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Theorem suplocexprlemloc 7522
Description: Lemma for suplocexpr 7526. The putative supremum is located. (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 ) ) )
suplocexpr.b  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
Assertion
Ref Expression
suplocexprlemloc  |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) )
Distinct variable groups:    u, A, z, w    x, A, y, u, z    u, q, z, w    x, q, y, ph    ph, r, w, q    ph, z, x, y   
u, r
Allowed substitution hints:    ph( u)    A( r,
q)    B( x, y, z, w, u, r, q)

Proof of Theorem suplocexprlemloc
Dummy variables  s  t  v  l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  ->  q  <Q  r )
2 ltbtwnnqq 7216 . . . . 5  |-  ( q 
<Q  r  <->  E. v  e.  Q.  ( q  <Q  v  /\  v  <Q  r ) )
31, 2sylib 121 . . . 4  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  ->  E. v  e.  Q.  ( q  <Q 
v  /\  v  <Q  r ) )
4 simplll 522 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  ph )
5 simprl 520 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  Q.  /\  r  e. 
Q. ) )  -> 
q  e.  Q. )
65ad2antrr 479 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  q  e.  Q. )
7 simprl 520 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  v  e.  Q. )
84, 6, 7jca32 308 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
) )
9 simprrl 528 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  q  <Q  v
)
10 ltnqpri 7395 . . . . . . . . 9  |-  ( q 
<Q  v  ->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )
1110adantl 275 . . . . . . . 8  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )
12 breq2 3928 . . . . . . . . . 10  |-  ( y  =  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >.  ->  ( <. { l  |  l 
<Q  q } ,  {
u  |  q  <Q  u } >.  <P  y  <->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )
)
13 breq2 3928 . . . . . . . . . . . 12  |-  ( y  =  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >.  ->  (
z  <P  y  <->  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. ) )
1413ralbidv 2435 . . . . . . . . . . 11  |-  ( y  =  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >.  ->  ( A. z  e.  A  z  <P  y  <->  A. z  e.  A  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. ) )
1514orbi2d 779 . . . . . . . . . 10  |-  ( y  =  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >.  ->  (
( E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z  \/  A. z  e.  A  z  <P  y )  <->  ( E. z  e.  A  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z  \/ 
A. z  e.  A  z  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )
) )
1612, 15imbi12d 233 . . . . . . . . 9  |-  ( y  =  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >.  ->  (
( <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  y  ->  ( E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z  \/  A. z  e.  A  z  <P  y ) )  <-> 
( <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  <. { l  |  l  <Q  v } ,  { u  |  v  <Q  u } >.  ->  ( E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  z  \/  A. z  e.  A  z  <P 
<. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. ) ) ) )
17 breq1 3927 . . . . . . . . . . . 12  |-  ( x  =  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  ->  (
x  <P  y  <->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  y ) )
18 breq1 3927 . . . . . . . . . . . . . 14  |-  ( x  =  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  ->  (
x  <P  z  <->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  z ) )
1918rexbidv 2436 . . . . . . . . . . . . 13  |-  ( x  =  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  ->  ( E. z  e.  A  x  <P  z  <->  E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  z ) )
2019orbi1d 780 . . . . . . . . . . . 12  |-  ( x  =  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  ->  (
( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y )  <->  ( E. z  e.  A  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z  \/ 
A. z  e.  A  z  <P  y ) ) )
2117, 20imbi12d 233 . . . . . . . . . . 11  |-  ( x  =  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  ->  (
( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) )  <-> 
( <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  y  ->  ( E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z  \/  A. z  e.  A  z  <P  y ) ) ) )
2221ralbidv 2435 . . . . . . . . . 10  |-  ( x  =  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  ->  ( A. y  e.  P.  ( x  <P  y  -> 
( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) )  <->  A. y  e.  P.  ( <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  y  ->  ( E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z  \/  A. z  e.  A  z  <P  y ) ) ) )
23 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 ) ) )
2423ad2antrr 479 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  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
) ) )
25 simplrl 524 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  q  e.  Q. )
26 nqprlu 7348 . . . . . . . . . . 11  |-  ( q  e.  Q.  ->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  e.  P. )
2725, 26syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  e.  P. )
2822, 24, 27rspcdva 2789 . . . . . . . . 9  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  A. y  e.  P.  ( <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  y  ->  ( E. z  e.  A  <. { l  |  l 
<Q  q } ,  {
u  |  q  <Q  u } >.  <P  z  \/ 
A. z  e.  A  z  <P  y ) ) )
29 simplrr 525 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  v  e.  Q. )
30 nqprlu 7348 . . . . . . . . . 10  |-  ( v  e.  Q.  ->  <. { l  |  l  <Q  v } ,  { u  |  v  <Q  u } >.  e.  P. )
3129, 30syl 14 . . . . . . . . 9  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  <. { l  |  l  <Q  v } ,  { u  |  v  <Q  u } >.  e.  P. )
3216, 28, 31rspcdva 2789 . . . . . . . 8  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  ( <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  <. { l  |  l  <Q  v } ,  { u  |  v  <Q  u } >.  ->  ( E. z  e.  A  <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  z  \/  A. z  e.  A  z  <P 
<. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. ) ) )
3311, 32mpd 13 . . . . . . 7  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  ( E. z  e.  A  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z  \/ 
A. z  e.  A  z  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )
)
34 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)
3527ad2antrr 479 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  e.  P. )
36 suplocexpr.m . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  x  e.  A )
37 suplocexpr.ub . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
3836, 37, 23suplocexprlemss 7516 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  C_  P. )
3938ad4antr 485 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  A  C_  P. )
40 simplr 519 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  z  e.  A
)
4139, 40sseldd 3093 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  z  e.  P. )
42 ltdfpr 7307 . . . . . . . . . . . . 13  |-  ( (
<. { l  |  l 
<Q  q } ,  {
u  |  q  <Q  u } >.  e.  P.  /\  z  e.  P. )  ->  ( <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z  <->  E. w  e.  Q.  (
w  e.  ( 2nd `  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >. )  /\  w  e.  ( 1st `  z ) ) ) )
4335, 41, 42syl2anc 408 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  ( <. { l  |  l  <Q  q } ,  { u  |  q  <Q  u } >.  <P  z  <->  E. w  e.  Q.  ( w  e.  ( 2nd `  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >. )  /\  w  e.  ( 1st `  z
) ) ) )
4434, 43mpbid 146 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  E. w  e.  Q.  ( w  e.  ( 2nd `  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >. )  /\  w  e.  ( 1st `  z ) ) )
45 vex 2684 . . . . . . . . . . . . . 14  |-  w  e. 
_V
46 breq2 3928 . . . . . . . . . . . . . 14  |-  ( u  =  w  ->  (
q  <Q  u  <->  q  <Q  w ) )
47 ltnqex 7350 . . . . . . . . . . . . . . 15  |-  { l  |  l  <Q  q }  e.  _V
48 gtnqex 7351 . . . . . . . . . . . . . . 15  |-  { u  |  q  <Q  u }  e.  _V
4947, 48op2nd 6038 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >. )  =  { u  |  q 
<Q  u }
5045, 46, 49elab2 2827 . . . . . . . . . . . . 13  |-  ( w  e.  ( 2nd `  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >. )  <->  q  <Q  w )
5150anbi1i 453 . . . . . . . . . . . 12  |-  ( ( w  e.  ( 2nd `  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >. )  /\  w  e.  ( 1st `  z ) )  <-> 
( q  <Q  w  /\  w  e.  ( 1st `  z ) ) )
5251rexbii 2440 . . . . . . . . . . 11  |-  ( E. w  e.  Q.  (
w  e.  ( 2nd `  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >. )  /\  w  e.  ( 1st `  z ) )  <->  E. w  e.  Q.  ( q  <Q  w  /\  w  e.  ( 1st `  z ) ) )
5344, 52sylib 121 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  E. w  e.  Q.  ( q  <Q  w  /\  w  e.  ( 1st `  z ) ) )
54 simpllr 523 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  z  e.  A
)
55 simprrl 528 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  q  <Q  w
)
