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Theorem ruclem12 12767
Description: Lemma for ruc 12769. 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 12766 . . . . 5  |-  ( ph  ->  ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1 ) )
76simp1d 969 . . . 4  |-  ( ph  ->  ran  ( 1st  o.  G )  C_  RR )
86simp2d 970 . . . 4  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
9 1re 9023 . . . . 5  |-  1  e.  RR
106simp3d 971 . . . . 5  |-  ( ph  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )
11 breq2 4157 . . . . . . 7  |-  ( n  =  1  ->  (
z  <_  n  <->  z  <_  1 ) )
1211ralbidv 2669 . . . . . 6  |-  ( n  =  1  ->  ( A. z  e.  ran  ( 1st  o.  G ) z  <_  n  <->  A. z  e.  ran  ( 1st  o.  G ) z  <_ 
1 ) )
1312rspcev 2995 . . . . 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 645 . . . 4  |-  ( ph  ->  E. n  e.  RR  A. z  e.  ran  ( 1st  o.  G ) z  <_  n )
15 suprcl 9900 . . . 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 1184 . . 3  |-  ( ph  ->  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )  e.  RR )
171, 16syl5eqel 2471 . 2  |-  ( ph  ->  S  e.  RR )
182adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  F : NN
--> RR )
193adantr 452 . . . . . . . . 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 12761 . . . . . . . . . . 11  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
21 nnm1nn0 10193 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
22 ffvelrn 5807 . . . . . . . . . . 11  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  ( n  -  1
)  e.  NN0 )  ->  ( G `  (
n  -  1 ) )  e.  ( RR 
X.  RR ) )
2320, 21, 22syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( n  - 
1 ) )  e.  ( RR  X.  RR ) )
24 xp1st 6315 . . . . . . . . . 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 6316 . . . . . . . . . 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 5809 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  e.  RR )
29 eqid 2387 . . . . . . . . 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 2387 . . . . . . . . 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 12763 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  -  1 )  e. 
NN0 )  ->  ( 1st `  ( G `  ( n  -  1
) ) )  < 
( 2nd `  ( G `  ( n  -  1 ) ) ) )
3221, 31sylan2 461 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  <  ( 2nd `  ( G `  ( n  -  1
) ) ) )
3318, 19, 25, 27, 28, 29, 30, 32ruclem3 12759 . . . . . . . 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 12762 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  -  1 )  e. 
NN0 )  ->  ( G `  ( (
n  -  1 )  +  1 ) )  =  ( ( G `
 ( n  - 
1 ) ) D ( F `  (
( n  -  1 )  +  1 ) ) ) )
3521, 34sylan2 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( ( n  -  1 )  +  1 ) )  =  ( ( G `  ( n  -  1
) ) D ( F `  ( ( n  -  1 )  +  1 ) ) ) )
36 nncn 9940 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  CC )
3736adantl 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
38 ax-1cn 8981 . . . . . . . . . . . . . 14  |-  1  e.  CC
39 npcan 9246 . . . . . . . . . . . . . 14  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
4037, 38, 39sylancl 644 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( n  -  1 )  +  1 )  =  n )
4140fveq2d 5672 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( ( n  -  1 )  +  1 ) )  =  ( G `  n
) )
42 1st2nd2 6325 . . . . . . . . . . . . . 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 5672 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 ( ( n  -  1 )  +  1 ) )  =  ( F `  n
) )
4543, 44oveq12d 6038 . . . . . . . . . . . 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 2427 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
4746fveq2d 5672 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  =  ( 1st `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) ) )
4847breq2d 4165 . . . . . . . . 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 5672 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n
) )  =  ( 2nd `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) ) )
5049breq1d 4163 . . . . . . . . 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 691 . . . . . . . 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 224 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  \/  ( 2nd `  ( G `  n
) )  <  ( F `  n )
) )
537adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ran  ( 1st  o.  G )  C_  RR )
548adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ran  ( 1st  o.  G )  =/=  (/) )
5514adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  E. n  e.  RR  A. z  e. 
ran  ( 1st  o.  G ) z  <_  n )
56 nnnn0 10160 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  NN0 )
57 fvco3 5739 . . . . . . . . . . . . 13  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  =  ( 1st `  ( G `  n
) ) )
5820, 56, 57syl2an 464 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  G ) `
 n )  =  ( 1st `  ( G `  n )
) )
5920adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN )  ->  G : NN0
--> ( RR  X.  RR ) )
60 1stcof 6313 . . . . . . . . . . . . . 14  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
61 ffn 5531 . . . . . . . . . . . . . 14  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
6259, 60, 613syl 19 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st 
o.  G )  Fn 
NN0 )
6356adantl 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  n  e. 
