ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  suplocexprlemmu Unicode version

Theorem suplocexprlemmu 7640
Description: Lemma for suplocexpr 7647. The upper cut of the putative supremum is inhabited. (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
suplocexprlemmu  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
Distinct variable groups:    A, s, u, w    x, A, y, s, u    B, s    ph, s, u, x, y
Allowed substitution hints:    ph( z, w)    A( z)    B( x, y, z, w, u)

Proof of Theorem suplocexprlemmu
Dummy variables  j  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.ub . . . 4  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
2 prop 7397 . . . . . . 7  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
3 prmu 7400 . . . . . . 7  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
42, 3syl 14 . . . . . 6  |-  ( x  e.  P.  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
54ad2antrl 482 . . . . 5  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
6 fo2nd 6108 . . . . . . . . . . . . 13  |-  2nd : _V -onto-> _V
7 fofun 5395 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
9 fvelima 5522 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  t  e.  ( 2nd " A
) )  ->  E. u  e.  A  ( 2nd `  u )  =  t )
108, 9mpan 421 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd " A
)  ->  E. u  e.  A  ( 2nd `  u )  =  t )
1110adantl 275 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  /\  t  e.  ( 2nd " A
) )  ->  E. u  e.  A  ( 2nd `  u )  =  t )
12 suplocexpr.m . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  x  e.  A )
13 suplocexpr.loc . . . . . . . . . . . . . . . 16  |-  ( 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 ) ) )
1412, 1, 13suplocexprlemss 7637 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  C_  P. )
1514ad5antr 488 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  A  C_  P. )
16 simprl 521 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  u  e.  A
)
1715, 16sseldd 3129 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  u  e.  P. )
18 simprl 521 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  x  e.  P. )
1918ad4antr 486 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  x  e.  P. )
20 breq1 3970 . . . . . . . . . . . . . . 15  |-  ( y  =  u  ->  (
y  <P  x  <->  u  <P  x ) )
21 simprr 522 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  A. y  e.  A  y  <P  x )
2221ad4antr 486 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  A. y  e.  A  y  <P  x )
2320, 22, 16rspcdva 2821 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  u  <P  x
)
24 ltsopr 7518 . . . . . . . . . . . . . . . . 17  |-  <P  Or  P.
25 so2nr 4283 . . . . . . . . . . . . . . . . 17  |-  ( ( 
<P  Or  P.  /\  (
u  e.  P.  /\  x  e.  P. )
)  ->  -.  (
u  <P  x  /\  x  <P  u ) )
2624, 25mpan 421 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  x  e.  P. )  ->  -.  ( u  <P  x  /\  x  <P  u
) )
2717, 19, 26syl2anc 409 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  -.  ( u  <P  x  /\  x  <P  u ) )
28 imnan 680 . . . . . . . . . . . . . . 15  |-  ( ( u  <P  x  ->  -.  x  <P  u )  <->  -.  ( u  <P  x  /\  x  <P  u ) )
2927, 28sylibr 133 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  ( u  <P  x  ->  -.  x  <P  u ) )
3023, 29mpd 13 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  -.  x  <P  u )
31 aptiprlemu 7562 . . . . . . . . . . . . 13  |-  ( ( u  e.  P.  /\  x  e.  P.  /\  -.  x  <P  u )  -> 
( 2nd `  x
)  C_  ( 2nd `  u ) )
3217, 19, 30, 31syl3anc 1220 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  ( 2nd `  x
)  C_  ( 2nd `  u ) )
33 simpllr 524 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  s  e.  ( 2nd `  x ) )
3432, 33sseldd 3129 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  s  e.  ( 2nd `  u ) )
35 simprr 522 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  ( 2nd `  u
)  =  t )
3634, 35eleqtrd 2236 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  s  e.  t )
3711, 36rexlimddv 2579 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  /\  t  e.  ( 2nd " A
) )  ->  s  e.  t )
3837ralrimiva 2530 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  ->  A. t  e.  ( 2nd " A
) s  e.  t )
39 vex 2715 . . . . . . . . 9  |-  s  e. 
_V
4039elint2 3816 . . . . . . . 8  |-  ( s  e.  |^| ( 2nd " A
)  <->  A. t  e.  ( 2nd " A ) s  e.  t )
4138, 40sylibr 133 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  ->  s  e.  |^| ( 2nd " A
) )
4241ex 114 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P.  /\  A. y  e.  A  y 
<P  x ) )  /\  s  e.  Q. )  ->  ( s  e.  ( 2nd `  x )  ->  s  e.  |^| ( 2nd " A ) ) )
4342reximdva 2559 . . . . 5  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  ( E. s  e.  Q.  s  e.  ( 2nd `  x )  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) ) )
445, 43mpd 13 . . . 4  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) )
451, 44rexlimddv 2579 . . 3  |-  ( ph  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) )
46 simprr 522 . . . . . . 7  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  e.  |^| ( 2nd " A ) )
47 simprl 521 . . . . . . . . 9  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  e.  Q. )
48 1nq 7288 . . . . . . . . 9  |-  1Q  e.  Q.
