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Theorem monoord2 12117
Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
Hypotheses
Ref Expression
monoord2.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
monoord2.2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  RR )
monoord2.3  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k )
)
Assertion
Ref Expression
monoord2  |-  ( ph  ->  ( F `  N
)  <_  ( F `  M ) )
Distinct variable groups:    k, F    k, M    k, N    ph, k

Proof of Theorem monoord2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 monoord2.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 monoord2.2 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  RR )
32renegcld 9992 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  -u ( F `
 k )  e.  RR )
4 eqid 2443 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)  =  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)
53, 4fmptd 6040 . . . . 5  |-  ( ph  ->  ( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) : ( M ... N
) --> RR )
65ffvelrnda 6016 . . . 4  |-  ( (
ph  /\  n  e.  ( M ... N ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  n )  e.  RR )
7 monoord2.3 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( k  +  1 ) )  <_  ( F `  k )
)
87ralrimiva 2857 . . . . . . . 8  |-  ( ph  ->  A. k  e.  ( M ... ( N  -  1 ) ) ( F `  (
k  +  1 ) )  <_  ( F `  k ) )
9 oveq1 6288 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
109fveq2d 5860 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
11 fveq2 5856 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
1210, 11breq12d 4450 . . . . . . . . 9  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
1312cbvralv 3070 . . . . . . . 8  |-  ( A. k  e.  ( M ... ( N  -  1 ) ) ( F `
 ( k  +  1 ) )  <_ 
( F `  k
)  <->  A. n  e.  ( M ... ( N  -  1 ) ) ( F `  (
n  +  1 ) )  <_  ( F `  n ) )
148, 13sylib 196 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( M ... ( N  -  1 ) ) ( F `  (
n  +  1 ) )  <_  ( F `  n ) )
1514r19.21bi 2812 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
)
16 fzp1elp1 11742 . . . . . . . . . 10  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  (
n  +  1 )  e.  ( M ... ( ( N  - 
1 )  +  1 ) ) )
1716adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... (
( N  -  1 )  +  1 ) ) )
18 eluzelz 11099 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
191, 18syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
2019zcnd 10975 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
21 ax-1cn 9553 . . . . . . . . . . . 12  |-  1  e.  CC
22 npcan 9834 . . . . . . . . . . . 12  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2320, 21, 22sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2423oveq2d 6297 . . . . . . . . . 10  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
2524adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( M ... ( ( N  - 
1 )  +  1 ) )  =  ( M ... N ) )
2617, 25eleqtrd 2533 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... N
) )
272ralrimiva 2857 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  ( M ... N ) ( F `  k
)  e.  RR )
2827adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A. k  e.  ( M ... N
) ( F `  k )  e.  RR )
29 fveq2 5856 . . . . . . . . . 10  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
3029eleq1d 2512 . . . . . . . . 9  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( n  +  1 ) )  e.  RR ) )
3130rspcv 3192 . . . . . . . 8  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( F `  k )  e.  RR  ->  ( F `  ( n  +  1 ) )  e.  RR ) )
3226, 28, 31sylc 60 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
33 fzssp1 11735 . . . . . . . . . 10  |-  ( M ... ( N  - 
1 ) )  C_  ( M ... ( ( N  -  1 )  +  1 ) )
3433, 24syl5sseq 3537 . . . . . . . . 9  |-  ( ph  ->  ( M ... ( N  -  1 ) )  C_  ( M ... N ) )
3534sselda 3489 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( M ... N ) )
3611eleq1d 2512 . . . . . . . . 9  |-  ( k  =  n  ->  (
( F `  k
)  e.  RR  <->  ( F `  n )  e.  RR ) )
