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Theorem dvlt0 19567
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 9050 . . 3  |-  <  Or  RR
6 cnvso 5317 . . 3  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
75, 6mpbi 199 . 2  |-  `'  <  Or  RR
81, 2, 3, 4dvgt0lem1 19564 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  (  -oo (,) 0
) )
9 eliooord 10863 . . . . . . . . 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 15 . . . . . . . 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 449 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  <  0 )
12 cncff 18611 . . . . . . . . . . . 12  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
133, 12syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F : ( A [,] B ) --> RR )
1413ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F :
( A [,] B
) --> RR )
15 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
16 ffvelrn 5770 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  y  e.  ( A [,] B ) )  ->  ( F `  y )  e.  RR )
1714, 15, 16syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  RR )
18 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
19 ffvelrn 5770 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
2014, 18, 19syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  RR )
2117, 20resubcld 9358 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  RR )
22 0re 8985 . . . . . . . . 9  |-  0  e.  RR
2322a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  e.  RR )
24 iccssre 10884 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
251, 2, 24syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( A [,] B
)  C_  RR )
2625ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
2726, 15sseldd 3267 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
2826, 18sseldd 3267 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
2927, 28resubcld 9358 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
30 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
3128, 27posdifd 9506 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  0  <  (
y  -  x ) ) )
3230, 31mpbid 201 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  0  <  ( y  -  x ) )
33 ltdivmul 9775 . . . . . . . 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 ) ) )
3421, 23, 29, 32, 33syl112anc 1187 . . . . . . 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 ) ) )
3511, 34mpbid 201 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  < 
( ( y  -  x )  x.  0 ) )
3629recnd 9008 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
3736mul01d 9158 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
y  -  x )  x.  0 )  =  0 )
3835, 37breqtrd 4149 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  <  0 )
3917, 20, 23ltsubaddd 9515 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  <  0  <->  ( F `  y )  <  (
0  +  ( F `
 x ) ) ) )
4038, 39mpbid 201 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  <  (
0  +  ( F `
 x ) ) )
4120recnd 9008 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
4241addid2d 9160 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( 0  +  ( F `  x ) )  =  ( F `  x
) )
4340, 42breqtrd 4149 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  <  ( F `  x )
)
44 fvex 5646 . . . 4  |-  ( F `
 x )  e. 
_V
45 fvex 5646 . . . 4  |-  ( F `
 y )  e. 
_V
4644, 45brcnv 4967 . . 3  |-  ( ( F `  x ) `'  <  ( F `  y )  <->  ( F `  y )  <  ( F `  x )
)
4743, 46sylibr 203 . 2  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x ) `'  <  ( F `  y ) )
481, 2, 3, 4, 7, 47dvgt0lem2 19565 1  |-  ( ph  ->  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1715    C_ wss 3238   class class class wbr 4125    Or wor 4416   `'ccnv 4791   ran crn 4793   -->wf 5354   ` cfv 5358    Isom wiso 5359  (class class class)co 5981   RRcr 8883   0cc0 8884    + caddc 8887    x. cmul 8889    -oocmnf 9012    < clt 9014    - cmin 9184    / cdiv 9570   (,)cioo 10809   [,]cicc 10812   -cn->ccncf 18594    _D cdv 19428
This theorem is referenced by:  dvne0  19573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-lp 17085  df-perf 17086  df-cn 17174  df-cnp 17175  df-haus 17260  df-cmp 17331  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596  df-limc 19431  df-dv 19432
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