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Theorem preqlu 7273
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 7272 . . . . 5  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
21sseli 3088 . . . 4  |-  ( A  e.  P.  ->  A  e.  ( ~P Q.  X.  ~P Q. ) )
3 1st2nd2 6066 . . . 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 3088 . . . 4  |-  ( B  e.  P.  ->  B  e.  ( ~P Q.  X.  ~P Q. ) )
6 1st2nd2 6066 . . . 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 2153 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  B ) ,  ( 2nd `  B
) >. ) )
9 xp1st 6056 . . . . 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 6057 . . . . 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 4155 . . . 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 1331    e. wcel 1480   ~Pcpw 3505   <.cop 3525    X. cxp 4532   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   P.cnp 7092
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-1st 6031  df-2nd 6032  df-inp 7267
This theorem is referenced by:  genpassg  7327  addnqpr  7362  mulnqpr  7378  distrprg  7389  1idpr  7393  ltexpri  7414  addcanprg  7417  recexprlemex  7438  aptipr  7442
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