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Theorem suplocexpr 7540
Description: An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m  |-  ( ph  ->  E. x  x  e.  A )
suplocexpr.ub  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
suplocexpr.loc  |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  (
x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )
Assertion
Ref Expression
suplocexpr  |-  ( ph  ->  E. x  e.  P.  ( A. y  e.  A  -.  x  <P  y  /\  A. y  e.  P.  (
y  <P  x  ->  E. z  e.  A  y  <P  z ) ) )
Distinct variable groups:    y, A, z, x    ph, y, z, x

Proof of Theorem suplocexpr
Dummy variables  a  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.m . . 3  |-  ( ph  ->  E. x  x  e.  A )
2 suplocexpr.ub . . 3  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
3 suplocexpr.loc . . 3  |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  (
x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )
4 breq1 3932 . . . . . 6  |-  ( a  =  w  ->  (
a  <Q  u  <->  w  <Q  u ) )
54cbvrexv 2655 . . . . 5  |-  ( E. a  e.  |^| ( 2nd " A ) a 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  u )
65rabbii 2672 . . . 4  |-  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u }  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u }
76opeq2i 3709 . . 3  |-  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >.  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
81, 2, 3, 7suplocexprlemex 7537 . 2  |-  ( ph  -> 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  e.  P. )
91, 2, 3, 7suplocexprlemub 7538 . 2  |-  ( ph  ->  A. y  e.  A  -.  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  <P  y )
101, 2, 3, 7suplocexprlemlub 7539 . . 3  |-  ( ph  ->  ( y  <P  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >.  ->  E. z  e.  A  y  <P  z ) )
1110ralrimivw 2506 . 2  |-  ( ph  ->  A. y  e.  P.  ( y  <P  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >.  ->  E. z  e.  A  y  <P  z ) )
12 breq1 3932 . . . . . 6  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( x  <P  y  <->  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  <P  y ) )
1312notbid 656 . . . . 5  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( -.  x  <P  y  <->  -.  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >.  <P  y ) )
1413ralbidv 2437 . . . 4  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( A. y  e.  A  -.  x  <P  y  <->  A. y  e.  A  -.  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >.  <P  y ) )
15 breq2 3933 . . . . . 6  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( y  <P  x  <->  y  <P  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >. ) )
1615imbi1d 230 . . . . 5  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( (
y  <P  x  ->  E. z  e.  A  y  <P  z )  <->  ( y  <P  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  E. z  e.  A  y  <P  z ) ) )
1716ralbidv 2437 . . . 4  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( A. y  e.  P.  (
y  <P  x  ->  E. z  e.  A  y  <P  z )  <->  A. y  e.  P.  ( y  <P  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A ) a  <Q  u } >.  ->  E. z  e.  A  y  <P  z ) ) )
1814, 17anbi12d 464 . . 3  |-  ( x  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  ( ( A. y  e.  A  -.  x  <P  y  /\  A. y  e.  P.  (
y  <P  x  ->  E. z  e.  A  y  <P  z ) )  <->  ( A. y  e.  A  -.  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  <P  y  /\  A. y  e.  P.  (
y  <P  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  E. z  e.  A  y  <P  z ) ) ) )
1918rspcev 2789 . 2  |-  ( (
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  e.  P.  /\  ( A. y  e.  A  -.  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  <P  y  /\  A. y  e.  P.  (
y  <P  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  ->  E. z  e.  A  y  <P  z ) ) )  ->  E. x  e.  P.  ( A. y  e.  A  -.  x  <P  y  /\  A. y  e.  P.  (
y  <P  x  ->  E. z  e.  A  y  <P  z ) ) )
208, 9, 11, 19syl12anc 1214 1  |-  ( ph  ->  E. x  e.  P.  ( A. y  e.  A  -.  x  <P  y  /\  A. y  e.  P.  (
y  <P  x  ->  E. z  e.  A  y  <P  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   U.cuni 3736   |^|cint 3771   class class class wbr 3929   "cima 4542   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095    <Q cltq 7100   P.cnp 7106    <P cltp 7110
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-in1 603  ax-in2 604  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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iltp 7285
This theorem is referenced by:  suplocsrlempr  7622
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