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Theorem monoord2 10201
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 8106 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  -u ( F `
 k )  e.  RR )
4 eqid 2115 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)  =  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)
53, 4fmptd 5540 . . . . 5  |-  ( ph  ->  ( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) : ( M ... N
) --> RR )
65ffvelrnda 5521 . . . 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 2480 . . . . . . . 8  |-  ( ph  ->  A. k  e.  ( M ... ( N  -  1 ) ) ( F `  (
k  +  1 ) )  <_  ( F `  k ) )
9 oveq1 5747 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
109fveq2d 5391 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
11 fveq2 5387 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
1210, 11breq12d 3910 . . . . . . . . 9  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
1312cbvralv 2629 . . . . . . . 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 121 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( M ... ( N  -  1 ) ) ( F `  (
n  +  1 ) )  <_  ( F `  n ) )
1514r19.21bi 2495 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
)
16 fveq2 5387 . . . . . . . . 9  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
1716eleq1d 2184 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( n  +  1 ) )  e.  RR ) )
182ralrimiva 2480 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  ( M ... N ) ( F `  k
)  e.  RR )
1918adantr 272 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A. k  e.  ( M ... N
) ( F `  k )  e.  RR )
20 fzp1elp1 9806 . . . . . . . . . 10  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  (
n  +  1 )  e.  ( M ... ( ( N  - 
1 )  +  1 ) ) )
2120adantl 273 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... (
( N  -  1 )  +  1 ) ) )
22 eluzelz 9287 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
231, 22syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
2423zcnd 9128 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
25 ax-1cn 7677 . . . . . . . . . . . 12  |-  1  e.  CC
26 npcan 7935 . . . . . . . . . . . 12  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2724, 25, 26sylancl 407 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2827oveq2d 5756 . . . . . . . . . 10  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
2928adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( M ... ( ( N  - 
1 )  +  1 ) )  =  ( M ... N ) )
3021, 29eleqtrd 2194 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... N
) )
3117, 19, 30rspcdva 2766 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
3211eleq1d 2184 . . . . . . . 8  |-  ( k  =  n  ->  (
( F `  k
)  e.  RR  <->  ( F `  n )  e.  RR ) )
33 fzssp1 9798 . . . . . . . . . 10  |-  ( M ... ( N  - 
1 ) )  C_  ( M ... ( ( N  -  1 )  +  1 ) )
3433, 28sseqtrid 3115 . . . . . . . . 9  |-  ( ph  ->  ( M ... ( N  -  1 ) )  C_  ( M ... N ) )
3534sselda 3065 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( M ... N ) )
3632, 19, 35rspcdva 2766 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  e.  RR )
3731, 36lenegd 8249 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( F `  ( n  +  1 ) )  <_  ( F `  n )  <->  -u ( F `
 n )  <_  -u ( F `  (
n  +  1 ) ) ) )
3815, 37mpbid 146 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 n )  <_  -u ( F `  (
n  +  1 ) ) )
3936renegcld 8106 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 n )  e.  RR )
4011negeqd 7921 . . . . . . 7  |-  ( k  =  n  ->  -u ( F `  k )  =  -u ( F `  n ) )
4140, 4fvmptg 5463 . . . . . 6  |-  ( ( n  e.  ( M ... N )  /\  -u ( F `  n
)  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  n )  =  -u ( F `  n ) )
4235, 39, 41syl2anc 406 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  n )  =  -u ( F `  n ) )
4331renegcld 8106 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 ( n  + 
1 ) )  e.  RR )
4416negeqd 7921 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  -u ( F `  k )  =  -u ( F `  ( n  +  1
) ) )
4544, 4fvmptg 5463 . . . . . 6  |-  ( ( ( n  +  1 )  e.  ( M ... N )  /\  -u ( F `  (
n  +  1 ) )  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  ( n  +  1 ) )  =  -u ( F `  ( n  +  1
) ) )
4630, 43, 45syl2anc 406 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  ( n  +  1 ) )  =  -u ( F `  ( n  +  1
) ) )
4738, 42, 463brtr4d 3928 . . . 4  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  n )  <_  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  ( n  +  1 ) ) )
481, 6, 47monoord 10200 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  <_  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  N ) )
49 eluzfz1 9762 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
501, 49syl 14 . . . 4  |-  ( ph  ->  M  e.  ( M ... N ) )
51 fveq2 5387 . . . . . . 7  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
5251eleq1d 2184 . . . . . 6  |-  ( k  =  M  ->  (
( F `  k
)  e.  RR  <->  ( F `  M )  e.  RR ) )
5352, 18, 50rspcdva 2766 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  RR )
5453renegcld 8106 . . . 4  |-  ( ph  -> 
-u ( F `  M )  e.  RR )
5551negeqd 7921 . . . . 5  |-  ( k  =  M  ->  -u ( F `  k )  =  -u ( F `  M ) )
5655, 4fvmptg 5463 . . . 4  |-  ( ( M  e.  ( M ... N )  /\  -u ( F `  M
)  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  M )  =  -u ( F `  M ) )
5750, 54, 56syl2anc 406 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  =  -u ( F `  M ) )
58 eluzfz2 9763 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
591, 58syl 14 . . . 4  |-  ( ph  ->  N  e.  ( M ... N ) )
60 fveq2 5387 . . . . . . 7  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
6160eleq1d 2184 . . . . . 6  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
6261, 18, 59rspcdva 2766 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  RR )
6362renegcld 8106 . . . 4  |-  ( ph  -> 
-u ( F `  N )  e.  RR )
6460negeqd 7921 . . . . 5  |-  ( k  =  N  ->  -u ( F `  k )  =  -u ( F `  N ) )
6564, 4fvmptg 5463 . . . 4  |-  ( ( N  e.  ( M ... N )  /\  -u ( F `  N
)  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  N )  =  -u ( F `  N ) )
6659, 63, 65syl2anc 406 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  N )  =  -u ( F `  N ) )
6748, 57, 663brtr3d 3927 . 2  |-  ( ph  -> 
-u ( F `  M )  <_  -u ( F `  N )
)
6862, 53lenegd 8249 . 2  |-  ( ph  ->  ( ( F `  N )  <_  ( F `  M )  <->  -u ( F `  M
)  <_  -u ( F `
 N ) ) )
6967, 68mpbird 166 1  |-  ( ph  ->  ( F `  N
)  <_  ( F `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   class class class wbr 3897    |-> cmpt 3957   ` cfv 5091  (class class class)co 5740   CCcc 7582   RRcr 7583   1c1 7585    + caddc 7587    <_ cle 7765    - cmin 7897   -ucneg 7898   ZZcz 9008   ZZ>=cuz 9278   ...cfz 9741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742
This theorem is referenced by: (None)
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