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Theorem dveq0 19884
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 18923 . . . 4  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
31, 2syl 16 . . 3  |-  ( ph  ->  F : ( A [,] B ) --> CC )
4 ffn 5591 . . 3  |-  ( F : ( A [,] B ) --> CC  ->  F  Fn  ( A [,] B ) )
53, 4syl 16 . 2  |-  ( ph  ->  F  Fn  ( A [,] B ) )
6 fvex 5742 . . 3  |-  ( F `
 A )  e. 
_V
7 fnconstg 5631 . . 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 5947 . . . 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 9134 . . . . . 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 9134 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
18 elicc2 10975 . . . . . . . . . 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 9227 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  B )
25 lbicc2 11013 . . . . . 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 5871 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
283ffvelrnda 5870 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
2927, 28subcld 9411 . . . . 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 5072 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  ( ( A (,) B )  X.  {
0 } ) )
34 c0ex 9085 . . . . . . . . . . . 12  |-  0  e.  _V
3534snnz 3922 . . . . . . . . . . 11  |-  { 0 }  =/=  (/)
36 dmxp 5088 . . . . . . . . . . 11  |-  ( { 0 }  =/=  (/)  ->  dom  ( ( A (,) B )  X.  {
0 } )  =  ( A (,) B
) )
3735, 36ax-mp 8 . . . . . . . . . 10  |-  dom  (
( A (,) B
)  X.  { 0 } )  =  ( A (,) B )
3833, 37syl6eq 2484 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
39 0re 9091 . . . . . . . . . 10  |-  0  e.  RR
4039a1i 11 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
4132fveq1d 5730 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F ) `  y
)  =  ( ( ( A (,) B
)  X.  { 0 } ) `  y
) )
4234fvconst2 5947 . . . . . . . . . . . 12  |-  ( y  e.  ( A (,) B )  ->  (
( ( A (,) B )  X.  {
0 } ) `  y )  =  0 )
4341, 42sylan9eq 2488 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  y )  =  0 )
4443abs00bd 12096 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  0 )
45 0le0 10081 . . . . . . . . . 10  |-  0  <_  0
4644, 45syl6eqbr 4249 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  <_  0
)
4712, 15, 1, 38, 40, 46dvlip 19877 . . . . . . . 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 9114 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  CC )
5021recnd 9114 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
5149, 50subcld 9411 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  -  x )  e.  CC )
5251abscld 12238 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  RR )
5352recnd 9114 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  CC )
5453mul02d 9264 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 0  x.  ( abs `  ( A  -  x )
) )  =  0 )
5548, 54breqtrd 4236 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  0
)
5629absge0d 12246 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  0  <_  ( abs `  ( ( F `  A )  -  ( F `  x ) ) ) )
5729abscld 12238 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  e.  RR )
58 letri3 9160 . . . . . . 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 12248 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  =  0 )
6227, 28, 61subeq0d 9419 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  =  ( F `  x ) )
6310, 62eqtr2d 2469 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  ( ( ( A [,] B )  X.  {
( F `  A
) } ) `  x ) )
645, 8, 63eqfnfvd 5830 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 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   (/)c0 3628   {csn 3814   class class class wbr 4212    X. cxp 4876   dom cdm 4878    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995   RR*cxr 9119    <_ cle 9121    - cmin 9291   (,)cioo 10916   [,]cicc 10919   abscabs 12039   -cn->ccncf 18906    _D cdv 19750
This theorem is referenced by:  ftc2  19928  ftc2nc  26289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-cmp 17450  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754
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