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Theorem preqlu 7248
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 7247 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 3063 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 1st2nd2 6041 . . . 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 3063 . . . 4  |-  ( B  e.  P.  ->  B  e.  ( ~P Q.  X.  ~P Q. ) )
6 1st2nd2 6041 . . . 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 2133 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
9 xp1st 6031 . . . . 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 6032 . . . . 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 4130 . . . 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 408 . . 3  |-  ( A  e.  P.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
1514adantr 274 . 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 187 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   ~Pcpw 3480   <.cop 3500    X. cxp 4507   ` cfv 5093   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   P.cnp 7067
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006  df-2nd 6007  df-inp 7242
This theorem is referenced by:  genpassg  7302  addnqpr  7337  mulnqpr  7353  distrprg  7364  1idpr  7368  ltexpri  7389  addcanprg  7392  recexprlemex  7413  aptipr  7417
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