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Theorem addext 8485
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 5833. 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 519 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  A  e.  CC )
2 simplr 520 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  B  e.  CC )
31, 2addcld 7897 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A  +  B
)  e.  CC )
4 simprl 521 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  C  e.  CC )
5 simprr 522 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  D  e.  CC )
64, 5addcld 7897 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  +  D
)  e.  CC )
74, 2addcld 7897 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( C  +  B
)  e.  CC )
8 apcotr 8482 . . 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 1220 . 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 8483 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  B  e.  CC )  ->  ( A #  C  <->  ( A  +  B ) #  ( C  +  B ) ) )
111, 4, 2, 10syl3anc 1220 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( A #  C  <->  ( A  +  B ) #  ( C  +  B ) ) )
12 apadd2 8484 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  C  e.  CC )  ->  ( B #  D  <->  ( C  +  B ) #  ( C  +  D ) ) )
132, 5, 4, 12syl3anc 1220 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( B #  D  <->  ( C  +  B ) #  ( C  +  D ) ) )
14 apsym 8481 . . . . 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 783 . 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 698    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   CCcc 7730    + caddc 7735   # cap 8456
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-ltxr 7917  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457
This theorem is referenced by:  mulext1  8487  abs00ap  10962  absext  10963
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