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Theorem ruclem12 12535
Description: Lemma for ruc 12537. 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 12534 . . . . 5  |-  ( ph  ->  ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1 ) )
76simp1d 967 . . . 4  |-  ( ph  ->  ran  ( 1st  o.  G )  C_  RR )
86simp2d 968 . . . 4  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
9 1re 8853 . . . . 5  |-  1  e.  RR
106simp3d 969 . . . . 5  |-  ( ph  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )
11 breq2 4043 . . . . . . 7  |-  ( n  =  1  ->  (
z  <_  n  <->  z  <_  1 ) )
1211ralbidv 2576 . . . . . 6  |-  ( n  =  1  ->  ( A. z  e.  ran  ( 1st  o.  G ) z  <_  n  <->  A. z  e.  ran  ( 1st  o.  G ) z  <_ 
1 ) )
1312rspcev 2897 . . . . 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 644 . . . 4  |-  ( ph  ->  E. n  e.  RR  A. z  e.  ran  ( 1st  o.  G ) z  <_  n )
15 suprcl 9730 . . . 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 1182 . . 3  |-  ( ph  ->  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  )  e.  RR )
171, 16syl5eqel 2380 . 2  |-  ( ph  ->  S  e.  RR )
182adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  F : NN
--> RR )
193adantr 451 . . . . . . . . 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 12529 . . . . . . . . . . 11  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
21 nnm1nn0 10021 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
22 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  ( n  -  1
)  e.  NN0 )  ->  ( G `  (
n  -  1 ) )  e.  ( RR 
X.  RR ) )
2320, 21, 22syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( n  - 
1 ) )  e.  ( RR  X.  RR ) )
24 xp1st 6165 . . . . . . . . . 10  |-  ( ( G `  ( n  -  1 ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  e.  RR )
2523, 24syl 15 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  e.  RR )
26 xp2nd 6166 . . . . . . . . . 10  |-  ( ( G `  ( n  -  1 ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  (
n  -  1 ) ) )  e.  RR )
2723, 26syl 15 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  (
n  -  1 ) ) )  e.  RR )
28 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( F : NN --> RR  /\  n  e.  NN )  ->  ( F `  n
)  e.  RR )
292, 28sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  e.  RR )
30 eqid 2296 . . . . . . . . 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 ) ) )
31 eqid 2296 . . . . . . . . 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 ) ) )
322, 3, 4, 5ruclem8 12531 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  -  1 )  e. 
NN0 )  ->  ( 1st `  ( G `  ( n  -  1
) ) )  < 
( 2nd `  ( G `  ( n  -  1 ) ) ) )
3321, 32sylan2 460 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  (
n  -  1 ) ) )  <  ( 2nd `  ( G `  ( n  -  1
) ) ) )
3418, 19, 25, 27, 29, 30, 31, 33ruclem3 12527 . . . . . . . 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 ) ) )
352, 3, 4, 5ruclem7 12530 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  -  1 )  e. 
NN0 )  ->  ( G `  ( (
n  -  1 )  +  1 ) )  =  ( ( G `
 ( n  - 
1 ) ) D ( F `  (
( n  -  1 )  +  1 ) ) ) )
3621, 35sylan2 460 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( ( n  -  1 )  +  1 ) )  =  ( ( G `  ( n  -  1
) ) D ( F `  ( ( n  -  1 )  +  1 ) ) ) )
37 nncn 9770 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  CC )
3837adantl 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  CC )
39 ax-1cn 8811 . . . . . . . . . . . . . 14  |-  1  e.  CC
40 npcan 9076 . . . . . . . . . . . . . 14  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
4138, 39, 40sylancl 643 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( n  -  1 )  +  1 )  =  n )
4241fveq2d 5545 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( ( n  -  1 )  +  1 ) )  =  ( G `  n
) )
43 1st2nd2 6175 . . . . . . . . . . . . . 14  |-  ( ( G `  ( n  -  1 ) )  e.  ( RR  X.  RR )  ->  ( G `
 ( n  - 
1 ) )  = 
<. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. )
4423, 43syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 ( n  - 
1 ) )  = 
<. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. )
4541fveq2d 5545 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 ( ( n  -  1 )  +  1 ) )  =  ( F `  n
) )
4644, 45oveq12d 5892 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( G `  ( n  -  1 ) ) D ( F `  ( ( n  - 
1 )  +  1 ) ) )  =  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
4736, 42, 463eqtr3d 2336 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )
4847fveq2d 5545 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  =  ( 1st `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) ) )
4948breq2d 4051 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  <->  ( F `  n )  <  ( 1st `  ( <. ( 1st `  ( G `  ( n  -  1
) ) ) ,  ( 2nd `  ( G `  ( n  -  1 ) ) ) >. D ( F `
 n ) ) ) ) )
5047fveq2d 5545 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n
) )  =  ( 2nd `  ( <.
( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) ) )
5150breq1d 4049 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  <  ( F `  n )  <->  ( 2nd `  ( <. ( 1st `  ( G `  ( n  -  1 ) ) ) ,  ( 2nd `  ( G `  (
n  -  1 ) ) ) >. D ( F `  n ) ) )  <  ( F `  n )
) )
5249, 51orbi12d 690 . . . . . . . 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 ) ) ) )
5334, 52mpbird 223 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  \/  ( 2nd `  ( G `  n
) )  <  ( F `  n )
) )
547adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ran  ( 1st  o.  G )  C_  RR )
558adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ran  ( 1st  o.  G )  =/=  (/) )
5614adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  E. n  e.  RR  A. z  e. 
ran  ( 1st  o.  G ) z  <_  n )
57 nnnn0 9988 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  NN0 )
58 fvco3 5612 . . . . . . . . . . . . 13  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  =  ( 1st `  ( G `  n
) ) )
5920, 57, 58syl2an 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  G ) `
 n )  =  ( 1st `  ( G `  n )
) )
6020adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN )  ->  G : NN0
--> ( RR  X.  RR ) )
61 1stcof 6163 . . . . . . . . . . . . . 14  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
62 ffn 5405 . . . . . . . . . . . . . 14  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
6360, 61, 623syl 18 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st 
o.  G )  Fn 
NN0 )
6457adantl 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  n  e. 
NN0 )
65 fnfvelrn 5678 . . . . . . . . . . . . 13  |-  ( ( ( 1st  o.  G
)  Fn  NN0  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  e.  ran  ( 1st  o.  G ) )
6663, 64, 65syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 1st  o.  G ) `
 n )  e. 
ran  ( 1st  o.  G ) )
6759, 66eqeltrrd 2371 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  e.  ran  ( 1st  o.  G ) )
68 suprub 9731 . . . . . . . . . . 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 ,  <  ) )
6954, 55, 56, 67, 68syl31anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  <_  sup ( ran  ( 1st  o.  G ) ,  RR ,  <  ) )
7069, 1syl6breqr 4079 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  <_  S
)
71 ffvelrn 5679 . . . . . . . . . . . 12  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( G `  n
)  e.  ( RR 
X.  RR ) )
7220, 57, 71syl2an 463 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  ( RR  X.  RR ) )
73 xp1st 6165 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
7472, 73syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( 1st `  ( G `  n
) )  e.  RR )
7517adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  S  e.  RR )
76 ltletr 8929 . . . . . . . . . 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 )
)
7729, 74, 75, 76syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( F `  n
)  <  ( 1st `  ( G `  n
) )  /\  ( 1st `  ( G `  n ) )  <_  S )  ->  ( F `  n )  <  S ) )
7870, 77mpan2d 655 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  ( 1st `  ( G `  n )
)  ->  ( F `  n )  <  S
) )
79 fvco3 5612 . . . . . . . . . . . . . . 15  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  k  e.  NN0 )  -> 
( ( 1st  o.  G ) `  k
)  =  ( 1st `  ( G `  k
) ) )
8060, 79sylan 457 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  (
( 1st  o.  G
) `  k )  =  ( 1st `  ( G `  k )
) )
81 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  k  e.  NN0 )  -> 
( G `  k
)  e.  ( RR 
X.  RR ) )
8260, 81sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( G `  k )  e.  ( RR  X.  RR ) )
83 xp1st 6165 . . . . . . . . . . . . . . . 16  |-  ( ( G `  k )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  k
) )  e.  RR )
8482, 83syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  e.  RR )
85 xp2nd 6166 . . . . . . . . . . . . . . . . 17  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
8672, 85syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  NN )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
8786adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 2nd `  ( G `  n ) )  e.  RR )
