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Theorem ruclem12 13523
Description: Lemma for ruc 13525. The supremum of the increasing sequence  1st  o.  G is a real number that is not in the range of  F. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1  |-  ( ph  ->  F : NN --> RR )
ruc.2  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
ruc.4  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
ruc.5  |-  G  =  seq 0 ( D ,  C )
ruc.6  |-  S  =  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )
Assertion
Ref Expression
ruclem12  |-  ( ph  ->  S  e.  ( RR 
\  ran  F )
)
Distinct variable groups:    x, m, y, F    m, G, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)    S( x, y, m)

Proof of Theorem ruclem12
Dummy variables  z  n  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.6 . . 3  |-  S  =  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )
2 ruc.1 . . . . . 6  |-  ( ph  ->  F : NN --> RR )
3 ruc.2 . . . . . 6  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
4 ruc.4 . . . . . 6  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
5 ruc.5 . . . . . 6  |-  G  =  seq 0 ( D ,  C )
62, 3, 4, 5ruclem11 13522 . . . . 5  |-  ( ph  ->  ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1 ) )
76simp1d 1000 . . . 4  |-  ( ph  ->  ran  ( 1st  o.  G )  C_  RR )
86simp2d 1001 . . . 4  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
9 1re 9385 . . . . 5  |-  1  e.  RR
106simp3d 1002 . . . . 5  |-  ( ph  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )
11 breq2 4296 . . . . . . 7  |-  ( n  =  1  ->  (
z  <_  n  <->  z  <_  1 ) )
1211ralbidv 2735 . . . . . 6  |-  ( n  =  1  ->  ( A. z  e.  ran  ( 1st  o.  G ) z  <_  n  <->  A. z  e.  ran  ( 1st  o.  G ) z  <_ 
1 ) )
1312rspcev 3073 . . . . 5  |-  ( ( 1  e.  RR  /\  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )  ->  E. n  e.  RR  A. z  e.  ran  ( 1st  o.  G ) z  <_  n )
149, 10, 13sylancr 663 . . . 4  |-  ( ph  ->  E. n  e.  RR  A. z  e.  ran  ( 1st  o.  G ) z  <_  n )
15 suprcl 10290 . . . 4  |-  ( ( ran  ( 1st  o.  G )  C_  RR  /\ 
ran  ( 1st  o.  G )  =/=  (/)  /\  E. n  e.  RR  A. z  e.  ran  ( 1st  o.  G ) z  <_  n )  ->  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )  e.  RR )
167, 8, 14, 15syl3anc 1218 . . 3  |-  ( ph  ->  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )  e.  RR )
171, 16syl5eqel 2527 . 2  |-  ( ph  ->  S  e.  RR )
182adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  F : NN
--> RR )
193adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
202, 3, 4, 5ruclem6 13517 . . . . . . . . . . 11  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
21 nnm1nn0 10621 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
22 ffvelrn 5841 . . . . . . . . . . 11  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  ( n  -  1
)  e.  NN0 )  ->  ( G `  (
n  -  1 ) )  e.  ( RR 
X.  RR ) )
2320, 21, 22syl2an 477 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( n  - 
1 ) )  e.  ( RR  X.  RR ) )
24 xp1st 6606 . . . . . . . . . 10  |-  ( ( G `  ( n  -  1 ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  e.  RR )
2523, 24syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  e.  RR )
26 xp2nd 6607 . . . . . . . . . 10  |-  ( ( G `  ( n  -  1 ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  (
n  -  1 ) ) )  e.  RR )
2723, 26syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  (
n  -  1 ) ) )  e.  RR )
282ffvelrnda 5843 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  e.  RR )
29 eqid 2443 . . . . . . . . 9  |-  ( 1st `  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
30 eqid 2443 . . . . . . . . 9  |-  ( 2nd `  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
312, 3, 4, 5ruclem8 13519 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  -  1 )  e. 
