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Theorem ruclem9 12516
Description: Lemma for ruc 12521. The first components of the  G sequence are increasing, and the second components are decreasing. (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 )
ruclem9.6  |-  ( ph  ->  M  e.  NN0 )
ruclem9.7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
Assertion
Ref Expression
ruclem9  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  N
) )  /\  ( 2nd `  ( G `  N ) )  <_ 
( 2nd `  ( G `  M )
) ) )
Distinct variable groups:    x, m, y, F    m, G, x, y    m, M, x, y    m, N, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem9
Dummy variables  n  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruclem9.7 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 fveq2 5525 . . . . . . 7  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
32fveq2d 5529 . . . . . 6  |-  ( k  =  M  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  M )
) )
43breq2d 4035 . . . . 5  |-  ( k  =  M  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  M ) ) ) )
52fveq2d 5529 . . . . . 6  |-  ( k  =  M  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  M )
) )
65breq1d 4033 . . . . 5  |-  ( k  =  M  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  M
) )  <_  ( 2nd `  ( G `  M ) ) ) )
74, 6anbi12d 691 . . . 4  |-  ( k  =  M  ->  (
( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  /\  ( 2nd `  ( G `  k ) )  <_ 
( 2nd `  ( G `  M )
) )  <->  ( ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  M )
)  /\  ( 2nd `  ( G `  M
) )  <_  ( 2nd `  ( G `  M ) ) ) ) )
87imbi2d 307 . . 3  |-  ( k  =  M  ->  (
( ph  ->  ( ( 1st `  ( G `
 M ) )  <_  ( 1st `  ( G `  k )
)  /\  ( 2nd `  ( G `  k
) )  <_  ( 2nd `  ( G `  M ) ) ) )  <->  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) )  /\  ( 2nd `  ( G `  M ) )  <_ 
( 2nd `  ( G `  M )
) ) ) ) )
9 fveq2 5525 . . . . . . 7  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
109fveq2d 5529 . . . . . 6  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
1110breq2d 4035 . . . . 5  |-  ( k  =  n  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  n ) ) ) )
129fveq2d 5529 . . . . . 6  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
1312breq1d 4033 . . . . 5  |-  ( k  =  n  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  n
) )  <_  ( 2nd `  ( G `  M ) ) ) )
1411, 13anbi12d 691 . . . 4  |-  ( k  =  n  ->  (
( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  /\  ( 2nd `  ( G `  k ) )  <_ 
( 2nd `  ( G `  M )
) )  <->  ( ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  n )
)  /\  ( 2nd `  ( G `  n
) )  <_  ( 2nd `  ( G `  M ) ) ) ) )
1514imbi2d 307 . . 3  |-  ( k  =  n  ->  (
( ph  ->  ( ( 1st `  ( G `
 M ) )  <_  ( 1st `  ( G `  k )
)  /\  ( 2nd `  ( G `  k
) )  <_  ( 2nd `  ( G `  M ) ) ) )  <->  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  <_ 
( 2nd `  ( G `  M )
) ) ) ) )
16 fveq2 5525 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1716fveq2d 5529 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1817breq2d 4035 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) ) )
1916fveq2d 5529 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
2019breq1d 4033 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
2118, 20anbi12d 691 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  /\  ( 2nd `  ( G `  k ) )  <_ 
( 2nd `  ( G `  M )
) )  <->  ( ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  ( n  +  1 ) ) )  /\  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  M ) ) ) ) )
2221imbi2d 307 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( ( 1st `  ( G `
 M ) )  <_  ( 1st `  ( G `  k )
)  /\  ( 2nd `  ( G `  k
) )  <_  ( 2nd `  ( G `  M ) ) ) )  <->  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  (
n  +  1 ) ) )  /\  ( 2nd `  ( G `  ( n  +  1
) ) )  <_ 
( 2nd `  ( G `  M )
) ) ) ) )
23 fveq2 5525 . . . . . . 7  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
2423fveq2d 5529 . . . . . 6  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
2524breq2d 4035 . . . . 5  |-  ( k  =  N  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  N ) ) ) )
2623fveq2d 5529 . . . . . 6  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
2726breq1d 4033 . . . . 5  |-  ( k  =  N  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  N
) )  <_  ( 2nd `  ( G `  M ) ) ) )
2825, 27anbi12d 691 . . . 4  |-  ( k  =  N  ->  (
( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  /\  ( 2nd `  ( G `  k ) )  <_ 
( 2nd `  ( G `  M )
) )  <->  ( ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  N )
)  /\  ( 2nd `  ( G `  N
) )  <_  ( 2nd `  ( G `  M ) ) ) ) )
2928imbi2d 307 . . 3  |-  ( k  =  N  ->  (
( ph  ->  ( ( 1st `  ( G `
 M ) )  <_  ( 1st `  ( G `  k )
)  /\  ( 2nd `  ( G `  k
