MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvlt0 Unicode version

Theorem dvlt0 19842
Description: A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvgt0.a  |-  ( ph  ->  A  e.  RR )
dvgt0.b  |-  ( ph  ->  B  e.  RR )
dvgt0.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
dvlt0.d  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> ( 
-oo (,) 0 ) )
Assertion
Ref Expression
dvlt0  |-  ( ph  ->  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) )

Proof of Theorem dvlt0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvgt0.a . 2  |-  ( ph  ->  A  e.  RR )
2 dvgt0.b . 2  |-  ( ph  ->  B  e.  RR )
3 dvgt0.f . 2  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
4 dvlt0.d . 2  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> ( 
-oo (,) 0 ) )
5 ltso 9112 . . 3  |-  <  Or  RR
6 cnvso 5370 . . 3  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
75, 6mpbi 200 . 2  |-  `'  <  Or  RR
81, 2, 3, 4dvgt0lem1 19839 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  (  -oo (,) 0
) )
9 eliooord 10926 . . . . . . . . 9  |-  ( ( ( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) )  e.  (  -oo (,) 0 )  ->  (  -oo  <  ( ( ( F `  y )  -  ( F `  x ) )  / 
( y  -  x
) )  /\  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) )  <  0 ) )
108, 9syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  (  -oo  <  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) )  /\  ( ( ( F `  y )  -  ( F `  x ) )  / 
( y  -  x
) )  <  0
) )
1110simprd 450 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  <  0 )
12 cncff 18876 . . . . . . . . . . . 12  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
133, 12syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F : ( A [,] B ) --> RR )
1413ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F :
( A [,] B
) --> RR )
15 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
1614, 15ffvelrnd 5830 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  RR )
17 simplrl 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
1814, 17ffvelrnd 5830 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  RR )
1916, 18resubcld 9421 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  RR )
20 0re 9047 . . . . . . . . 9  |-  0  e.  RR
2120a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  e.  RR )
22 iccssre 10948 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
231, 2, 22syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( A [,] B
)  C_  RR )
2423ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
2524, 15sseldd 3309 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
2624, 17sseldd 3309 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
2725, 26resubcld 9421 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
28 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
2926, 25posdifd 9569 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  0  <  (
y  -  x ) ) )
3028, 29mpbid 202 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  <  ( y  -  x ) )
31 ltdivmul 9838 . . . . . . . 8  |-  ( ( ( ( F `  y )  -  ( F `  x )
)  e.  RR  /\  0  e.  RR  /\  (
( y  -  x
)  e.  RR  /\  0  <  ( y  -  x ) ) )  ->  ( ( ( ( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  <  0  <->  ( ( F `
 y )  -  ( F `  x ) )  <  ( ( y  -  x )  x.  0 ) ) )
3219, 21, 27, 30, 31syl112anc 1188 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) )  <  0  <->  ( ( F `  y )  -  ( F `  x ) )  < 
( ( y  -  x )  x.  0 ) ) )
3311, 32mpbid 202 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  < 
( ( y  -  x )  x.  0 ) )
3427recnd 9070 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
3534mul01d 9221 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
y  -  x )  x.  0 )  =  0 )
3633, 35breqtrd 4196 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  <  0 )
3716, 18, 21ltsubaddd 9578 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  <  0  <->  ( F `  y )  <  (
0  +  ( F `
 x ) ) ) )
3836, 37mpbid 202 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  <  (
0  +  ( F `
 x ) ) )
3918recnd 9070 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
4039addid2d 9223 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( 0  +  ( F `  x ) )  =  ( F `  x
) )
4138, 40breqtrd 4196 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  <  ( F `  x )
)
42 fvex 5701 . . . 4  |-  ( F `
 x )  e. 
_V
43 fvex 5701 . . . 4  |-  ( F `
 y )  e. 
_V
4442, 43brcnv 5014 . . 3  |-  ( ( F `  x ) `'  <  ( F `  y )  <->  ( F `  y )  <  ( F `  x )
)
4541, 44sylibr 204 . 2  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x ) `'  <  ( F `  y ) )
461, 2, 3, 4, 7, 45dvgt0lem2 19840 1  |-  ( ph  ->  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    C_ wss 3280   class class class wbr 4172    Or wor 4462   `'ccnv 4836   ran crn 4838   -->wf 5409   ` cfv 5413    Isom wiso 5414  (class class class)co 6040   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    -oocmnf 9074    < clt 9076    - cmin 9247    / cdiv 9633   (,)cioo 10872   [,]cicc 10875   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvne0  19848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
  Copyright terms: Public domain W3C validator