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Theorem ovolicc2lem2 19414
Description: Lemma for ovolicc2 19418. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1  |-  ( ph  ->  A  e.  RR )
ovolicc.2  |-  ( ph  ->  B  e.  RR )
ovolicc.3  |-  ( ph  ->  A  <_  B )
ovolicc2.4  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
ovolicc2.5  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
ovolicc2.6  |-  ( ph  ->  U  e.  ( ~P
ran  ( (,)  o.  F )  i^i  Fin ) )
ovolicc2.7  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
ovolicc2.8  |-  ( ph  ->  G : U --> NN )
ovolicc2.9  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
ovolicc2.10  |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B
) )  =/=  (/) }
ovolicc2.11  |-  ( ph  ->  H : T --> T )
ovolicc2.12  |-  ( (
ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) )  <_  B ,  ( 2nd `  ( F `
 ( G `  t ) ) ) ,  B )  e.  ( H `  t
) )
ovolicc2.13  |-  ( ph  ->  A  e.  C )
ovolicc2.14  |-  ( ph  ->  C  e.  T )
ovolicc2.15  |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )
ovolicc2.16  |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }
Assertion
Ref Expression
ovolicc2lem2  |-  ( (
ph  /\  ( N  e.  NN  /\  -.  N  e.  W ) )  -> 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B )
Distinct variable groups:    t, n, u, A    B, n, t, u    t, H    C, n, t    n, F, t   
n, K, t, u   
n, G, t    n, W    ph, n, t    T, n, t    n, N, t, u    U, n, t, u
Allowed substitution hints:    ph( u)    C( u)    S( u, t, n)    T( u)    F( u)    G( u)    H( u, n)    W( u, t)

Proof of Theorem ovolicc2lem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovolicc.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
21adantr 452 . . . . 5  |-  ( (
ph  /\  N  e.  NN )  ->  B  e.  RR )
3 ovolicc2.5 . . . . . . . . 9  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3562 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5599 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR ) )  ->  F : NN --> ( RR  X.  RR ) )
63, 4, 5sylancl 644 . . . . . . . 8  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
76adantr 452 . . . . . . 7  |-  ( (
ph  /\  N  e.  NN )  ->  F : NN
--> ( RR  X.  RR ) )
8 ovolicc2.8 . . . . . . . . 9  |-  ( ph  ->  G : U --> NN )
98adantr 452 . . . . . . . 8  |-  ( (
ph  /\  N  e.  NN )  ->  G : U
--> NN )
10 nnuz 10521 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
11 ovolicc2.15 . . . . . . . . . . . 12  |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )
12 1z 10311 . . . . . . . . . . . . 13  |-  1  e.  ZZ
1312a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
14 ovolicc2.14 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  T )
15 ovolicc2.11 . . . . . . . . . . . 12  |-  ( ph  ->  H : T --> T )
1610, 11, 13, 14, 15algrf 13064 . . . . . . . . . . 11  |-  ( ph  ->  K : NN --> T )
1716ffvelrnda 5870 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  NN )  ->  ( K `
 N )  e.  T )
18 ineq1 3535 . . . . . . . . . . . 12  |-  ( u  =  ( K `  N )  ->  (
u  i^i  ( A [,] B ) )  =  ( ( K `  N )  i^i  ( A [,] B ) ) )
1918neeq1d 2614 . . . . . . . . . . 11  |-  ( u  =  ( K `  N )  ->  (
( u  i^i  ( A [,] B ) )  =/=  (/)  <->  ( ( K `
 N )  i^i  ( A [,] B
) )  =/=  (/) ) )
20 ovolicc2.10 . . . . . . . . . . 11  |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B
) )  =/=  (/) }
2119, 20elrab2 3094 . . . . . . . . . 10  |-  ( ( K `  N )  e.  T  <->  ( ( K `  N )  e.  U  /\  (
( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) ) )
2217, 21sylib 189 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( K `  N )  e.  U  /\  (
( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) ) )
2322simpld 446 . . . . . . . 8  |-  ( (
ph  /\  N  e.  NN )  ->  ( K `
 N )  e.  U )
249, 23ffvelrnd 5871 . . . . . . 7  |-  ( (
ph  /\  N  e.  NN )  ->  ( G `
 ( K `  N ) )  e.  NN )
257, 24ffvelrnd 5871 . . . . . 6  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 ( G `  ( K `  N ) ) )  e.  ( RR  X.  RR ) )
26 xp2nd 6377 . . . . . 6  |-  ( ( F `  ( G `
 ( K `  N ) ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
2725, 26syl 16 . . . . 5  |-  ( (
ph  /\  N  e.  NN )  ->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
282, 27ltnled 9220 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <->  -.  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B ) )
29 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  N  e.  NN )
301adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  B  e.  RR )
3122adantrr 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
( K `  N
)  e.  U  /\  ( ( K `  N )  i^i  ( A [,] B ) )  =/=  (/) ) )
3231simprd 450 . . . . . . . . 9  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) )
33 n0 3637 . . . . . . . . 9  |-  ( ( ( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) 
<->  E. x  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )
3432, 33sylib 189 . . . . . . . 8  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  E. x  x  e.  ( ( K `  N )  i^i  ( A [,] B
) ) )
35 xp1st 6376 . . . . . . . . . . . 12  |-  ( ( F `  ( G `
 ( K `  N ) ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