) )  <_  ( 2nd `  ( G `  M ) ) ) )  <->  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  N
) )  /\  ( 2nd `  ( G `  N ) )  <_ 
( 2nd `  ( G `  M )
) ) ) ) )
30 ruc.1 . . . . . . . . 9  |-  ( ph  ->  F : NN --> RR )
31 ruc.2 . . . . . . . . 9  |-  ( 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
) >. ) ) )
32 ruc.4 . . . . . . . . 9  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
33 ruc.5 . . . . . . . . 9  |-  G  =  seq  0 ( D ,  C )
3430, 31, 32, 33ruclem6 12513 . . . . . . . 8  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
35 ruclem9.6 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
36 ffvelrn 5663 . . . . . . . 8  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  M  e.  NN0 )  -> 
( G `  M
)  e.  ( RR 
X.  RR ) )
3734, 35, 36syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
38 xp1st 6149 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
3937, 38syl 15 . . . . . 6  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
4039leidd 9339 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) ) )
41 xp2nd 6150 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  M
) )  e.  RR )
4237, 41syl 15 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( G `  M )
)  e.  RR )
4342leidd 9339 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  M )
)  <_  ( 2nd `  ( G `  M
) ) )
4440, 43jca 518 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) )  /\  ( 2nd `  ( G `  M ) )  <_ 
( 2nd `  ( G `  M )
) ) )
4544a1i 10 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  M ) )  /\  ( 2nd `  ( G `
 M ) )  <_  ( 2nd `  ( G `  M )
) ) ) )
4630adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  F : NN
--> RR )
4731adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  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
) >. ) ) )
4834adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  G : NN0
--> ( RR  X.  RR ) )
49 eluznn0 10288 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  n  e.  ( ZZ>= `  M ) )  ->  n  e.  NN0 )
5035, 49sylan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  NN0 )
51 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( G `  n
)  e.  ( RR 
X.  RR ) )
5248, 50, 51syl2anc 642 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  e.  ( RR  X.  RR ) )
53 xp1st 6149 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
5452, 53syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  e.  RR )
55 xp2nd 6150 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
5652, 55syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  n
) )  e.  RR )
57 nn0p1nn 10003 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
5850, 57syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e.  NN )
59 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : NN --> RR  /\  ( n  +  1
)  e.  NN )  ->  ( F `  ( n  +  1
) )  e.  RR )
6046, 58, 59syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  ( n  +  1 ) )  e.  RR )
61 eqid 2283 . . . . . . . . . 10  |-  ( 1st `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
62 eqid 2283 . . . . . . . . . 10  |-  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
6330, 31, 32, 33ruclem8 12515 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) )
6450, 63syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) )
6546, 47, 54, 56, 60, 61, 62, 64ruclem2 12510 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( 1st `  ( G `  n ) )  <_ 
( 1st `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  /\  ( 1st `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) )  <  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  /\  ( 2nd `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) )  <_  ( 2nd `  ( G `  n
) ) ) )
6665simp1d 967 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <_  ( 1st `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) ) )
6730, 31, 32, 33ruclem7 12514 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
6850, 67syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
69 1st2nd2 6159 . . . . . . . . . . . 12  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( G `
 n )  = 
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. )
7052, 69syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  =  <. ( 1st `  ( G `
 n ) ) ,  ( 2nd `  ( G `  n )
) >. )
7170oveq1d 5873 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( G `  n ) D ( F `  ( n  +  1
) ) )  =  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
7268, 71eqtrd 2315 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  =  (
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
7372fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
7466, 73breqtrrd 4049 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) )
7539adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  M
) )  e.  RR )
76 peano2nn0 10004 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e. 
