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Theorem monoord2 12120
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 9982 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  -u ( F `
 k )  e.  RR )
4 eqid 2454 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)  =  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)
53, 4fmptd 6031 . . . . 5  |-  ( ph  ->  ( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) : ( M ... N
) --> RR )
65ffvelrnda 6007 . . . 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 2868 . . . . . . . 8  |-  ( ph  ->  A. k  e.  ( M ... ( N  -  1 ) ) ( F `  (
k  +  1 ) )  <_  ( F `  k ) )
9 oveq1 6277 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
109fveq2d 5852 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
11 fveq2 5848 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
1210, 11breq12d 4452 . . . . . . . . 9  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
1312cbvralv 3081 . . . . . . . 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 2823 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
)
16 fzp1elp1 11737 . . . . . . . . . 10  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  (
n  +  1 )  e.  ( M ... ( ( N  - 
1 )  +  1 ) ) )
1716adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... (
( N  -  1 )  +  1 ) ) )
18 eluzelz 11091 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
191, 18syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
2019zcnd 10966 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
21 ax-1cn 9539 . . . . . . . . . . . 12  |-  1  e.  CC
22 npcan 9820 . . . . . . . . . . . 12  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2320, 21, 22sylancl 660 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2423oveq2d 6286 . . . . . . . . . 10  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
2524adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( M ... ( ( N  - 
1 )  +  1 ) )  =  ( M ... N ) )
2617, 25eleqtrd 2544 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... N
) )
272ralrimiva 2868 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  ( M ... N ) ( F `  k
)  e.  RR )
2827adantr 463 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A. k  e.  ( M ... N
) ( F `  k )  e.  RR )
29 fveq2 5848 . . . . . . . . . 10  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
3029eleq1d 2523 . . . . . . . . 9  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( n  +  1 ) )  e.  RR ) )
3130rspcv 3203 . . . . . . . 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 11730 . . . . . . . . . 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 2523 . . . . . . . . 9  |-  ( k  =  n  ->  (
( F `  k
)  e.  RR  <->  ( F `  n )  e.  RR ) )
3736rspcv 3203 . . . . . . . 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 10127 . . . . . 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 9805 . . . . . . 7  |-  ( k  =  n  ->  -u ( F `  k )  =  -u ( F `  n ) )
42 negex 9809 . . . . . . 7  |-  -u ( F `  n )  e.  _V
4341, 4, 42fvmpt 5931 . . . . . 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 9805 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  -u ( F `  k )  =  -u ( F `  ( n  +  1
) ) )
46 negex 9809 . . . . . . 7  |-  -u ( F `  ( n  +  1 ) )  e.  _V
4745, 4, 46fvmpt 5931 . . . . . 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 4469 . . . 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 12119 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  <_  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  N ) )
51 eluzfz1 11696 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
521, 51syl 16 . . . 4  |-  ( ph  ->  M  e.  ( M ... N ) )
53 fveq2 5848 . . . . . 6  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
5453negeqd 9805 . . . . 5  |-  ( k  =  M  ->  -u ( F `  k )  =  -u ( F `  M ) )
55 negex 9809 . . . . 5  |-  -u ( F `  M )  e.  _V
5654, 4, 55fvmpt 5931 . . . 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 11697 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
591, 58syl 16 . . . 4  |-  ( ph  ->  N  e.  ( M ... N ) )
60 fveq2 5848 . . . . . 6  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
6160negeqd 9805 . . . . 5  |-  ( k  =  N  ->  -u ( F `  k )  =  -u ( F `  N ) )
62 negex 9809 . . . . 5  |-  -u ( F `  N )  e.  _V
6361, 4, 62fvmpt 5931 . . . 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 4468 . 2  |-  ( ph  -> 
-u ( F `  M )  <_  -u ( F `  N )
)
6660eleq1d 2523 . . . . 5  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
6766rspcv 3203 . . . 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 2523 . . . . 5  |-  ( k  =  M  ->  (
( F `  k
)  e.  RR  <->  ( F `  M )  e.  RR ) )
7069rspcv 3203 . . . 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 10127 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9796   -ucneg 9797   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by:  iseraltlem1  13586  climcndslem1  13743  climcndslem2  13744  dvfsumlem3  22595  emcllem7  23529  climinf  31851
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