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Theorem preqlu 6570
Description: Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
preqlu  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <-> 
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )

Proof of Theorem preqlu
StepHypRef Expression
1 npsspw 6569 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 2941 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 1st2nd2 5801 . . . 4  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
42, 3syl 14 . . 3  |-  ( A  e.  P.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
51sseli 2941 . . . 4  |-  ( B  e.  P.  ->  B  e.  ( ~P Q.  X.  ~P Q. ) )
6 1st2nd2 5801 . . . 4  |-  ( B  e.  ( ~P Q.  X.  ~P Q. )  ->  B  =  <. ( 1st `  B ) ,  ( 2nd `  B )
>. )
75, 6syl 14 . . 3  |-  ( B  e.  P.  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
84, 7eqeqan12d 2055 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
9 xp1st 5792 . . . . 5  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 1st `  A
)  e.  ~P Q. )
102, 9syl 14 . . . 4  |-  ( A  e.  P.  ->  ( 1st `  A )  e. 
~P Q. )
11 xp2nd 5793 . . . . 5  |-  ( A  e.  ( ~P Q.  X.  ~P Q. )  -> 
( 2nd `  A
)  e.  ~P Q. )
122, 11syl 14 . . . 4  |-  ( A  e.  P.  ->  ( 2nd `  A )  e. 
~P Q. )
13 opthg 3975 . . . 4  |-  ( ( ( 1st `  A
)  e.  ~P Q.  /\  ( 2nd `  A
)  e.  ~P Q. )  ->  ( <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
1410, 12, 13syl2anc 391 . . 3  |-  ( A  e.  P.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
1514adantr 261 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
168, 15bitrd 177 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <-> 
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   ~Pcpw 3359   <.cop 3378    X. cxp 4343   ` cfv 4902   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   P.cnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768  df-inp 6564
This theorem is referenced by:  genpassg  6624  addnqpr  6659  mulnqpr  6675  distrprg  6686  1idpr  6690  ltexpri  6711  addcanprg  6714  recexprlemex  6735  aptipr  6739
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