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Theorem suplocexpr 7674
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 3990 . . . . . 6  |-  ( a  =  w  ->  (
a  <Q  u  <->  w  <Q  u ) )
54cbvrexv 2697 . . . . 5  |-  ( E. a  e.  |^| ( 2nd " A ) a 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  u )
65rabbii 2716 . . . 4  |-  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u }  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u }
76opeq2i 3767 . . 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 7671 . 2  |-  ( ph  -> 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. a  e.  |^| ( 2nd " A
) a  <Q  u } >.  e.  P. )
91, 2, 3, 7suplocexprlemub 7672 . 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 7673 . . 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 2544 . 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 3990 . . . . . 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 662 . . . . 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 2470 . . . 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 3991 . . . . . 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 2470 . . . 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 470 . . 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 2834 . 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 1231 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 703    = wceq 1348   E.wex 1485    e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3584   U.cuni 3794   |^|cint 3829   class class class wbr 3987   "cima 4612   1stc1st 6114   2ndc2nd 6115   Q.cnq 7229    <Q cltq 7234   P.cnp 7240    <P cltp 7244
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-iltp 7419
This theorem is referenced by:  suplocsrlempr  7756
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