NN0 )
64 fnfvelrn 5806 . . . . . . . . . . . . 13  |-  ( ( ( 1st  o.  G
)  Fn  NN0  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  e.  ran  ( 1st  o.  G ) )
6562, 63, 64syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  G ) `
 n )  e. 
ran  ( 1st  o.  G ) )
6658, 65eqeltrrd 2462 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  e.  ran  ( 1st  o.  G ) )
67 suprub 9901 . . . . . . . . . . 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 1187 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  <_  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  ) )
6968, 1syl6breqr 4193 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  <_  S
)
70 ffvelrn 5807 . . . . . . . . . . . 12  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( G `  n
)  e.  ( RR 
X.  RR ) )
7120, 56, 70syl2an 464 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  ( RR  X.  RR ) )
72 xp1st 6315 . . . . . . . . . . 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 452 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  S  e.  RR )
75 ltletr 9099 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( F `  n
)  <  ( 1st `  ( G `  n
) )  /\  ( 1st `  ( G `  n ) )  <_  S )  ->  ( F `  n )  <  S ) )
7769, 76mpan2d 656 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  ->  ( F `  n )  <  S
) )
78 fvco3 5739 . . . . . . . . . . . . . . 15  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  k  e.  NN0 )  -> 
( ( 1st  o.  G ) `  k
)  =  ( 1st `  ( G `  k
) ) )
7959, 78sylan 458 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  (
( 1st  o.  G
) `  k )  =  ( 1st `  ( G `  k )
) )
8059ffvelrnda 5809 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( G `  k )  e.  ( RR  X.  RR ) )
81 xp1st 6315 . . . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 2nd `  ( G `  n ) )  e.  RR )
8618adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  F : NN --> RR )
8719adantr 452 . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
8963adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  n  e.  NN0 )
9086, 87, 4, 5, 88, 89ruclem10 12765 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  < 
( 2nd `  ( G `  n )
) )
9182, 85, 90ltled 9153 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  <_ 
( 2nd `  ( G `  n )
) )
9279, 91eqbrtrd 4173 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  (
( 1st  o.  G
) `  k )  <_  ( 2nd `  ( G `  n )
) )
9392ralrimiva 2732 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A. k  e.  NN0  ( ( 1st 
o.  G ) `  k )  <_  ( 2nd `  ( G `  n ) ) )
94 breq1 4156 . . . . . . . . . . . . . 14  |-  ( z  =  ( ( 1st 
o.  G ) `  k )  ->  (
z  <_  ( 2nd `  ( G `  n
) )  <->  ( ( 1st  o.  G ) `  k )  <_  ( 2nd `  ( G `  n ) ) ) )
9594ralrn 5812 . . . . . . . . . . . . 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 224 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_ 
( 2nd `  ( G `  n )
) )
98 suprleub 9904 . . . . . . . . . . . 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 1187 . . . . . . . . . . 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 224 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  sup ( ran  ( 1st  o.  G
) ,  RR ,  <  )  <_  ( 2nd `  ( G `  n
) ) )
1011, 100syl5eqbr 4186 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  S  <_ 
( 2nd `  ( G `  n )
) )
102 lelttr 9098 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( S  <_  ( 2nd `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  < 
( F `  n
) )  ->  S  <  ( F `  n
) ) )
104101, 103mpand 657 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  <  ( F `  n )  ->  S  <  ( F `  n
) ) )
10577, 104orim12d 812 . . . . . . 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 9144 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  =/=  S  <->  ( ( F `  n )  <  S  \/  S  < 
( F `  n
) ) ) )
108106, 107mpbird 224 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =/= 
S )
109108neneqd 2566 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  -.  ( F `  n )  =  S )
110109nrexdv 2752 . . 3  |-  ( ph  ->  -.  E. n  e.  NN  ( F `  n )  =  S )
111 risset 2696 . . . 4  |-  ( S  e.  ran  F  <->  E. z  e.  ran  F  z  =  S )
112 ffn 5531 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
113 eqeq1 2393 . . . . . 6  |-  ( z  =  ( F `  n )  ->  (
z  =  S  <->  ( F `  n )  =  S ) )
114113rexrn 5811 . . . . 5  |-  ( F  Fn  NN  ->  ( E. z  e.  ran  F  z  =  S  <->  E. n  e.  NN  ( F `  n )  =  S ) )
1152, 112, 1143syl 19 . . . 4  |-  ( ph  ->  ( E. z  e. 
ran  F  z  =  S 
<->  E. n  e.  NN  ( F `  n )  =  S ) )
116111, 115syl5bb 249 . . 3  |-  ( ph  ->  ( S  e.  ran  F  <->  E. n  e.  NN  ( F `  n )  =  S ) )
117110, 116mtbird 293 . 2  |-  ( ph  ->  -.  S  e.  ran  F )
11817, 117eldifd 3274 1  |-  ( ph  ->  S  e.  ( RR 
\  ran  F )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   [_csb 3194    \ cdif 3260    u. cun 3261    C_ wss 3263   (/)c0 3571   ifcif 3682   {csn 3757   <.cop 3760   class class class wbr 4153    X. cxp 4816   ran crn 4819    o. ccom 4822    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287   supcsup 7380   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   2c2 9981   NN0cn0 10153    seq cseq 11250
This theorem is referenced by:  ruclem13  12768
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-seq 11251
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