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Theorem suplocexprlemrl 7532
Description: Lemma for suplocexpr 7540. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-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
suplocexprlemrl  |-  ( ph  ->  A. q  e.  Q.  ( q  e.  U. ( 1st " A )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) ) )
Distinct variable groups:    A, r    x, A, y    ph, q, r    ph, x, y
Allowed substitution hints:    ph( z)    A( z,
q)

Proof of Theorem suplocexprlemrl
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 suplocexprlemell 7528 . . . . . . 7  |-  ( q  e.  U. ( 1st " A )  <->  E. s  e.  A  q  e.  ( 1st `  s ) )
21biimpi 119 . . . . . 6  |-  ( q  e.  U. ( 1st " A )  ->  E. s  e.  A  q  e.  ( 1st `  s ) )
32adantl 275 . . . . 5  |-  ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A
) )  ->  E. s  e.  A  q  e.  ( 1st `  s ) )
4 suplocexpr.m . . . . . . . . . . 11  |-  ( ph  ->  E. x  x  e.  A )
5 suplocexpr.ub . . . . . . . . . . 11  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
6 suplocexpr.loc . . . . . . . . . . 11  |-  ( 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 ) ) )
74, 5, 6suplocexprlemss 7530 . . . . . . . . . 10  |-  ( ph  ->  A  C_  P. )
87ad3antrrr 483 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  ->  A  C_  P. )
9 simprl 520 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  -> 
s  e.  A )
108, 9sseldd 3098 . . . . . . . 8  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  -> 
s  e.  P. )
11 prop 7290 . . . . . . . 8  |-  ( s  e.  P.  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
1210, 11syl 14 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  ->  <. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P. )
13 simprr 521 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  -> 
q  e.  ( 1st `  s ) )
14 prnmaxl 7303 . . . . . . 7  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  q  e.  ( 1st `  s ) )  ->  E. r  e.  ( 1st `  s ) q 
<Q  r )
1512, 13, 14syl2anc 408 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  ->  E. r  e.  ( 1st `  s ) q 
<Q  r )
16 ltrelnq 7180 . . . . . . . . 9  |-  <Q  C_  ( Q.  X.  Q. )
1716brel 4591 . . . . . . . 8  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
1817simprd 113 . . . . . . 7  |-  ( q 
<Q  r  ->  r  e. 
Q. )
1918ad2antll 482 . . . . . 6  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  r  e.  Q. )
20 simprr 521 . . . . . . 7  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  q  <Q  r )
21 simplrl 524 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  s  e.  A )
22 simprl 520 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  r  e.  ( 1st `  s
) )
23 rspe 2481 . . . . . . . . 9  |-  ( ( s  e.  A  /\  r  e.  ( 1st `  s ) )  ->  E. s  e.  A  r  e.  ( 1st `  s ) )
2421, 22, 23syl2anc 408 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  E. s  e.  A  r  e.  ( 1st `  s ) )
25 suplocexprlemell 7528 . . . . . . . 8  |-  ( r  e.  U. ( 1st " A )  <->  E. s  e.  A  r  e.  ( 1st `  s ) )
2624, 25sylibr 133 . . . . . . 7  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  r  e.  U. ( 1st " A
) )
2720, 26jca 304 . . . . . 6  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  /\  ( r  e.  ( 1st `  s )  /\  q  <Q  r
) )  ->  (
q  <Q  r  /\  r  e.  U. ( 1st " A
) ) )
2815, 19, 27reximssdv 2536 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A ) )  /\  ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )
293, 28rexlimddv 2554 . . . 4  |-  ( ( ( ph  /\  q  e.  Q. )  /\  q  e.  U. ( 1st " A
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  U. ( 1st " A
) ) )
3029ex 114 . . 3  |-  ( (
ph  /\  q  e.  Q. )  ->  ( q  e.  U. ( 1st " A )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  U. ( 1st " A
) ) ) )
31 simprr 521 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  r  e.  U. ( 1st " A
) ) )  -> 
r  e.  U. ( 1st " A ) )
3231, 25sylib 121 . . . . . 6  |-  ( ( ( ph  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  r  e.  U. ( 1st " A
) ) )  ->  E. s  e.  A  r  e.  ( 1st `  s ) )
33 simprl 520 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
s  e.  A )
34 simplrl 524 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
q  <Q  r )
357ad3antrrr 483 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  ->  A  C_  P. )
3635, 33sseldd 3098 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
s  e.  P. )
3736, 11syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  ->  <. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P. )
38 simprr 521 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
r  e.  ( 1st `  s ) )
39 prcdnql 7299 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  r  e.  ( 1st `  s ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  s ) ) )
4037, 38, 39syl2anc 408 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
( q  <Q  r  ->  q  e.  ( 1st `  s ) ) )
4134, 40mpd 13 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
q  e.  ( 1st `  s ) )
42 19.8a 1569 . . . . . . . . 9  |-  ( ( s  e.  A  /\  q  e.  ( 1st `  s ) )  ->  E. s ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )
4333, 41, 42syl2anc 408 . . . . . . . 8  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  ->  E. s ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )
44 df-rex 2422 . . . . . . . 8  |-  ( E. s  e.  A  q  e.  ( 1st `  s
)  <->  E. s ( s  e.  A  /\  q  e.  ( 1st `  s
) ) )
4543, 44sylibr 133 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  ->  E. s  e.  A  q  e.  ( 1st `  s ) )
4645, 1sylibr 133 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  ( s  e.  A  /\  r  e.  ( 1st `  s
) ) )  -> 
q  e.  U. ( 1st " A ) )
4732, 46rexlimddv 2554 . . . . 5  |-  ( ( ( ph  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  r  e.  U. ( 1st " A
) ) )  -> 
q  e.  U. ( 1st " A ) )
4847ex 114 . . . 4  |-  ( (
ph  /\  q  e.  Q. )  ->  ( ( q  <Q  r  /\  r  e.  U. ( 1st " A ) )  ->  q  e.  U. ( 1st " A ) ) )
4948rexlimdvw 2553 . . 3  |-  ( (
ph  /\  q  e.  Q. )  ->  ( E. r  e.  Q.  (
q  <Q  r  /\  r  e.  U. ( 1st " A
) )  ->  q  e.  U. ( 1st " A
) ) )
5030, 49impbid 128 . 2  |-  ( (
ph  /\  q  e.  Q. )  ->  ( q  e.  U. ( 1st " A )  <->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  U. ( 1st " A
) ) ) )
5150ralrimiva 2505 1  |-  ( ph  ->  A. q  e.  Q.  ( q  e.  U. ( 1st " A )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417    C_ wss 3071   <.cop 3530   U.cuni 3736   class class class wbr 3929   "cima 4542   ` cfv 5123   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-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  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-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-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-id 4215  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-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-ltnqqs 7168  df-inp 7281  df-iltp 7285
This theorem is referenced by:  suplocexprlemex  7537
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