NN0 )  ->  ( 1st `  ( G `  ( n  -  1
) ) )  < 
( 2nd `  ( G `  ( n  -  1 ) ) ) )
3221, 31sylan2 474 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  <  ( 2nd `  ( G `  ( n  -  1
) ) ) )
3318, 19, 25, 27, 28, 29, 30, 32ruclem3 13515 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )  \/  ( 2nd `  ( <. ( 1st `  ( G `  ( n  -  1
) ) ) ,  ( 2nd `  ( G `  ( n  -  1 ) ) ) >. D ( F `
 n ) ) )  <  ( F `
 n ) ) )
342, 3, 4, 5ruclem7 13518 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  -  1 )  e. 
NN0 )  ->  ( G `  ( (
n  -  1 )  +  1 ) )  =  ( ( G `
 ( n  - 
1 ) ) D ( F `  (
( n  -  1 )  +  1 ) ) ) )
3521, 34sylan2 474 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( ( n  -  1 )  +  1 ) )  =  ( ( G `  ( n  -  1
) ) D ( F `  ( ( n  -  1 )  +  1 ) ) ) )
36 nncn 10330 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  CC )
3736adantl 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
38 ax-1cn 9340 . . . . . . . . . . . . . 14  |-  1  e.  CC
39 npcan 9619 . . . . . . . . . . . . . 14  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
4037, 38, 39sylancl 662 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( n  -  1 )  +  1 )  =  n )
4140fveq2d 5695 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( ( n  -  1 )  +  1 ) )  =  ( G `  n
) )
42 1st2nd2 6613 . . . . . . . . . . . . . 14  |-  ( ( G `  ( n  -  1 ) )  e.  ( RR  X.  RR )  ->  ( G `
 ( n  - 
1 ) )  = 
<. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. )
4323, 42syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( n  - 
1 ) )  = 
<. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. )
4440fveq2d 5695 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 ( ( n  -  1 )  +  1 ) )  =  ( F `  n
) )
4543, 44oveq12d 6109 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( G `  ( n  -  1 ) ) D ( F `  ( ( n  - 
1 )  +  1 ) ) )  =  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
4635, 41, 453eqtr3d 2483 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
4746fveq2d 5695 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  =  ( 1st `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) ) )
4847breq2d 4304 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  <->  ( F `  n )  <  ( 1st `  ( <. ( 1st `  ( G `  ( n  -  1
) ) ) ,  ( 2nd `  ( G `  ( n  -  1 ) ) ) >. D ( F `
 n ) ) ) ) )
4946fveq2d 5695 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n
) )  =  ( 2nd `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) ) )
5049breq1d 4302 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  <  ( F `  n )  <->  ( 2nd `  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )  <  ( F `  n )
) )
5148, 50orbi12d 709 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( F `  n
)  <  ( 1st `  ( G `  n
) )  \/  ( 2nd `  ( G `  n ) )  < 
( F `  n
) )  <->  ( ( F `  n )  <  ( 1st `  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )  \/  ( 2nd `  ( <. ( 1st `  ( G `  ( n  -  1
) ) ) ,  ( 2nd `  ( G `  ( n  -  1 ) ) ) >. D ( F `
 n ) ) )  <  ( F `
 n ) ) ) )
5233, 51mpbird 232 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  \/  ( 2nd `  ( G `  n
) )  <  ( F `  n )
) )
537adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ran  ( 1st  o.  G )  C_  RR )
548adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ran  ( 1st  o.  G )  =/=  (/) )
5514adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  E. n  e.  RR  A. z  e. 
ran  ( 1st  o.  G ) z  <_  n )
56 nnnn0 10586 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  NN0 )
57 fvco3 5768 . . . . . . . . . . . . 13  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  =  ( 1st `  ( G `  n
) ) )
5820, 56, 57syl2an 477 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  G ) `
 n )  =  ( 1st `  ( G `  n )
) )
5920adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN )  ->  G : NN0
--> ( RR  X.  RR ) )
60 1stcof 6604 . . . . . . . . . . . . . 14  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
61 ffn 5559 . . . . . . . . . . . . . 14  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
6259, 60, 613syl 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st 
o.  G )  Fn 
NN0 )
6356adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  n  e. 