49 addclnq 7297 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  1Q  e.  Q. )  -> 
( s  +Q  1Q )  e.  Q. )
5047, 48, 49sylancl 410 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
( s  +Q  1Q )  e.  Q. )
51 ltaddnq 7329 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  1Q  e.  Q. )  -> 
s  <Q  ( s  +Q  1Q ) )
5247, 48, 51sylancl 410 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  <Q  ( s  +Q  1Q ) )
53 breq2 3971 . . . . . . . . 9  |-  ( j  =  ( s  +Q  1Q )  ->  (
s  <Q  j  <->  s  <Q  ( s  +Q  1Q ) ) )
5453rspcev 2816 . . . . . . . 8  |-  ( ( ( s  +Q  1Q )  e.  Q.  /\  s  <Q  ( s  +Q  1Q ) )  ->  E. j  e.  Q.  s  <Q  j
)
5550, 52, 54syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  s  <Q  j )
56 breq1 3970 . . . . . . . . 9  |-  ( w  =  s  ->  (
w  <Q  j  <->  s  <Q  j ) )
5756rexbidv 2458 . . . . . . . 8  |-  ( w  =  s  ->  ( E. j  e.  Q.  w  <Q  j  <->  E. j  e.  Q.  s  <Q  j
) )
5857rspcev 2816 . . . . . . 7  |-  ( ( s  e.  |^| ( 2nd " A )  /\  E. j  e.  Q.  s  <Q  j )  ->  E. w  e.  |^| ( 2nd " A
) E. j  e. 
Q.  w  <Q  j
)
5946, 55, 58syl2anc 409 . . . . . 6  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. w  e.  |^| ( 2nd " A ) E. j  e.  Q.  w  <Q  j )
60 rexcom 2621 . . . . . 6  |-  ( E. w  e.  |^| ( 2nd " A ) E. j  e.  Q.  w  <Q  j  <->  E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w 
<Q  j )
6159, 60sylib 121 . . . . 5  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w 
<Q  j )
62 ssid 3148 . . . . . 6  |-  Q.  C_  Q.
63 rexss 3195 . . . . . 6  |-  ( Q.  C_  Q.  ->  ( E. j  e.  Q.  E. w  e.  |^| ( 2nd " A
) w  <Q  j  <->  E. j  e.  Q.  (
j  e.  Q.  /\  E. w  e.  |^| ( 2nd " A ) w 
<Q  j ) ) )
6462, 63ax-mp 5 . . . . 5  |-  ( E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w  <Q  j  <->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
6561, 64sylib 121 . . . 4  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
66 suplocexpr.b . . . . . . . . . 10  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
6766suplocexprlem2b 7636 . . . . . . . . 9  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
6814, 67syl 14 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  B
)  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
6968eleq2d 2227 . . . . . . 7  |-  ( ph  ->  ( j  e.  ( 2nd `  B )  <-> 
j  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
70 breq2 3971 . . . . . . . . 9  |-  ( u  =  j  ->  (
w  <Q  u  <->  w  <Q  j ) )
7170rexbidv 2458 . . . . . . . 8  |-  ( u  =  j  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
7271elrab 2868 . . . . . . 7  |-  ( j  e.  { u  e. 
Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } 
<->  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
7369, 72bitrdi 195 . . . . . 6  |-  ( ph  ->  ( j  e.  ( 2nd `  B )  <-> 
( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7473rexbidv 2458 . . . . 5  |-  ( ph  ->  ( E. j  e. 
Q.  j  e.  ( 2nd `  B )  <->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7574adantr 274 . . . 4  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
( E. j  e. 
Q.  j  e.  ( 2nd `  B )  <->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7665, 75mpbird 166 . . 3  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  j  e.  ( 2nd `  B ) )
7745, 76rexlimddv 2579 . 2  |-  ( ph  ->  E. j  e.  Q.  j  e.  ( 2nd `  B ) )
78 eleq1w 2218 . . 3  |-  ( j  =  s  ->  (
j  e.  ( 2nd `  B )  <->  s  e.  ( 2nd `  B ) ) )
7978cbvrexv 2681 . 2  |-  ( E. j  e.  Q.  j  e.  ( 2nd `  B
)  <->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
8077, 79sylib 121 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1335   E.wex 1472    e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439   _Vcvv 2712    C_ wss 3102   <.cop 3564   U.cuni 3774   |^|cint 3809   class class class wbr 3967    Or wor 4257   "cima 4591   Fun wfun 5166   -onto->wfo 5170   ` cfv 5172  (class class class)co 5826   1stc1st 6088   2ndc2nd 6089   Q.cnq 7202   1Qc1q 7203    +Q cplq 7204    <Q cltq 7207   P.cnp 7213    <P cltp 7217
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388  df-iltp 7392
This theorem is referenced by:  suplocexprlemex  7644
  Copyright terms: Public domain W3C validator