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Theorem monoord2 9870
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 7837 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  -u ( F `
 k )  e.  RR )
4 eqid 2088 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)  =  ( k  e.  ( M ... N )  |->  -u ( F `  k )
)
53, 4fmptd 5436 . . . . 5  |-  ( ph  ->  ( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) : ( M ... N
) --> RR )
65ffvelrnda 5418 . . . 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 2446 . . . . . . . 8  |-  ( ph  ->  A. k  e.  ( M ... ( N  -  1 ) ) ( F `  (
k  +  1 ) )  <_  ( F `  k ) )
9 oveq1 5641 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
109fveq2d 5293 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  ( k  +  1 ) )  =  ( F `  ( n  +  1
) ) )
11 fveq2 5289 . . . . . . . . . 10  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
1210, 11breq12d 3850 . . . . . . . . 9  |-  ( k  =  n  ->  (
( F `  (
k  +  1 ) )  <_  ( F `  k )  <->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
) )
1312cbvralv 2590 . . . . . . . 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 120 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( M ... ( N  -  1 ) ) ( F `  (
n  +  1 ) )  <_  ( F `  n ) )
1514r19.21bi 2461 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  <_  ( F `  n )
)
16 fveq2 5289 . . . . . . . . 9  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
1716eleq1d 2156 . . . . . . . 8  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  RR  <->  ( F `  ( n  +  1 ) )  e.  RR ) )
182ralrimiva 2446 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  ( M ... N ) ( F `  k
)  e.  RR )
1918adantr 270 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A. k  e.  ( M ... N
) ( F `  k )  e.  RR )
20 fzp1elp1 9456 . . . . . . . . . 10  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  (
n  +  1 )  e.  ( M ... ( ( N  - 
1 )  +  1 ) ) )
2120adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... (
( N  -  1 )  +  1 ) ) )
22 eluzelz 8997 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
231, 22syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
2423zcnd 8839 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
25 ax-1cn 7417 . . . . . . . . . . . 12  |-  1  e.  CC
26 npcan 7670 . . . . . . . . . . . 12  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
2724, 25, 26sylancl 404 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
2827oveq2d 5650 . . . . . . . . . 10  |-  ( ph  ->  ( M ... (
( N  -  1 )  +  1 ) )  =  ( M ... N ) )
2928adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( M ... ( ( N  - 
1 )  +  1 ) )  =  ( M ... N ) )
3021, 29eleqtrd 2166 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  +  1 )  e.  ( M ... N
) )
3117, 19, 30rspcdva 2727 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
3211eleq1d 2156 . . . . . . . 8  |-  ( k  =  n  ->  (
( F `  k
)  e.  RR  <->  ( F `  n )  e.  RR ) )
33 fzssp1 9449 . . . . . . . . . 10  |-  ( M ... ( N  - 
1 ) )  C_  ( M ... ( ( N  -  1 )  +  1 ) )
3433, 28syl5sseq 3072 . . . . . . . . 9  |-  ( ph  ->  ( M ... ( N  -  1 ) )  C_  ( M ... N ) )
3534sselda 3023 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( M ... N ) )
3632, 19, 35rspcdva 2727 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  e.  RR )
3731, 36lenegd 7977 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( F `  ( n  +  1 ) )  <_  ( F `  n )  <->  -u ( F `
 n )  <_  -u ( F `  (
n  +  1 ) ) ) )
3815, 37mpbid 145 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 n )  <_  -u ( F `  (
n  +  1 ) ) )
3936renegcld 7837 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 n )  e.  RR )
4011negeqd 7656 . . . . . . 7  |-  ( k  =  n  ->  -u ( F `  k )  =  -u ( F `  n ) )
4140, 4fvmptg 5364 . . . . . 6  |-  ( ( n  e.  ( M ... N )  /\  -u ( F `  n
)  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  n )  =  -u ( F `  n ) )
4235, 39, 41syl2anc 403 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  n )  =  -u ( F `  n ) )
4331renegcld 7837 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -u ( F `
 ( n  + 
1 ) )  e.  RR )
4416negeqd 7656 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  -u ( F `  k )  =  -u ( F `  ( n  +  1
) ) )
4544, 4fvmptg 5364 . . . . . 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 403 . . . . 5  |-  ( (
ph  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( (
k  e.  ( M ... N )  |->  -u ( F `  k ) ) `  ( n  +  1 ) )  =  -u ( F `  ( n  +  1
) ) )
4738, 42, 463brtr4d 3867 . . . 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 9869 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  <_  (
( k  e.  ( M ... N ) 
|->  -u ( F `  k ) ) `  N ) )
49 eluzfz1 9414 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
501, 49syl 14 . . . 4  |-  ( ph  ->  M  e.  ( M ... N ) )
51 fveq2 5289 . . . . . . 7  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
5251eleq1d 2156 . . . . . 6  |-  ( k  =  M  ->  (
( F `  k
)  e.  RR  <->  ( F `  M )  e.  RR ) )
5352, 18, 50rspcdva 2727 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  RR )
5453renegcld 7837 . . . 4  |-  ( ph  -> 
-u ( F `  M )  e.  RR )
5551negeqd 7656 . . . . 5  |-  ( k  =  M  ->  -u ( F `  k )  =  -u ( F `  M ) )
5655, 4fvmptg 5364 . . . 4  |-  ( ( M  e.  ( M ... N )  /\  -u ( F `  M
)  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  M )  =  -u ( F `  M ) )
5750, 54, 56syl2anc 403 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  M )  =  -u ( F `  M ) )
58 eluzfz2 9415 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
591, 58syl 14 . . . 4  |-  ( ph  ->  N  e.  ( M ... N ) )
60 fveq2 5289 . . . . . . 7  |-  ( k  =  N  ->  ( F `  k )  =  ( F `  N ) )
6160eleq1d 2156 . . . . . 6  |-  ( k  =  N  ->  (
( F `  k
)  e.  RR  <->  ( F `  N )  e.  RR ) )
6261, 18, 59rspcdva 2727 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  RR )
6362renegcld 7837 . . . 4  |-  ( ph  -> 
-u ( F `  N )  e.  RR )
6460negeqd 7656 . . . . 5  |-  ( k  =  N  ->  -u ( F `  k )  =  -u ( F `  N ) )
6564, 4fvmptg 5364 . . . 4  |-  ( ( N  e.  ( M ... N )  /\  -u ( F `  N
)  e.  RR )  ->  ( ( k  e.  ( M ... N )  |->  -u ( F `  k )
) `  N )  =  -u ( F `  N ) )
6659, 63, 65syl2anc 403 . . 3  |-  ( ph  ->  ( ( k  e.  ( M ... N
)  |->  -u ( F `  k ) ) `  N )  =  -u ( F `  N ) )
6748, 57, 663brtr3d 3866 . 2  |-  ( ph  -> 
-u ( F `  M )  <_  -u ( F `  N )
)
6862, 53lenegd 7977 . 2  |-  ( ph  ->  ( ( F `  N )  <_  ( F `  M )  <->  -u ( F `  M
)  <_  -u ( F `
 N ) ) )
6967, 68mpbird 165 1  |-  ( ph  ->  ( F `  N
)  <_  ( F `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   class class class wbr 3837    |-> cmpt 3891   ` cfv 5002  (class class class)co 5634   CCcc 7327   RRcr 7328   1c1 7330    + caddc 7332    <_ cle 7502    - cmin 7632   -ucneg 7633   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394
This theorem is referenced by: (None)
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