NN0 )
64 fnfvelrn 5840 . . . . . . . . . . . . 13  |-  ( ( ( 1st  o.  G
)  Fn  NN0  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  e.  ran  ( 1st  o.  G ) )
6562, 63, 64syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  G ) `
 n )  e. 
ran  ( 1st  o.  G ) )
6658, 65eqeltrrd 2518 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  e.  ran  ( 1st  o.  G ) )
67 suprub 10291 . . . . . . . . . . 11  |-  ( ( ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  E. n  e.  RR  A. z  e. 
ran  ( 1st  o.  G ) z  <_  n )  /\  ( 1st `  ( G `  n ) )  e. 
ran  ( 1st  o.  G ) )  -> 
( 1st `  ( G `  n )
)  <_  sup ( ran  ( 1st  o.  G
) ,  RR ,  <  ) )
6853, 54, 55, 66, 67syl31anc 1221 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  <_  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  ) )
6968, 1syl6breqr 4332 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  <_  S
)
70 ffvelrn 5841 . . . . . . . . . . . 12  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( G `  n
)  e.  ( RR 
X.  RR ) )
7120, 56, 70syl2an 477 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  ( RR  X.  RR ) )
72 xp1st 6606 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
7371, 72syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  e.  RR )
7417adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  S  e.  RR )
75 ltletr 9466 . . . . . . . . . 10  |-  ( ( ( F `  n
)  e.  RR  /\  ( 1st `  ( G `
 n ) )  e.  RR  /\  S  e.  RR )  ->  (
( ( F `  n )  <  ( 1st `  ( G `  n ) )  /\  ( 1st `  ( G `
 n ) )  <_  S )  -> 
( F `  n
)  <  S )
)
7628, 73, 74, 75syl3anc 1218 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( F `  n
)  <  ( 1st `  ( G `  n
) )  /\  ( 1st `  ( G `  n ) )  <_  S )  ->  ( F `  n )  <  S ) )
7769, 76mpan2d 674 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  ->  ( F `  n )  <  S
) )
78 fvco3 5768 . . . . . . . . . . . . . . 15  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  k  e.  NN0 )  -> 
( ( 1st  o.  G ) `  k
)  =  ( 1st `  ( G `  k
) ) )
7959, 78sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  (
( 1st  o.  G
) `  k )  =  ( 1st `  ( G `  k )
) )
8059ffvelrnda 5843 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( G `  k )  e.  ( RR  X.  RR ) )
81 xp1st 6606 . . . . . . . . . . . . . . . 16  |-  ( ( G `  k )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  k
) )  e.  RR )
8280, 81syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  e.  RR )
83 xp2nd 6607 . . . . . . . . . . . . . . . . 17  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
8471, 83syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
8584adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 2nd `  ( G `  n ) )  e.  RR )
8618adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  F : NN --> RR )
8719adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
88 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
8963adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  n  e.  NN0 )
9086, 87, 4, 5, 88, 89ruclem10 13521 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  < 
( 2nd `  ( G `  n )
) )
9182, 85, 90ltled 9522 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  <_ 
( 2nd `  ( G `  n )
) )
9279, 91eqbrtrd 4312 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  (
( 1st  o.  G
) `  k )  <_  ( 2nd `  ( G `  n )
) )
9392ralrimiva 2799 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A. k  e.  NN0  ( ( 1st 
o.  G ) `  k )  <_  ( 2nd `  ( G `  n ) ) )
94 breq1 4295 . . . . . . . . . . . . . 14  |-  ( z  =  ( ( 1st 
o.  G ) `  k )  ->  (
z  <_  ( 2nd `  ( G `  n
) )  <->  ( ( 1st  o.  G ) `  k )  <_  ( 2nd `  ( G `  n ) ) ) )
9594ralrn 5846 . . . . . . . . . . . . 13  |-  ( ( 1st  o.  G )  Fn  NN0  ->  ( A. z  e.  ran  ( 1st 
o.  G ) z  <_  ( 2nd `  ( G `  n )
)  <->  A. k  e.  NN0  ( ( 1st  o.  G ) `  k
)  <_  ( 2nd `  ( G `  n
) ) ) )
9662, 95syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( A. z  e.  ran  ( 1st 
o.  G ) z  <_  ( 2nd `  ( G `  n )
)  <->  A. k  e.  NN0  ( ( 1st  o.  G ) `  k
)  <_  ( 2nd `  ( G `  n
) ) ) )
9793, 96mpbird 232 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_ 
( 2nd `  ( G `  n )
) )
98 suprleub 10294 . . . . . . . . . . . 12  |-  ( ( ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  E. n  e.  RR  A. z  e. 
