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Theorem ovolicc2lem2 19367
Description: Lemma for ovolicc2 19371. (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 3522 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5558 . . . . . . . . 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 10477 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
11 ovolicc2.15 . . . . . . . . . . . 12  |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )
12 1z 10267 . . . . . . . . . . . . 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 13019 . . . . . . . . . . 11  |-  ( ph  ->  K : NN --> T )
1716ffvelrnda 5829 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  NN )  ->  ( K `
 N )  e.  T )
18 ineq1 3495 . . . . . . . . . . . 12  |-  ( u  =  ( K `  N )  ->  (
u  i^i  ( A [,] B ) )  =  ( ( K `  N )  i^i  ( A [,] B ) ) )
1918neeq1d 2580 . . . . . . . . . . 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 3054 . . . . . . . . . 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 5830 . . . . . . 7  |-  ( (
ph  /\  N  e.  NN )  ->  ( G `
 ( K `  N ) )  e.  NN )
257, 24ffvelrnd 5830 . . . . . 6  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 ( G `  ( K `  N ) ) )  e.  ( RR  X.  RR ) )
26 xp2nd 6336 . . . . . 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 9176 . . . 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 3597 . . . . . . . . 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 6335 . . . . . . . . . . . 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 3490 . . . . . . . . . . . . 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 10931 . . . . . . . . . . . . 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 19366 . . . . . . . . . . . . 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 9186 . . . . . . . 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 1645 . . . . . . 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 19366 . . . . . . . 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 5687 . . . . . . . 8  |-  ( n  =  N  ->  ( K `  n )  =  ( K `  N ) )
7069eleq2d 2471 . . . . . . 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 3054 . . . . . 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 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670    i^i cin 3279    C_ wss 3280   (/)c0 3588   ifcif 3699   ~Pcpw 3759   {csn 3774   U.cuni 3975   class class class wbr 4172    X. cxp 4835   ran crn 4838    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Fincfn 7068   RRcr 8945   1c1 8947    + caddc 8949    < clt 9076    <_ cle 9077    - cmin 9247   NNcn 9956   ZZcz 10238   (,)cioo 10872   [,]cicc 10875    seq cseq 11278   abscabs 11994
This theorem is referenced by:  ovolicc2lem3  19368  ovolicc2lem4  19369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-ioo 10876  df-icc 10879  df-fz 11000  df-seq 11279
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