5641adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  z  e.  P. )
57 prop 7276 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  P.  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  e.  P. )
5856, 57syl 14 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  P. )
59 simprrr 529 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  w  e.  ( 1st `  z ) )
60 prcdnql 7285 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  P.  /\  w  e.  ( 1st `  z ) )  -> 
( q  <Q  w  ->  q  e.  ( 1st `  z ) ) )
6158, 59, 60syl2anc 408 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  ( q  <Q  w  ->  q  e.  ( 1st `  z ) ) )
6255, 61mpd 13 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  q  e.  ( 1st `  z ) )
6354, 62jca 304 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  ( z  e.  A  /\  q  e.  ( 1st `  z
) ) )
646319.8ad 1570 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  E. z ( z  e.  A  /\  q  e.  ( 1st `  z
) ) )
65 df-rex 2420 . . . . . . . . . . . 12  |-  ( E. z  e.  A  q  e.  ( 1st `  z
)  <->  E. z ( z  e.  A  /\  q  e.  ( 1st `  z
) ) )
6664, 65sylibr 133 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  E. z  e.  A  q  e.  ( 1st `  z ) )
67 suplocexprlemell 7514 . . . . . . . . . . 11  |-  ( q  e.  U. ( 1st " A )  <->  E. z  e.  A  q  e.  ( 1st `  z ) )
6866, 67sylibr 133 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q  q } ,  { u  |  q 
<Q  u } >.  <P  z
)  /\  ( w  e.  Q.  /\  ( q 
<Q  w  /\  w  e.  ( 1st `  z
) ) ) )  ->  q  e.  U. ( 1st " A ) )
6953, 68rexlimddv 2552 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  z  e.  A )  /\  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z )  ->  q  e.  U. ( 1st " A ) )
7069rexlimdva2 2550 . . . . . . . 8  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  ( E. z  e.  A  <. { l  |  l  <Q 
q } ,  {
u  |  q  <Q  u } >.  <P  z  -> 
q  e.  U. ( 1st " A ) ) )
71 fo2nd 6049 . . . . . . . . . . . . . . 15  |-  2nd : _V -onto-> _V
72 fofun 5341 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
7371, 72ax-mp 5 . . . . . . . . . . . . . 14  |-  Fun  2nd
74 fvelima 5466 . . . . . . . . . . . . . 14  |-  ( ( Fun  2nd  /\  s  e.  ( 2nd " A
) )  ->  E. t  e.  A  ( 2nd `  t )  =  s )
7573, 74mpan 420 . . . . . . . . . . . . 13  |-  ( s  e.  ( 2nd " A
)  ->  E. t  e.  A  ( 2nd `  t )  =  s )
7675adantl 275 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  A. z  e.  A  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. )  /\  s  e.  ( 2nd " A
) )  ->  E. t  e.  A  ( 2nd `  t )  =  s )
77 breq1 3927 . . . . . . . . . . . . . . 15  |-  ( z  =  t  ->  (
z  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >.  <->  t  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. ) )
78 simpllr 523 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  A. z  e.  A  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. )
79 simprl 520 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  t  e.  A )
8077, 78, 79rspcdva 2789 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  t  <P 
<. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. )
8129ad3antrrr 483 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  v  e.  Q. )
8238ad5antr 487 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  A  C_ 
P. )
8382, 79sseldd 3093 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  t  e.  P. )
84 nqpru 7353 . . . . . . . . . . . . . . 15  |-  ( ( v  e.  Q.  /\  t  e.  P. )  ->  ( v  e.  ( 2nd `  t )  <-> 
t  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )
)
8581, 83, 84syl2anc 408 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  (
v  e.  ( 2nd `  t )  <->  t  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. ) )
8680, 85mpbird 166 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  v  e.  ( 2nd `  t
) )
87 simprr 521 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  ( 2nd `  t )  =  s )
8886, 87eleqtrd 2216 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( q  e.  Q.  /\  v  e. 
Q. ) )  /\  q  <Q  v )  /\  A. z  e.  A  z 
<P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  /\  s  e.  ( 2nd " A ) )  /\  ( t  e.  A  /\  ( 2nd `  t )  =  s ) )  ->  v  e.  s )
8976, 88rexlimddv 2552 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  A. z  e.  A  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. )  /\  s  e.  ( 2nd " A
) )  ->  v  e.  s )
9089ralrimiva 2503 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  A. z  e.  A  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. )  ->  A. s  e.  ( 2nd " A
) v  e.  s )
91 vex 2684 . . . . . . . . . . 11  |-  v  e. 