ran  ( 1st  o.  G ) z  <_  n )  /\  ( 2nd `  ( G `  n ) )  e.  RR )  ->  ( sup ( ran  ( 1st 
o.  G ) ,  RR ,  <  )  <_  ( 2nd `  ( G `  n )
)  <->  A. z  e.  ran  ( 1st  o.  G ) z  <_  ( 2nd `  ( G `  n
) ) ) )
9953, 54, 55, 84, 98syl31anc 1221 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( sup ( ran  ( 1st 
o.  G ) ,  RR ,  <  )  <_  ( 2nd `  ( G `  n )
)  <->  A. z  e.  ran  ( 1st  o.  G ) z  <_  ( 2nd `  ( G `  n
) ) ) )
10097, 99mpbird 232 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  sup ( ran  ( 1st  o.  G
) ,  RR ,  <  )  <_  ( 2nd `  ( G `  n
) ) )
1011, 100syl5eqbr 4325 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  S  <_ 
( 2nd `  ( G `  n )
) )
102 lelttr 9465 . . . . . . . . . 10  |-  ( ( S  e.  RR  /\  ( 2nd `  ( G `
 n ) )  e.  RR  /\  ( F `  n )  e.  RR )  ->  (
( S  <_  ( 2nd `  ( G `  n ) )  /\  ( 2nd `  ( G `
 n ) )  <  ( F `  n ) )  ->  S  <  ( F `  n ) ) )
10374, 84, 28, 102syl3anc 1218 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( S  <_  ( 2nd `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  < 
( F `  n
) )  ->  S  <  ( F `  n
) ) )
104101, 103mpand 675 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  <  ( F `  n )  ->  S  <  ( F `  n
) ) )
10577, 104orim12d 834 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( F `  n
)  <  ( 1st `  ( G `  n
) )  \/  ( 2nd `  ( G `  n ) )  < 
( F `  n
) )  ->  (
( F `  n
)  <  S  \/  S  <  ( F `  n ) ) ) )
10652, 105mpd 15 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  S  \/  S  <  ( F `  n
) ) )
10728, 74lttri2d 9513 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  =/=  S  <->  ( ( F `  n )  <  S  \/  S  < 
( F `  n
) ) ) )
108106, 107mpbird 232 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =/= 
S )
109108neneqd 2624 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  -.  ( F `  n )  =  S )
110109nrexdv 2819 . . 3  |-  ( ph  ->  -.  E. n  e.  NN  ( F `  n )  =  S )
111 risset 2763 . . . 4  |-  ( S  e.  ran  F  <->  E. z  e.  ran  F  z  =  S )
112 ffn 5559 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
113 eqeq1 2449 . . . . . 6  |-  ( z  =  ( F `  n )  ->  (
z  =  S  <->  ( F `  n )  =  S ) )
114113rexrn 5845 . . . . 5  |-  ( F  Fn  NN  ->  ( E. z  e.  ran  F  z  =  S  <->  E. n  e.  NN  ( F `  n )  =  S ) )
1152, 112, 1143syl 20 . . . 4  |-  ( ph  ->  ( E. z  e. 
ran  F  z  =  S 
<->  E. n  e.  NN  ( F `  n )  =  S ) )
116111, 115syl5bb 257 . . 3  |-  ( ph  ->  ( S  e.  ran  F  <->  E. n  e.  NN  ( F `  n )  =  S ) )
117110, 116mtbird 301 . 2  |-  ( ph  ->  -.  S  e.  ran  F )
11817, 117eldifd 3339 1  |-  ( ph  ->  S  e.  ( RR 
\  ran  F )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   [_csb 3288    \ cdif 3325    u. cun 3326    C_ wss 3328   (/)c0 3637   ifcif 3791   {csn 3877   <.cop 3883   class class class wbr 4292    X. cxp 4838   ran crn 4841    o. ccom 4844    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419    - cmin 9595    / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579    seqcseq 11806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-seq 11807
This theorem is referenced by:  ruclem13  13524
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