3736rspcv 3192 . . . . . . . 8  |-  ( n  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( F `  k )  e.  RR  ->  ( F `  n )  e.  RR ) )
3835, 28, 37sylc 60 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  e.  RR )
3932, 38lenegd 10137 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( F `  ( n  +  1 ) )  <_  ( F `  n )  <->  -u ( F `
 n )  <_  -u ( F `  (
n  +  1 ) ) ) )
4015, 39mpbid 210 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 n )  <_  -u ( F `  (
n  +  1 ) ) )
4111negeqd 9819 . . . . . . 7  |-  ( k  =  n  ->  -u ( F `  k )  =  -u ( F `  n ) )
42 negex 9823 . . . . . . 7  |-  -u ( F `  n )  e.  _V
4341, 4, 42fvmpt 5941 . . . . . 6  |-  ( n  e.  ( M ... N )  ->  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  n )  =  -u ( F `  n ) )
4435, 43syl 16 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  n )  =  -u ( F `  n ) )
4529negeqd 9819 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  -u ( F `  k )  =  -u ( F `  ( n  +  1
) ) )
46 negex 9823 . . . . . . 7  |-  -u ( F `  ( n  +  1 ) )  e.  _V
4745, 4, 46fvmpt 5941 . . . . . 6  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  ( n  +  1
) )  =  -u ( F `  ( n  +  1 ) ) )
4826, 47syl 16 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  ( n  +  1 ) )  =  -u ( F `  ( n  +  1
) ) )
4940, 44, 483brtr4d 4467 . . . 4  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  n )  <_  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  ( n  +  1 ) ) )
501, 6, 49monoord 12116 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  <_  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  N ) )
51 eluzfz1 11702 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
521, 51syl 16 . . . 4  |-  ( ph  ->  M  e.  ( M ... N ) )
53 fveq2 5856 . . . . . 6  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
5453negeqd 9819 . . . . 5  |-  ( k  =  M  ->  -u ( F `  k )  =  -u ( F `  M ) )
55 negex 9823 . . . . 5  |-  -u ( F `  M )  e.  _V
5654, 4, 55fvmpt 5941 . . . 4  |-  ( M  e.  ( M ... N )  ->  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  M )  =  -u ( F `  M ) )
5752, 56syl 16 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  =  -u ( F `  M ) )
58 eluzfz2 11703 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
591, 58syl 16 . . . 4  |-  ( ph  ->  N  e.  ( M ... N ) )
60 fveq2 5856 . . . . . 6  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
6160negeqd 9819 . . . . 5  |-  ( k  =  N  ->  -u ( F `  k )  =  -u ( F `  N ) )
62 negex 9823 . . . . 5  |-  -u ( F `  N )  e.  _V
6361, 4, 62fvmpt 5941 . . . 4  |-  ( N  e.  ( M ... N )  ->  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  N )  =  -u ( F `  N ) )
6459, 63syl 16 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  N )  =  -u ( F `  N ) )
6550, 57, 643brtr3d 4466 . 2  |-  ( ph  -> 
-u ( F `  M )  <_  -u ( F `  N )
)
6660eleq1d 2512 . . . . 5  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
6766rspcv 3192 . . . 4  |-  ( N  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( F `  k )  e.  RR  ->  ( F `  N )  e.  RR ) )
6859, 27, 67sylc 60 . . 3  |-  ( ph  ->  ( F `  N
)  e.  RR )
6953eleq1d 2512 . . . . 5  |-  ( k  =  M  ->  (
( F `  k
)  e.  RR  <->  ( F `  M )  e.  RR ) )
7069rspcv 3192 . . . 4  |-  ( M  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( F `  k )  e.  RR  ->  ( F `  M )  e.  RR ) )
7152, 27, 70sylc 60 . . 3  |-  ( ph  ->  ( F `  M
)  e.  RR )
7268, 71lenegd 10137 . 2  |-  ( ph  ->  ( ( F `  N )  <_  ( F `  M )  <->  -u ( F `  M
)  <_  -u ( F `
 N ) ) )
7365, 72mpbird 232 1  |-  ( ph  ->  ( F `  N
)  <_  ( F `  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   class class class wbr 4437    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   1c1 9496    + caddc 9498    <_ cle 9632    - cmin 9810   -ucneg 9811   ZZcz 10870   ZZ>=cuz 11090   ...cfz 11681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682
This theorem is referenced by:  iseraltlem1  13483  climcndslem1  13640  climcndslem2  13641  dvfsumlem3  22302  emcllem7  23203  climinf  31520
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