MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixxlb Structured version   Unicode version

Theorem ixxlb 10938
Description: Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxub.2  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w S B ) )
ixxub.3  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w  <_  B ) )
ixxub.4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )
ixxub.5  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )
Assertion
Ref Expression
ixxlb  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  =  A )
Distinct variable groups:    x, w, y, z, A    w, O    w, B, x, y, z   
x, R, y, z   
x, S, y, z
Allowed substitution hints:    R( w)    S( w)    O( x, y, z)

Proof of Theorem ixxlb
StepHypRef Expression
1 simprr 734 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) )
2 ixx.1 . . . . . . . . . . . . . . 15  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
32elixx1 10925 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
433adant3 977 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
54biimpa 471 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w  e.  RR*  /\  A R w  /\  w S B ) )
65simp1d 969 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
76ex 424 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
87ssrdv 3354 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
98ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( A O B )  C_  RR* )
10 qre 10579 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
1110rexrd 9134 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
1211ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  e.  RR* )
13 simprl 733 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  A  <  w )
14 simp1 957 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
1514ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  A  e.  RR* )
16 ixxub.4 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )
1715, 12, 16syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( A  <  w  ->  A R w ) )
1813, 17mpd 15 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  A R w )
19 infmxrcl 10895 . . . . . . . . . . . . 13  |-  ( ( A O B ) 
C_  RR*  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
208, 19syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
2120ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
22 simpll2 997 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  B  e.  RR* )
23 simp3 959 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
24 n0 3637 . . . . . . . . . . . . . 14  |-  ( ( A O B )  =/=  (/)  <->  E. w  w  e.  ( A O B ) )
2523, 24sylib 189 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  E. w  w  e.  ( A O B ) )
2620adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
27 simpl2 961 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  B  e.  RR* )
28 infmxrlb 10912 . . . . . . . . . . . . . . 15  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
298, 28sylan 458 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
305simp3d 971 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w S B )
31 ixxub.3 . . . . . . . . . . . . . . . 16  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w  <_  B ) )
326, 27, 31syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w S B  ->  w  <_  B ) )
3330, 32mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  <_  B )
3426, 6, 27, 29, 33xrletrd 10752 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  B
)
3525, 34exlimddv 1648 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  <_  B )
3635ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  B
)
3712, 21, 22, 1, 36xrltletrd 10751 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  <  B )
38 ixxub.2 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w S B ) )
3912, 22, 38syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( w  <  B  ->  w S B ) )
4037, 39mpd 15 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w S B )
414ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( w  e.  ( A O B )  <-> 
( w  e.  RR*  /\  A R w  /\  w S B ) ) )
4212, 18, 40, 41mpbir3and 1137 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  e.  ( A O B ) )
439, 42, 28syl2anc 643 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
44 xrlenlt 9143 . . . . . . . 8  |-  ( ( sup ( ( A O B ) , 
RR* ,  `'  <  )  e.  RR*  /\  w  e.  RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w  <->  -.  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )
4521, 12, 44syl2anc 643 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w  <->  -.  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
4643, 45mpbid 202 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  -.  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) )
471, 46pm2.65da 560 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
4847nrexdv 2809 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
49 qbtwnxr 10786 . . . . . 6  |-  ( ( A  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR*  /\  A  <  sup (
( A O B ) ,  RR* ,  `'  <  ) )  ->  E. w  e.  QQ  ( A  < 
w  /\  w  <  sup ( ( A O B ) ,  RR* ,  `'  <  ) ) )
50493expia 1155 . . . . 5  |-  ( ( A  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )  ->  ( A  <  sup ( ( A O B ) ,  RR* ,  `'  <  )  ->  E. w  e.  QQ  ( A  < 
w  /\  w  <  sup ( ( A O B ) ,  RR* ,  `'  <  ) ) ) )
5114, 20, 50syl2anc 643 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A  <  sup ( ( A O B ) , 
RR* ,  `'  <  )  ->  E. w  e.  QQ  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) ) )
5248, 51mtod 170 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  A  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) )
53 xrlenlt 9143 . . . 4  |-  ( ( sup ( ( A O B ) , 
RR* ,  `'  <  )  e.  RR*  /\  A  e. 
RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  <->  -.  A  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )
5420, 14, 53syl2anc 643 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  <->  -.  A  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )
5552, 54mpbird 224 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  <_  A )
565simp2d 970 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A R w )
5714adantr 452 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  e.  RR* )
58 ixxub.5 . . . . . 6  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )
5957, 6, 58syl2anc 643 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  ( A R w  ->  A  <_  w ) )
6056, 59mpd 15 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  w )
6160ralrimiva 2789 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) A  <_  w
)
62 infmxrgelb 10913 . . . 4  |-  ( ( ( A O B )  C_  RR*  /\  A  e.  RR* )  ->  ( A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  )  <->  A. w  e.  ( A O B ) A  <_  w ) )
638, 14, 62syl2anc 643 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  )  <->  A. w  e.  ( A O B ) A  <_  w ) )
6461, 63mpbird 224 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  <_  sup ( ( A O B ) ,  RR* ,  `'  <  ) )
65 xrletri3 10745 . . 3  |-  ( ( sup ( ( A O B ) , 
RR* ,  `'  <  )  e.  RR*  /\  A  e. 
RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  =  A  <-> 
( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  /\  A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) ) )
6620, 14, 65syl2anc 643 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  =  A  <-> 
( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  /\  A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) ) )
6755, 64, 66mpbir2and 889 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3320   (/)c0 3628   class class class wbr 4212   `'ccnv 4877  (class class class)co 6081    e. cmpt2 6083   supcsup 7445   RR*cxr 9119    < clt 9120    <_ cle 9121   QQcq 10574
This theorem is referenced by:  ioorf  19465  ioorinv2  19467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575
  Copyright terms: Public domain W3C validator