_V
9291elint2 3773 . . . . . . . . . 10  |-  ( v  e.  |^| ( 2nd " A
)  <->  A. s  e.  ( 2nd " A ) v  e.  s )
9390, 92sylibr 133 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  /\  A. z  e.  A  z  <P  <. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >. )  ->  v  e.  |^| ( 2nd " A
) )
9493ex 114 . . . . . . . 8  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  ( A. z  e.  A  z  <P 
<. { l  |  l 
<Q  v } ,  {
u  |  v  <Q  u } >.  ->  v  e. 
|^| ( 2nd " A
) ) )
9570, 94orim12d 775 . . . . . . 7  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  ( ( E. z  e.  A  <. { l  |  l 
<Q  q } ,  {
u  |  q  <Q  u } >.  <P  z  \/ 
A. z  e.  A  z  <P  <. { l  |  l  <Q  v } ,  { u  |  v 
<Q  u } >. )  ->  ( q  e.  U. ( 1st " A )  \/  v  e.  |^| ( 2nd " A ) ) ) )
9633, 95mpd 13 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  v  e.  Q. )
)  /\  q  <Q  v )  ->  ( q  e.  U. ( 1st " A
)  \/  v  e. 
|^| ( 2nd " A
) ) )
978, 9, 96syl2anc 408 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  ( q  e. 
U. ( 1st " A
)  \/  v  e. 
|^| ( 2nd " A
) ) )
98 breq2 3928 . . . . . . . . . 10  |-  ( u  =  r  ->  (
w  <Q  u  <->  w  <Q  r ) )
9998rexbidv 2436 . . . . . . . . 9  |-  ( u  =  r  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  r )
)
100 simprr 521 . . . . . . . . . 10  |-  ( (
ph  /\  ( q  e.  Q.  /\  r  e. 
Q. ) )  -> 
r  e.  Q. )
101100ad3antrrr 483 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  r  e.  Q. )
102 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  v  e.  |^| ( 2nd " A
) )
103 simprrr 529 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  v  <Q  r
)
104103adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  v  <Q  r )
105 breq1 3927 . . . . . . . . . . 11  |-  ( w  =  v  ->  (
w  <Q  r  <->  v  <Q  r ) )
106105rspcev 2784 . . . . . . . . . 10  |-  ( ( v  e.  |^| ( 2nd " A )  /\  v  <Q  r )  ->  E. w  e.  |^| ( 2nd " A ) w 
<Q  r )
107102, 104, 106syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  E. w  e.  |^| ( 2nd " A
) w  <Q  r
)
10899, 101, 107elrabd 2837 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  r  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
109 suplocexpr.b . . . . . . . . . . . 12  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
110109suplocexprlem2b 7515 . . . . . . . . . . 11  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
11138, 110syl 14 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  B
)  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
112111eleq2d 2207 . . . . . . . . 9  |-  ( ph  ->  ( r  e.  ( 2nd `  B )  <-> 
r  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
113112ad4antr 485 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  ( r  e.  ( 2nd `  B
)  <->  r  e.  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
114108, 113mpbird 166 . . . . . . 7  |-  ( ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  /\  v  e.  |^| ( 2nd " A ) )  ->  r  e.  ( 2nd `  B ) )
115114ex 114 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  ( v  e. 
|^| ( 2nd " A
)  ->  r  e.  ( 2nd `  B ) ) )
116115orim2d 777 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  ( ( q  e.  U. ( 1st " A )  \/  v  e.  |^| ( 2nd " A
) )  ->  (
q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) )
11797, 116mpd 13 . . . 4  |-  ( ( ( ( ph  /\  ( q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  /\  ( v  e.  Q.  /\  (
q  <Q  v  /\  v  <Q  r ) ) )  ->  ( q  e. 
U. ( 1st " A
)  \/  r  e.  ( 2nd `  B
) ) )
1183, 117rexlimddv 2552 . . 3  |-  ( ( ( ph  /\  (
q  e.  Q.  /\  r  e.  Q. )
)  /\  q  <Q  r )  ->  ( q  e.  U. ( 1st " A
)  \/  r  e.  ( 2nd `  B
) ) )
119118ex 114 . 2  |-  ( (
ph  /\  ( q  e.  Q.  /\  r  e. 
Q. ) )  -> 
( q  <Q  r  ->  ( q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) )
120119ralrimivva 2512 1  |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   _Vcvv 2681    C_ wss 3066   <.cop 3525   U.cuni 3731   |^|cint 3766   class class class wbr 3924   "cima 4537   Fun wfun 5112   -onto->wfo 5116   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081    <Q cltq 7086   P.cnp 7092    <P cltp 7096
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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271
This theorem is referenced by:  suplocexprlemex  7523
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