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Theorem ruclem9 13849
Description: Lemma for ruc 13854. 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 5872 . . . . . . 7  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
32fveq2d 5876 . . . . . 6  |-  ( k  =  M  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  M )
) )
43breq2d 4465 . . . . 5  |-  ( k  =  M  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  M ) ) ) )
52fveq2d 5876 . . . . . 6  |-  ( k  =  M  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  M )
) )
65breq1d 4463 . . . . 5  |-  ( k  =  M  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  M
) )  <_  ( 2nd `  ( G `  M ) ) ) )
74, 6anbi12d 710 . . . 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 316 . . 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 5872 . . . . . . 7  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
109fveq2d 5876 . . . . . 6  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
1110breq2d 4465 . . . . 5  |-  ( k  =  n  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  n ) ) ) )
129fveq2d 5876 . . . . . 6  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
1312breq1d 4463 . . . . 5  |-  ( k  =  n  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  n
) )  <_  ( 2nd `  ( G `  M ) ) ) )
1411, 13anbi12d 710 . . . 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 316 . . 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 5872 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1716fveq2d 5876 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1817breq2d 4465 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) ) )
1916fveq2d 5876 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
2019breq1d 4463 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
2118, 20anbi12d 710 . . . 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 316 . . 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 5872 . . . . . . 7  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
2423fveq2d 5876 . . . . . 6  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
2524breq2d 4465 . . . . 5  |-  ( k  =  N  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  N ) ) ) )
2623fveq2d 5876 . . . . . 6  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
2726breq1d 4463 . . . . 5  |-  ( k  =  N  ->  (
( 2nd `  ( G `  k )
)  <_  ( 2nd `  ( G `  M
) )  <->  ( 2nd `  ( G `  N
) )  <_  ( 2nd `  ( G `  M ) ) ) )
2825, 27anbi12d 710 . . . 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 316 . . 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 13846 . . . . . . . 8  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
35 ruclem9.6 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
3634, 35ffvelrnd 6033 . . . . . . 7  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
37 xp1st 6825 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
3938leidd 10131 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) ) )
40 xp2nd 6826 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  M
) )  e.  RR )
4136, 40syl 16 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( G `  M )
)  e.  RR )
4241leidd 10131 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  M )
)  <_  ( 2nd `  ( G `  M
) ) )
4339, 42jca 532 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) )  /\  ( 2nd `  ( G `  M ) )  <_ 
( 2nd `  ( G `  M )
) ) )
4443a1i 11 . . 3  |-  ( M  e.  ZZ  ->  ( ph  ->  ( ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  M ) )  /\  ( 2nd `  ( G `
 M ) )  <_  ( 2nd `  ( G `  M )
) ) ) )
4530adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  F : NN
--> RR )
4631adantr 465 . . . . . . . . . 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
) >. ) ) )
4734adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  G : NN0
--> ( RR  X.  RR ) )
48 eluznn0 11163 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN0  /\  n  e.  ( ZZ>= `  M ) )  ->  n  e.  NN0 )
4935, 48sylan 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  n  e.  NN0 )
5047, 49ffvelrnd 6033 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  e.  ( RR  X.  RR ) )
51 xp1st 6825 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
5250, 51syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  e.  RR )
53 xp2nd 6826 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
5450, 53syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  n
) )  e.  RR )
55 nn0p1nn 10847 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
5649, 55syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e.  NN )
5745, 56ffvelrnd 6033 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  ( n  +  1 ) )  e.  RR )
58 eqid 2467 . . . . . . . . . 10  |-  ( 1st `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
59 eqid 2467 . . . . . . . . . 10  |-  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
6030, 31, 32, 33ruclem8 13848 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) )
6149, 60syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) )
6245, 46, 52, 54, 57, 58, 59, 61ruclem2 13843 . . . . . . . . 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
) ) ) )
6362simp1d 1008 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <_  ( 1st `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) ) )
6430, 31, 32, 33ruclem7 13847 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
6549, 64syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
66 1st2nd2 6832 . . . . . . . . . . . 12  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( G `
 n )  = 
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. )
6750, 66syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  =  <. ( 1st `  ( G `
 n ) ) ,  ( 2nd `  ( G `  n )
) >. )
6867oveq1d 6310 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( G `  n ) D ( F `  ( n  +  1
) ) )  =  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
6965, 68eqtrd 2508 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  =  (
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
7069fveq2d 5876 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
7163, 70breqtrrd 4479 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) )
7238adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  M
) )  e.  RR )
73 peano2nn0 10848 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e. 
NN0 )
7449, 73syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e. 
NN0 )
7547, 74ffvelrnd 6033 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR ) )
76 xp1st 6825 . . . . . . . . 9  |-  ( ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  e.  RR )
7775, 76syl 16 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  e.  RR )
78 letr 9690 . . . . . . . 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 ) ) ) ) )
7972, 52, 77, 78syl3anc 1228 . . . . . . 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 ) ) ) ) )
8071, 79mpan2d 674 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( 1st `  ( G `  M ) )  <_ 
( 1st `  ( G `  n )
)  ->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) ) )
8169fveq2d 5876 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
8262simp3d 1010 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  <_  ( 2nd `  ( G `  n ) ) )
8381, 82eqbrtrd 4473 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  n ) ) )
84 xp2nd 6826 . . . . . . . . 9  |-  ( ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  e.  RR )
8575, 84syl 16 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  e.  RR )
8641adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  M
) )  e.  RR )
87 letr 9690 . . . . . . . 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 )
) ) )
8885, 54, 86, 87syl3anc 1228 . . . . . . 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 )
) ) )
8983, 88mpand 675 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( 2nd `  ( G `  n ) )  <_ 
( 2nd `  ( G `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
9080, 89anim12d 563 . . . . 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 )
) ) ) )
9190expcom 435 . . . 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 )
) ) ) ) )
9291a2d 26 . . 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 ) ) ) ) ) )
938, 15, 22, 29, 44, 92uzind4 11151 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  N
) )  /\  ( 2nd `  ( G `  N ) )  <_ 
( 2nd `  ( G `  M )
) ) ) )
941, 93mpcom 36 1  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  N
) )  /\  ( 2nd `  ( G `  N ) )  <_ 
( 2nd `  ( G `  M )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   [_csb 3440    u. cun 3479   ifcif 3945   {csn 4033   <.cop 4039   class class class wbr 4453    X. cxp 5003   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    < clt 9640    <_ cle 9641    / cdiv 10218   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094    seqcseq 12087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-seq 12088
This theorem is referenced by:  ruclem10  13850
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