NN0 )
7750, 76syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e. 
NN0 )
78 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  ( n  +  1
)  e.  NN0 )  ->  ( G `  (
n  +  1 ) )  e.  ( RR 
X.  RR ) )
7948, 77, 78syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR ) )
80 xp1st 6149 . . . . . . . . 9  |-  ( ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  e.  RR )
8179, 80syl 15 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  e.  RR )
82 letr 8914 . . . . . . . 8  |-  ( ( ( 1st `  ( G `  M )
)  e.  RR  /\  ( 1st `  ( G `
 n ) )  e.  RR  /\  ( 1st `  ( G `  ( n  +  1
) ) )  e.  RR )  ->  (
( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  n
) )  /\  ( 1st `  ( G `  n ) )  <_ 
( 1st `  ( G `  ( n  +  1 ) ) ) )  ->  ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  ( n  +  1 ) ) ) ) )
8375, 54, 81, 82syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  n
) )  /\  ( 1st `  ( G `  n ) )  <_ 
( 1st `  ( G `  ( n  +  1 ) ) ) )  ->  ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  ( n  +  1 ) ) ) ) )
8474, 83mpan2d 655 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  n )
)  ->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) ) )
8572fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
8665simp3d 969 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  <_  ( 2nd `  ( G `  n ) ) )
8785, 86eqbrtrd 4043 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  n ) ) )
88 xp2nd 6150 . . . . . . . . 9  |-  ( ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  e.  RR )
8979, 88syl 15 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  e.  RR )
9042adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  M
) )  e.  RR )
91 letr 8914 . . . . . . . 8  |-  ( ( ( 2nd `  ( G `  ( n  +  1 ) ) )  e.  RR  /\  ( 2nd `  ( G `
 n ) )  e.  RR  /\  ( 2nd `  ( G `  M ) )  e.  RR )  ->  (
( ( 2nd `  ( G `  ( n  +  1 ) ) )  <_  ( 2nd `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  <_ 
( 2nd `  ( G `  M )
) )  ->  ( 2nd `  ( G `  ( n  +  1
) ) )  <_ 
( 2nd `  ( G `  M )
) ) )
9289, 56, 90, 91syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( (
( 2nd `  ( G `  ( n  +  1 ) ) )  <_  ( 2nd `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  <_ 
( 2nd `  ( G `  M )
) )  ->  ( 2nd `  ( G `  ( n  +  1
) ) )  <_ 
( 2nd `  ( G `  M )
) ) )
9387, 92mpand 656 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( 2nd `  ( G `  n ) )  <_ 
( 2nd `  ( G `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
9484, 93anim12d 546 . . . . 5  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  n
) )  /\  ( 2nd `  ( G `  n ) )  <_ 
( 2nd `  ( G `  M )
) )  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  (
n  +  1 ) ) )  /\  ( 2nd `  ( G `  ( n  +  1
) ) )  <_ 
( 2nd `  ( G `  M )
) ) ) )
9594expcom 424 . . . 4  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( ( ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  n ) )  /\  ( 2nd `  ( G `
 n ) )  <_  ( 2nd `  ( G `  M )
) )  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  (
n  +  1 ) ) )  /\  ( 2nd `  ( G `  ( n  +  1
) ) )  <_ 
( 2nd `  ( G `  M )
) ) ) ) )
9695a2d 23 . . 3  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ( ph  ->  ( ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  n ) )  /\  ( 2nd `  ( G `
 n ) )  <_  ( 2nd `  ( G `  M )
) ) )  -> 
( ph  ->  ( ( 1st `  ( G `
 M ) )  <_  ( 1st `  ( G `  ( n  +  1 ) ) )  /\  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  M ) ) ) ) ) )
978, 15, 22, 29, 45, 96uzind4 10276 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  N
) )  /\  ( 2nd `  ( G `  N ) )  <_ 
( 2nd `  ( G `  M )
) ) ) )
981, 97mpcom 32 1  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  N
) )  /\  ( 2nd `  ( G `  N ) )  <_ 
( 2nd `  ( G `  M )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   [_csb 3081    u. cun 3150   ifcif 3565   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046
This theorem is referenced by:  ruclem10  12517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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