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Theorem preqlu 6627
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 6626 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 2968 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 1st2nd2 5828 . . . 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 2968 . . . 4  |-  ( B  e.  P.  ->  B  e.  ( ~P Q.  X.  ~P Q. ) )
6 1st2nd2 5828 . . . 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 2071 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
9 xp1st 5819 . . . . 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 5820 . . . . 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 4002 . . . 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 397 . . 3  |-  ( A  e.  P.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
1514adantr 265 . 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 181 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   ~Pcpw 3386   <.cop 3405    X. cxp 4370   ` cfv 4929   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435   P.cnp 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fv 4937  df-1st 5794  df-2nd 5795  df-inp 6621
This theorem is referenced by:  genpassg  6681  addnqpr  6716  mulnqpr  6732  distrprg  6743  1idpr  6747  ltexpri  6768  addcanprg  6771  recexprlemex  6792  aptipr  6796
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