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Theorem mulext 8188
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 5699. 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 497 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  A  e.  CC )
2 simplr 498 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  B  e.  CC )
31, 2mulcld 7605 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  x.  B
)  e.  CC )
4 simprl 499 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  C  e.  CC )
5 simprr 500 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  D  e.  CC )
64, 5mulcld 7605 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  x.  D
)  e.  CC )
74, 2mulcld 7605 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  x.  B
)  e.  CC )
8 apcotr 8181 . . 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 1181 . 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 8186 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  (
( A  x.  B
) #  ( C  x.  B )  ->  A #  C ) )
111, 4, 2, 10syl3anc 1181 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  x.  B ) #  ( C  x.  B )  ->  A #  C ) )
12 mulext2 8187 . . . . 5  |-  ( ( D  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  x.  D
) #  ( C  x.  B )  ->  D #  B ) )
135, 2, 4, 12syl3anc 1181 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  x.  D ) #  ( C  x.  B )  ->  D #  B ) )
14 apsym 8180 . . . . 5  |-  ( ( D  e.  CC  /\  B  e.  CC )  ->  ( D #  B  <->  B #  D
) )
155, 2, 14syl2anc 404 . . . 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 738 . 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 667    e. wcel 1445   class class class wbr 3867  (class class class)co 5690   CCcc 7445    x. cmul 7452   # cap 8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-ltxr 7624  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156
This theorem is referenced by:  mulap0r  8189  lt2msq  8444  absext  10611
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