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Theorem addext 8529
Description: Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5862. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
Assertion
Ref Expression
addext  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B ) #  ( C  +  D )  ->  ( A #  C  \/  B #  D ) ) )

Proof of Theorem addext
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, 2addcld 7939 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  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, 5addcld 7939 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  +  D
)  e.  CC )
74, 2addcld 7939 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  +  B
)  e.  CC )
8 apcotr 8526 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  ( C  +  D
)  e.  CC  /\  ( C  +  B
)  e.  CC )  ->  ( ( A  +  B ) #  ( C  +  D )  ->  ( ( A  +  B ) #  ( C  +  B )  \/  ( C  +  D ) #  ( C  +  B ) ) ) )
93, 6, 7, 8syl3anc 1233 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B ) #  ( C  +  D )  ->  (
( A  +  B
) #  ( C  +  B )  \/  ( C  +  D ) #  ( C  +  B
) ) ) )
10 apadd1 8527 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  ( A #  C  <->  ( A  +  B ) #  ( C  +  B ) ) )
111, 4, 2, 10syl3anc 1233 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A #  C  <->  ( A  +  B ) #  ( C  +  B ) ) )
12 apadd2 8528 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  C  e.  CC )  ->  ( B #  D  <->  ( C  +  B ) #  ( C  +  D ) ) )
132, 5, 4, 12syl3anc 1233 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( B #  D  <->  ( C  +  B ) #  ( C  +  D ) ) )
14 apsym 8525 . . . . 5  |-  ( ( ( C  +  B
)  e.  CC  /\  ( C  +  D
)  e.  CC )  ->  ( ( C  +  B ) #  ( C  +  D )  <-> 
( C  +  D
) #  ( C  +  B ) ) )
157, 6, 14syl2anc 409 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( C  +  B ) #  ( C  +  D )  <->  ( C  +  D ) #  ( C  +  B ) ) )
1613, 15bitrd 187 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( B #  D  <->  ( C  +  D ) #  ( C  +  B ) ) )
1711, 16orbi12d 788 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A #  C  \/  B #  D )  <->  ( ( A  +  B
) #  ( C  +  B )  \/  ( C  +  D ) #  ( C  +  B
) ) ) )
189, 17sylibrd 168 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B ) #  ( C  +  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 3989  (class class class)co 5853   CCcc 7772    + caddc 7777   # cap 8500
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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501
This theorem is referenced by:  mulext1  8531  abs00ap  11026  absext  11027
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