8818adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  F : NN --> RR )
8919adantr 451 . . . . . . . . . . . . . . . 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
) >. ) ) )
90 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
9164adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  n  e.  NN0 )
9288, 89, 4, 5, 90, 91ruclem10 12533 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  < 
( 2nd `  ( G `  n )
) )
9384, 87, 92ltled 8983 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  ( 1st `  ( G `  k ) )  <_ 
( 2nd `  ( G `  n )
) )
9480, 93eqbrtrd 4059 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN )  /\  k  e.  NN0 )  ->  (
( 1st  o.  G
) `  k )  <_  ( 2nd `  ( G `  n )
) )
9594ralrimiva 2639 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A. k  e.  NN0  ( ( 1st 
o.  G ) `  k )  <_  ( 2nd `  ( G `  n ) ) )
96 breq1 4042 . . . . . . . . . . . . . 14  |-  ( z  =  ( ( 1st 
o.  G ) `  k )  ->  (
z  <_  ( 2nd `  ( G `  n
) )  <->  ( ( 1st  o.  G ) `  k )  <_  ( 2nd `  ( G `  n ) ) ) )
9796ralrn 5684 . . . . . . . . . . . . 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
) ) ) )
9863, 97syl 15 . . . . . . . . . . . 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
) ) ) )
9995, 98mpbird 223 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_ 
( 2nd `  ( G `  n )
) )
100 suprleub 9734 . . . . . . . . . . . 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
) ) ) )
10154, 55, 56, 86, 100syl31anc 1185 . . . . . . . . . . 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
) ) ) )
10299, 101mpbird 223 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  sup ( ran  ( 1st  o.  G
) ,  RR ,  <  )  <_  ( 2nd `  ( G `  n
) ) )
1031, 102syl5eqbr 4072 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  S  <_ 
( 2nd `  ( G `  n )
) )
104 lelttr 8928 . . . . . . . . . 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 ) ) )
10575, 86, 29, 104syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( S  <_  ( 2nd `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  < 
( F `  n
) )  ->  S  <  ( F `  n
) ) )
106103, 105mpand 656 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( 2nd `  ( G `
 n ) )  <  ( F `  n )  ->  S  <  ( F `  n
) ) )
10778, 106orim12d 811 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( F `  n
)  <  ( 1st `  ( G `  n
) )  \/  ( 2nd `  ( G `  n ) )  < 
( F `  n
) )  ->  (
( F `  n
)  <  S  \/  S  <  ( F `  n ) ) ) )
10853, 107mpd 14 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  <  S  \/  S  <  ( F `  n
) ) )
10929, 75lttri2d 8974 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( F `  n )  =/=  S  <->  ( ( F `  n )  <  S  \/  S  < 
( F `  n
) ) ) )
110108, 109mpbird 223 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =/= 
S )
111110neneqd 2475 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  -.  ( F `  n )  =  S )
112111nrexdv 2659 . . 3  |-  ( ph  ->  -.  E. n  e.  NN  ( F `  n )  =  S )
113 risset 2603 . . . 4  |-  ( S  e.  ran  F  <->  E. z  e.  ran  F  z  =  S )
114 ffn 5405 . . . . 5  |-  ( F : NN --> RR  ->  F  Fn  NN )
115 eqeq1 2302 . . . . . 6  |-  ( z  =  ( F `  n )  ->  (
z  =  S  <->  ( F `  n )  =  S ) )
116115rexrn 5683 . . . . 5  |-  ( F  Fn  NN  ->  ( E. z  e.  ran  F  z  =  S  <->  E. n  e.  NN  ( F `  n )  =  S ) )
1172, 114, 1163syl 18 . . . 4  |-  ( ph  ->  ( E. z  e. 
ran  F  z  =  S 
<->  E. n  e.  NN  ( F `  n )  =  S ) )
118113, 117syl5bb 248 . . 3  |-  ( ph  ->  ( S  e.  ran  F  <->  E. n  e.  NN  ( F `  n )  =  S ) )
119112, 118mtbird 292 . 2  |-  ( ph  ->  -.  S  e.  ran  F )
120 eldif 3175 . 2  |-  ( S  e.  ( RR  \  ran  F )  <->  ( S  e.  RR  /\  -.  S  e.  ran  F ) )
12117, 119, 120sylanbrc 645 1  |-  ( ph  ->  S  e.  ( RR 
\  ran  F )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   [_csb 3094    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981    seq cseq 11062
This theorem is referenced by:  ruclem13  12536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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