3625, 35syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  N  e.  NN )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
3736adantrr 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `
 N ) ) ) )  e.  RR )
3837adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
39 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )
40 elin 3530 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( K `
 N )  i^i  ( A [,] B
) )  <->  ( x  e.  ( K `  N
)  /\  x  e.  ( A [,] B ) ) )
4139, 40sylib 189 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  ( K `  N
)  /\  x  e.  ( A [,] B ) ) )
4241simprd 450 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  ( A [,] B ) )
43 ovolicc.1 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
44 elicc2 10975 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4543, 1, 44syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4645ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  ( A [,] B
)  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) ) )
4742, 46mpbid 202 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4847simp1d 969 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  RR )
491ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  B  e.  RR )
5041simpld 446 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  ( K `  N ) )
5131simpld 446 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( K `  N )  e.  U )
52 ovolicc.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  <_  B )
53 ovolicc2.4 . . . . . . . . . . . . . 14  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
54 ovolicc2.6 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  ( ~P
ran  ( (,)  o.  F )  i^i  Fin ) )
55 ovolicc2.7 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
56 ovolicc2.9 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
5743, 1, 52, 53, 3, 54, 55, 8, 56ovolicc2lem1 19413 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( K `  N )  e.  U
)  ->  ( x  e.  ( K `  N
)  <->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
5851, 57syldan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
x  e.  ( K `
 N )  <->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
5958adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  ( K `  N
)  <->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
6050, 59mpbid 202 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )
6160simp2d 970 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x )
6247simp3d 971 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  <_  B )
6338, 48, 49, 61, 62ltletrd 9230 . . . . . . . 8  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  B )
6434, 63exlimddv 1648 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `
 N ) ) ) )  <  B
)
65 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) )
6643, 1, 52, 53, 3, 54, 55, 8, 56ovolicc2lem1 19413 . . . . . . . 8  |-  ( (
ph  /\  ( K `  N )  e.  U
)  ->  ( B  e.  ( K `  N
)  <->  ( B  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  B  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
6751, 66syldan 457 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( B  e.  ( K `  N )  <->  ( B  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  B  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
6830, 64, 65, 67mpbir3and 1137 . . . . . 6  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  B  e.  ( K `  N
) )
69 fveq2 5728 . . . . . . . 8  |-  ( n  =  N  ->  ( K `  n )  =  ( K `  N ) )
7069eleq2d 2503 . . . . . . 7  |-  ( n  =  N  ->  ( B  e.  ( K `  n )  <->  B  e.  ( K `  N ) ) )
71 ovolicc2.16 . . . . . . 7  |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }
7270, 71elrab2 3094 . . . . . 6  |-  ( N  e.  W  <->  ( N  e.  NN  /\  B  e.  ( K `  N
) ) )
7329, 68, 72sylanbrc 646 . . . . 5  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  N  e.  W )
7473expr 599 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  ->  N  e.  W
) )
7528, 74sylbird 227 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  ( -.  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B  ->  N  e.  W ) )
7675con1d 118 . 2  |-  ( (
ph  /\  N  e.  NN )  ->  ( -.  N  e.  W  -> 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B ) )
7776impr 603 1  |-  ( (
ph  /\  ( N  e.  NN  /\  -.  N  e.  W ) )  -> 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B )
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   {crab 2709    i^i cin 3319    C_ wss 3320   (/)c0 3628   ifcif 3739   ~Pcpw 3799   {csn 3814   U.cuni 4015   class class class wbr 4212    X. cxp 4876   ran crn 4879    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Fincfn 7109   RRcr 8989   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   ZZcz 10282   (,)cioo 10916   [,]cicc 10919    seq cseq 11323   abscabs 12039
This theorem is referenced by:  ovolicc2lem3  19415  ovolicc2lem4  19416
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
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-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-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-ioo 10920  df-icc 10923  df-fz 11044  df-seq 11324
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