C ) )
111, 4, 2, 10syl3anc 1233 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  B ) #  ( C  x.  B )  ->  A #  C ) )
12 mulext2 8525 . . . . 5  |-  ( ( D  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  x.  D
) #  ( C  x.  B )  ->  D #  B ) )
135, 2, 4, 12syl3anc 1233 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  x.  D ) #  ( C  x.  B )  ->  D #  B ) )
14 apsym 8518 . . . . 5  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( D #  B  <->  B #  D
) )
155, 2, 14syl2anc 409 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( D #  B  <->  B #  D
) )
1613, 15sylibd 148 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  x.  D ) #  ( C  x.  B )  ->  B #  D ) )
1711, 16orim12d 781 . 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 103    <-> wb 104    \/ wo 703    e. wcel 2141   class class class wbr 3987  (class class class)co 5851   CCcc 7765    x. cmul 7772   # cap 8493
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-ltxr 7952  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494
This theorem is referenced by:  mulap0r  8527  lt2msq  8795  apexp1  10645  absext  11020
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