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Theorem dveq0 19363
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 18413 . . . 4  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
31, 2syl 15 . . 3  |-  ( ph  ->  F : ( A [,] B ) --> CC )
4 ffn 5405 . . 3  |-  ( F : ( A [,] B ) --> CC  ->  F  Fn  ( A [,] B ) )
53, 4syl 15 . 2  |-  ( ph  ->  F  Fn  ( A [,] B ) )
6 fvex 5555 . . 3  |-  ( F `
 A )  e. 
_V
7 fnconstg 5445 . . 3  |-  ( ( F `  A )  e.  _V  ->  (
( A [,] B
)  X.  { ( F `  A ) } )  Fn  ( A [,] B ) )
86, 7mp1i 11 . 2  |-  ( ph  ->  ( ( A [,] B )  X.  {
( F `  A
) } )  Fn  ( A [,] B
) )
96fvconst2 5745 . . . 4  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  {
( F `  A
) } ) `  x )  =  ( F `  A ) )
109adantl 452 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
( A [,] B
)  X.  { ( F `  A ) } ) `  x
)  =  ( F `
 A ) )
113adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F :
( A [,] B
) --> CC )
12 dveq0.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
1413rexrd 8897 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR* )
15 dveq0.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
1615adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
1716rexrd 8897 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
18 elicc2 10731 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1912, 15, 18syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
2019biimpa 470 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
2120simp1d 967 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
2220simp2d 968 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
2320simp3d 969 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
2413, 21, 16, 22, 23letrd 8989 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  B )
25 lbicc2 10768 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2614, 17, 24, 25syl3anc 1182 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  ( A [,] B ) )
27 ffvelrn 5679 . . . . 5  |-  ( ( F : ( A [,] B ) --> CC 
/\  A  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
2811, 26, 27syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
29 ffvelrn 5679 . . . . 5  |-  ( ( F : ( A [,] B ) --> CC 
/\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
303, 29sylan 457 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
3128, 30subcld 9173 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  e.  CC )
32 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
3326, 32jca 518 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  e.  ( A [,] B
)  /\  x  e.  ( A [,] B ) ) )
34 dveq0.d . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( ( A (,) B )  X.  { 0 } ) )
3534dmeqd 4897 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  ( ( A (,) B )  X.  {
0 } ) )
36 c0ex 8848 . . . . . . . . . . . 12  |-  0  e.  _V
3736snnz 3757 . . . . . . . . . . 11  |-  { 0 }  =/=  (/)
38 dmxp 4913 . . . . . . . . . . 11  |-  ( { 0 }  =/=  (/)  ->  dom  ( ( A (,) B )  X.  {
0 } )  =  ( A (,) B
) )
3937, 38ax-mp 8 . . . . . . . . . 10  |-  dom  (
( A (,) B
)  X.  { 0 } )  =  ( A (,) B )
4035, 39syl6eq 2344 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
41 0re 8854 . . . . . . . . . 10  |-  0  e.  RR
4241a1i 10 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
4334fveq1d 5543 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  F ) `  y
)  =  ( ( ( A (,) B
)  X.  { 0 } ) `  y
) )
4436fvconst2 5745 . . . . . . . . . . . . 13  |-  ( y  e.  ( A (,) B )  ->  (
( ( A (,) B )  X.  {
0 } ) `  y )  =  0 )
4543, 44sylan9eq 2348 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  y )  =  0 )
4645fveq2d 5545 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  ( abs `  0 ) )
47 abs0 11786 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
4846, 47syl6eq 2344 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  0 )
49 0le0 9843 . . . . . . . . . 10  |-  0  <_  0
5048, 49syl6eqbr 4076 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  <_  0
)
5112, 15, 1, 40, 42, 50dvlip 19356 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  ( A [,] B
)  /\  x  e.  ( A [,] B ) ) )  ->  ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
5233, 51syldan 456 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
5313recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  CC )
5421recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
5553, 54subcld 9173 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  -  x )  e.  CC )
5655abscld 11934 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  RR )
5756recnd 8877 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  CC )
5857mul02d 9026 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 0  x.  ( abs `  ( A  -  x )
) )  =  0 )
5952, 58breqtrd 4063 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  0
)
6031absge0d 11942 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  0  <_  ( abs `  ( ( F `  A )  -  ( F `  x ) ) ) )
6131abscld 11934 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  e.  RR )
62 letri3 8923 . . . . . . 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 ) ) ) ) ) )
6361, 41, 62sylancl 643 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  =  0  <-> 
( ( abs `  (
( F `  A
)  -  ( F `
 x ) ) )  <_  0  /\  0  <_  ( abs `  (
( F `  A
)  -  ( F `
 x ) ) ) ) ) )
6459, 60, 63mpbir2and 888 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  =  0 )
6531, 64abs00d 11944 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  =  0 )
6628, 30, 65subeq0d 9181 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  =  ( F `  x ) )
6710, 66eqtr2d 2329 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  ( ( ( A [,] B )  X.  {
( F `  A
) } ) `  x ) )
685, 8, 67eqfnfvd 5641 1  |-  ( ph  ->  F  =  ( ( A [,] B )  X.  { ( F `
 A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653   class class class wbr 4039    X. cxp 4703   dom cdm 4705    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758   RR*cxr 8882    <_ cle 8884    - cmin 9053   (,)cioo 10672   [,]cicc 10675   abscabs 11735   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  ftc2  19407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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