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Theorem ruclem9 13642
Description: Lemma for ruc 13647. 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 5802 . . . . . . 7  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
32fveq2d 5806 . . . . . 6  |-  ( k  =  M  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  M )
) )
43breq2d 4415 . . . . 5  |-  ( k  =  M  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  M ) ) ) )
52fveq2d 5806 . . . . . 6  |-  ( k  =  M  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  M )
) )
65breq1d 4413 . . . . 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 5802 . . . . . . 7  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
109fveq2d 5806 . . . . . 6  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
1110breq2d 4415 . . . . 5  |-  ( k  =  n  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  n ) ) ) )
129fveq2d 5806 . . . . . 6  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
1312breq1d 4413 . . . . 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 5802 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1716fveq2d 5806 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1817breq2d 4415 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  ( n  +  1
) ) ) ) )
1916fveq2d 5806 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
2019breq1d 4413 . . . . 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 5802 . . . . . . 7  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
2423fveq2d 5806 . . . . . 6  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
2524breq2d 4415 . . . . 5  |-  ( k  =  N  ->  (
( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  k
) )  <->  ( 1st `  ( G `  M
) )  <_  ( 1st `  ( G `  N ) ) ) )
2623fveq2d 5806 . . . . . 6  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
2726breq1d 4413 . . . . 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 13639 . . . . . . . 8  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
35 ruclem9.6 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
3634, 35ffvelrnd 5956 . . . . . . 7  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
37 xp1st 6719 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
3938leidd 10021 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  M
) ) )
40 xp2nd 6720 . . . . . . 7  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  M
) )  e.  RR )
4136, 40syl 16 . . . . . 6  |-  ( ph  ->  ( 2nd `  ( G `  M )
)  e.  RR )
4241leidd 10021 . . . . 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 11039 . . . . . . . . . . . . 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 5956 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  n )  e.  ( RR  X.  RR ) )
51 xp1st 6719 . . . . . . . . . . 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 6720 . . . . . . . . . . 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 10734 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
5649, 55syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e.  NN )
5745, 56ffvelrnd 5956 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( F `  ( n  +  1 ) )  e.  RR )
58 eqid 2454 . . . . . . . . . 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 2454 . . . . . . . . . 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 13641 . . . . . . . . . . 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 13636 . . . . . . . . 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 1000 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  n
) )  <_  ( 1st `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) ) )
6430, 31, 32, 33ruclem7 13640 . . . . . . . . . . 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 6726 . . . . . . . . . . . 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 6218 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( ( G `  n ) D ( F `  ( n  +  1
) ) )  =  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
6965, 68eqtrd 2495 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  =  (
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
7069fveq2d 5806 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
7163, 70breqtrrd 4429 . . . . . . 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 10735 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e. 
NN0 )
7449, 73syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( n  +  1 )  e. 
NN0 )
7547, 74ffvelrnd 5956 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( G `  ( n  +  1 ) )  e.  ( RR  X.  RR ) )
76 xp1st 6719 . . . . . . . . 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 9583 . . . . . . . 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 1219 . . . . . . 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 5806 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
8262simp3d 1002 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  <_  ( 2nd `  ( G `  n ) ) )
8381, 82eqbrtrd 4423 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  <_  ( 2nd `  ( G `  n ) ) )
84 xp2nd 6720 . . . . . . . . 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 9583 . . . . . . . 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 1219 . . . . . . 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 11027 . 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 1370    e. wcel 1758   [_csb 3398    u. cun 3437   ifcif 3902   {csn 3988   <.cop 3994   class class class wbr 4403    X. cxp 4949   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   RRcr 9396   0cc0 9397   1c1 9398    + caddc 9400    < clt 9533    <_ cle 9534    / cdiv 10108   NNcn 10437   2c2 10486   NN0cn0 10694   ZZcz 10761   ZZ>=cuz 10976    seqcseq 11927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-seq 11928
This theorem is referenced by:  ruclem10  13643
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