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Theorem preqlu 7031
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 7030 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 3021 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 1st2nd2 5945 . . . 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 3021 . . . 4  |-  ( B  e.  P.  ->  B  e.  ( ~P Q.  X.  ~P Q. ) )
6 1st2nd2 5945 . . . 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 2103 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
9 xp1st 5936 . . . . 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 5937 . . . . 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 4065 . . . 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 403 . . 3  |-  ( A  e.  P.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
1514adantr 270 . 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 186 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   ~Pcpw 3429   <.cop 3449    X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839   P.cnp 6850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-2nd 5912  df-inp 7025
This theorem is referenced by:  genpassg  7085  addnqpr  7120  mulnqpr  7136  distrprg  7147  1idpr  7151  ltexpri  7172  addcanprg  7175  recexprlemex  7196  aptipr  7200
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