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Theorem dveq0 19752
Description: If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)
Hypotheses
Ref Expression
dveq0.a  |-  ( ph  ->  A  e.  RR )
dveq0.b  |-  ( ph  ->  B  e.  RR )
dveq0.c  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
dveq0.d  |-  ( ph  ->  ( RR  _D  F
)  =  ( ( A (,) B )  X.  { 0 } ) )
Assertion
Ref Expression
dveq0  |-  ( ph  ->  F  =  ( ( A [,] B )  X.  { ( F `
 A ) } ) )

Proof of Theorem dveq0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dveq0.c . . . 4  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
2 cncff 18795 . . . 4  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
31, 2syl 16 . . 3  |-  ( ph  ->  F : ( A [,] B ) --> CC )
4 ffn 5532 . . 3  |-  ( F : ( A [,] B ) --> CC  ->  F  Fn  ( A [,] B ) )
53, 4syl 16 . 2  |-  ( ph  ->  F  Fn  ( A [,] B ) )
6 fvex 5683 . . 3  |-  ( F `
 A )  e. 
_V
7 fnconstg 5572 . . 3  |-  ( ( F `  A )  e.  _V  ->  (
( A [,] B
)  X.  { ( F `  A ) } )  Fn  ( A [,] B ) )
86, 7mp1i 12 . 2  |-  ( ph  ->  ( ( A [,] B )  X.  {
( F `  A
) } )  Fn  ( A [,] B
) )
96fvconst2 5887 . . . 4  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  {
( F `  A
) } ) `  x )  =  ( F `  A ) )
109adantl 453 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
( A [,] B
)  X.  { ( F `  A ) } ) `  x
)  =  ( F `
 A ) )
113adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F :
( A [,] B
) --> CC )
12 dveq0.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1312adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
1413rexrd 9068 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR* )
15 dveq0.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
1615adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
1716rexrd 9068 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
18 elicc2 10908 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1912, 15, 18syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
2019biimpa 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
2120simp1d 969 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
2220simp2d 970 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
2320simp3d 971 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
2413, 21, 16, 22, 23letrd 9160 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  B )
25 lbicc2 10946 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2614, 17, 24, 25syl3anc 1184 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  ( A [,] B ) )
2711, 26ffvelrnd 5811 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
283ffvelrnda 5810 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
2927, 28subcld 9344 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  e.  CC )
30 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
3126, 30jca 519 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  e.  ( A [,] B
)  /\  x  e.  ( A [,] B ) ) )
32 dveq0.d . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( ( A (,) B )  X.  { 0 } ) )
3332dmeqd 5013 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  ( ( A (,) B )  X.  {
0 } ) )
34 c0ex 9019 . . . . . . . . . . . 12  |-  0  e.  _V
3534snnz 3866 . . . . . . . . . . 11  |-  { 0 }  =/=  (/)
36 dmxp 5029 . . . . . . . . . . 11  |-  ( { 0 }  =/=  (/)  ->  dom  ( ( A (,) B )  X.  {
0 } )  =  ( A (,) B
) )
3735, 36ax-mp 8 . . . . . . . . . 10  |-  dom  (
( A (,) B
)  X.  { 0 } )  =  ( A (,) B )
3833, 37syl6eq 2436 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
39 0re 9025 . . . . . . . . . 10  |-  0  e.  RR
4039a1i 11 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
4132fveq1d 5671 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F ) `  y
)  =  ( ( ( A (,) B
)  X.  { 0 } ) `  y
) )
4234fvconst2 5887 . . . . . . . . . . . 12  |-  ( y  e.  ( A (,) B )  ->  (
( ( A (,) B )  X.  {
0 } ) `  y )  =  0 )
4341, 42sylan9eq 2440 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  y )  =  0 )
4443abs00bd 12024 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  0 )
45 0le0 10014 . . . . . . . . . 10  |-  0  <_  0
4644, 45syl6eqbr 4191 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  <_  0
)
4712, 15, 1, 38, 40, 46dvlip 19745 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  ( A [,] B
)  /\  x  e.  ( A [,] B ) ) )  ->  ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
4831, 47syldan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
4913recnd 9048 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  CC )
5021recnd 9048 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
5149, 50subcld 9344 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  -  x )  e.  CC )
5251abscld 12166 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  RR )
5352recnd 9048 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  CC )
5453mul02d 9197 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 0  x.  ( abs `  ( A  -  x )
) )  =  0 )
5548, 54breqtrd 4178 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  0
)
5629absge0d 12174 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  0  <_  ( abs `  ( ( F `  A )  -  ( F `  x ) ) ) )
5729abscld 12166 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  e.  RR )
58 letri3 9094 . . . . . . 7  |-  ( ( ( abs `  (
( F `  A
)  -  ( F `
 x ) ) )  e.  RR  /\  0  e.  RR )  ->  ( ( abs `  (
( F `  A
)  -  ( F `
 x ) ) )  =  0  <->  (
( abs `  (
( F `  A
)  -  ( F `
 x ) ) )  <_  0  /\  0  <_  ( abs `  (
( F `  A
)  -  ( F `
 x ) ) ) ) ) )
5957, 39, 58sylancl 644 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  =  0  <-> 
( ( abs `  (
( F `  A
)  -  ( F `
 x ) ) )  <_  0  /\  0  <_  ( abs `  (
( F `  A
)  -  ( F `
 x ) ) ) ) ) )
6055, 56, 59mpbir2and 889 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  =  0 )
6129, 60abs00d 12176 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  =  0 )
6227, 28, 61subeq0d 9352 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  =  ( F `  x ) )
6310, 62eqtr2d 2421 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  ( ( ( A [,] B )  X.  {
( F `  A
) } ) `  x ) )
645, 8, 63eqfnfvd 5770 1  |-  ( ph  ->  F  =  ( ( A [,] B )  X.  { ( F `
 A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900   (/)c0 3572   {csn 3758   class class class wbr 4154    X. cxp 4817   dom cdm 4819    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924    x. cmul 8929   RR*cxr 9053    <_ cle 9055    - cmin 9224   (,)cioo 10849   [,]cicc 10852   abscabs 11967   -cn->ccncf 18778    _D cdv 19618
This theorem is referenced by:  ftc2  19796
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-cmp 17373  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622
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