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Theorem ruclem9 12532
Description: Lemma for ruc 12537. 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 5541 . . . . . . 7  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
32fveq2d 5545 . . . . . 6  |-  ( k  =  M  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  M )
) )
43breq2d 4051 . . . . 5  |-  ( k  =  M  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  M ) ) ) )
52fveq2d 5545 . . . . . 6  |-  ( k  =  M  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  M )
) )
65breq1d 4049 . . . . 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 5541 . . . . . . 7  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
109fveq2d 5545 . . . . . 6  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
1110breq2d 4051 . . . . 5  |-  ( k  =  n  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  n ) ) ) )
129fveq2d 5545 . . . . . 6  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
1312breq1d 4049 . . . . 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 5541 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1716fveq2d 5545 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1817breq2d 4051 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) ) )
1916fveq2d 5545 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
2019breq1d 4049 . . . . 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 5541 . . . . . . 7  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
2423fveq2d 5545 . . . . . 6  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
2524breq2d 4051 . . . . 5  |-  ( k  =  N  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  N ) ) ) )
2623fveq2d 5545 . . . . . 6  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
2726breq1d 4049 . . . . 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 12529 . . . . . . . 8  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
35 ruclem9.6 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
36 ffvelrn 5679 . . . . . . . 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 6165 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
3937, 38syl 15 . . . . . 6  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
4039leidd 9355 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) ) )
41 xp2nd 6166 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  M
) )  e.  RR )
4237, 41syl 15 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( G `  M )
)  e.  RR )
4342leidd 9355 . . . . 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 10304 . . . . . . . . . . . . 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 5679 . . . . . . . . . . . 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 6165 . . . . . . . . . . 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 6166 . . . . . . . . . . 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 10019 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
5850, 57syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e.  NN )
59 ffvelrn 5679 . . . . . . . . . . 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 2296 . . . . . . . . . 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 2296 . . . . . . . . . 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 12531 . . . . . . . . . . 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 12526 . . . . . . . . 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 12530 . . . . . . . . . . 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 6175 . . . . . . . . . . . 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 5889 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( G `  n ) D ( F `  ( n  +  1
) ) )  =  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
7268, 71eqtrd 2328 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  =  (
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
7372fveq2d 5545 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
7466, 73breqtrrd 4065 . . . . . . 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 10020 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e. 
NN0 )
7750, 76syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e. 
NN0 )
78 ffvelrn 5679 . . . . . . . . . 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 6165 . . . . . . . . 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 8930 . . . . . . . 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 5545 . . . . . . . 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 4059 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  n ) ) )
88 xp2nd 6166 . . . . . . . . 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 8930 . . . . . . . 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 10292 . 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 1632    e. wcel 1696   [_csb 3094    u. cun 3163   ifcif 3578   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062
This theorem is referenced by:  ruclem10  12533
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
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-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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