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Theorem suplocexprlemex 7723
Description: Lemma for suplocexpr 7726. The putative supremum is a positive real. (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 ) ) )
suplocexpr.b  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
Assertion
Ref Expression
suplocexprlemex  |-  ( ph  ->  B  e.  P. )
Distinct variable groups:    u, A, w, z    x, A, u, y, z    w, B    ph, u, w, z    ph, x, y
Allowed substitution hints:    B( x, y, z, u)

Proof of Theorem suplocexprlemex
Dummy variables  q  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.b . . 3  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
2 suplocexpr.m . . . . . 6  |-  ( ph  ->  E. x  x  e.  A )
3 suplocexpr.ub . . . . . 6  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
4 suplocexpr.loc . . . . . 6  |-  ( 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 ) ) )
52, 3, 4suplocexprlemss 7716 . . . . 5  |-  ( ph  ->  A  C_  P. )
61suplocexprlem2b 7715 . . . . 5  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
75, 6syl 14 . . . 4  |-  ( ph  ->  ( 2nd `  B
)  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
87opeq2d 3787 . . 3  |-  ( ph  -> 
<. U. ( 1st " A
) ,  ( 2nd `  B ) >.  =  <. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >. )
91, 8eqtr4id 2229 . 2  |-  ( ph  ->  B  =  <. U. ( 1st " A ) ,  ( 2nd `  B
) >. )
10 suplocexprlemell 7714 . . . . . . . . 9  |-  ( s  e.  U. ( 1st " A )  <->  E. t  e.  A  s  e.  ( 1st `  t ) )
1110biimpi 120 . . . . . . . 8  |-  ( s  e.  U. ( 1st " A )  ->  E. t  e.  A  s  e.  ( 1st `  t ) )
1211adantl 277 . . . . . . 7  |-  ( (
ph  /\  s  e.  U. ( 1st " A
) )  ->  E. t  e.  A  s  e.  ( 1st `  t ) )
135ad2antrr 488 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  U. ( 1st " A
) )  /\  (
t  e.  A  /\  s  e.  ( 1st `  t ) ) )  ->  A  C_  P. )
14 simprl 529 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  U. ( 1st " A
) )  /\  (
t  e.  A  /\  s  e.  ( 1st `  t ) ) )  ->  t  e.  A
)
1513, 14sseldd 3158 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  U. ( 1st " A
) )  /\  (
t  e.  A  /\  s  e.  ( 1st `  t ) ) )  ->  t  e.  P. )
16 prop 7476 . . . . . . . . 9  |-  ( t  e.  P.  ->  <. ( 1st `  t ) ,  ( 2nd `  t
) >.  e.  P. )
1715, 16syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  U. ( 1st " A
) )  /\  (
t  e.  A  /\  s  e.  ( 1st `  t ) ) )  ->  <. ( 1st `  t
) ,  ( 2nd `  t ) >.  e.  P. )
18 simprr 531 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  U. ( 1st " A
) )  /\  (
t  e.  A  /\  s  e.  ( 1st `  t ) ) )  ->  s  e.  ( 1st `  t ) )
19 elprnql 7482 . . . . . . . 8  |-  ( (
<. ( 1st `  t
) ,  ( 2nd `  t ) >.  e.  P.  /\  s  e.  ( 1st `  t ) )  -> 
s  e.  Q. )
2017, 18, 19syl2anc 411 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  U. ( 1st " A
) )  /\  (
t  e.  A  /\  s  e.  ( 1st `  t ) ) )  ->  s  e.  Q. )
2112, 20rexlimddv 2599 . . . . . 6  |-  ( (
ph  /\  s  e.  U. ( 1st " A
) )  ->  s  e.  Q. )
2221ex 115 . . . . 5  |-  ( ph  ->  ( s  e.  U. ( 1st " A )  ->  s  e.  Q. ) )
2322ssrdv 3163 . . . 4  |-  ( ph  ->  U. ( 1st " A
)  C_  Q. )
24 ssrab2 3242 . . . . 5  |-  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u }  C_  Q.
257, 24eqsstrdi 3209 . . . 4  |-  ( ph  ->  ( 2nd `  B
)  C_  Q. )
262, 3, 4suplocexprlemml 7717 . . . . 5  |-  ( ph  ->  E. q  e.  Q.  q  e.  U. ( 1st " A ) )
272, 3, 4, 1suplocexprlemmu 7719 . . . . 5  |-  ( ph  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
2826, 27jca 306 . . . 4  |-  ( ph  ->  ( E. q  e. 
Q.  q  e.  U. ( 1st " A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  B ) ) )
2923, 25, 28jca31 309 . . 3  |-  ( ph  ->  ( ( U. ( 1st " A )  C_  Q.  /\  ( 2nd `  B
)  C_  Q. )  /\  ( E. q  e. 
Q.  q  e.  U. ( 1st " A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  B ) ) ) )
302, 3, 4suplocexprlemrl 7718 . . . . 5  |-  ( ph  ->  A. q  e.  Q.  ( q  e.  U. ( 1st " A )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) ) )
312, 3, 4, 1suplocexprlemru 7720 . . . . 5  |-  ( ph  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  B )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )
3230, 31jca 306 . . . 4  |-  ( ph  ->  ( A. q  e. 
Q.  ( q  e. 
U. ( 1st " A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) ) )
332, 3, 4, 1suplocexprlemdisj 7721 . . . 4  |-  ( ph  ->  A. q  e.  Q.  -.  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )
342, 3, 4, 1suplocexprlemloc 7722 . . . 4  |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) )
3532, 33, 343jca 1177 . . 3  |-  ( ph  ->  ( ( A. q  e.  Q.  ( q  e. 
U. ( 1st " A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) )  /\  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) ) )
36 elinp 7475 . . 3  |-  ( <. U. ( 1st " A
) ,  ( 2nd `  B ) >.  e.  P.  <->  ( ( ( U. ( 1st " A )  C_  Q.  /\  ( 2nd `  B
)  C_  Q. )  /\  ( E. q  e. 
Q.  q  e.  U. ( 1st " A )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  B ) ) )  /\  (
( A. q  e. 
Q.  ( q  e. 
U. ( 1st " A
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) )  /\  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) ) ) )
3729, 35, 36sylanbrc 417 . 2  |-  ( ph  -> 
<. U. ( 1st " A
) ,  ( 2nd `  B ) >.  e.  P. )
389, 37eqeltrd 2254 1  |-  ( ph  ->  B  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3131   <.cop 3597   U.cuni 3811   |^|cint 3846   class class class wbr 4005   "cima 4631   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281    <Q cltq 7286   P.cnp 7292    <P cltp 7296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iltp 7471
This theorem is referenced by:  suplocexprlemub  7724  suplocexpr  7726
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