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Theorem dveq0 19341
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 ) } ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem dveq0
StepHypRef Expression
1 dveq0.c . . . 4  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
2 cncff 18391 . . . 4  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
31, 2syl 17 . . 3  |-  ( ph  ->  F : ( A [,] B ) --> CC )
4 ffn 5354 . . 3  |-  ( F : ( A [,] B ) --> CC  ->  F  Fn  ( A [,] B ) )
53, 4syl 17 . 2  |-  ( ph  ->  F  Fn  ( A [,] B ) )
6 fvex 5499 . . 3  |-  ( F `
 A )  e. 
_V
7 fnconstg 5394 . . 3  |-  ( ( F `  A )  e.  _V  ->  (
( A [,] B
)  X.  { ( F `  A ) } )  Fn  ( A [,] B ) )
86, 7mp1i 13 . 2  |-  ( ph  ->  ( ( A [,] B )  X.  {
( F `  A
) } )  Fn  ( A [,] B
) )
96fvconst2 5690 . . . 4  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  {
( F `  A
) } ) `  x )  =  ( F `  A ) )
109adantl 454 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
( A [,] B
)  X.  { ( F `  A ) } ) `  x
)  =  ( F `
 A ) )
113adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F :
( A [,] B
) --> CC )
12 dveq0.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1312adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
1413rexrd 8876 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR* )
15 dveq0.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
1615adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
1716rexrd 8876 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
18 elicc2 10709 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1912, 15, 18syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
2019biimpa 472 . . . . . . . 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 8968 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  B )
25 lbicc2 10746 . . . . . 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 ) )
27 ffvelrn 5624 . . . . 5  |-  ( ( F : ( A [,] B ) --> CC 
/\  A  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
2811, 26, 27syl2anc 644 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
29 ffvelrn 5624 . . . . 5  |-  ( ( F : ( A [,] B ) --> CC 
/\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
303, 29sylan 459 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
3128, 30subcld 9152 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  e.  CC )
32 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
3326, 32jca 520 . . . . . . . 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 4880 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  ( ( A (,) B )  X.  {
0 } ) )
36 c0ex 8827 . . . . . . . . . . . 12  |-  0  e.  _V
3736snnz 3745 . . . . . . . . . . 11  |-  { 0 }  =/=  (/)
38 dmxp 4896 . . . . . . . . . . 11  |-  ( { 0 }  =/=  (/)  ->  dom  ( ( A (,) B )  X.  {
0 } )  =  ( A (,) B
) )
3937, 38ax-mp 10 . . . . . . . . . 10  |-  dom  (
( A (,) B
)  X.  { 0 } )  =  ( A (,) B )
4035, 39syl6eq 2332 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
41 0re 8833 . . . . . . . . . 10  |-  0  e.  RR
4241a1i 12 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
4334fveq1d 5487 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  F ) `  y
)  =  ( ( ( A (,) B
)  X.  { 0 } ) `  y
) )
4436fvconst2 5690 . . . . . . . . . . . . 13  |-  ( y  e.  ( A (,) B )  ->  (
( ( A (,) B )  X.  {
0 } ) `  y )  =  0 )
4543, 44sylan9eq 2336 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  y )  =  0 )
4645fveq2d 5489 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  ( abs `  0 ) )
47 abs0 11764 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
4846, 47syl6eq 2332 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  0 )
49 0le0 9822 . . . . . . . . . 10  |-  0  <_  0
5048, 49syl6eqbr 4061 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  <_  0
)
5112, 15, 1, 40, 42, 50dvlip 19334 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  ( A [,] B
)  /\  x  e.  ( A [,] B ) ) )  ->  ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
5233, 51syldan 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
5313recnd 8856 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  CC )
5421recnd 8856 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
5553, 54subcld 9152 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  -  x )  e.  CC )
5655abscld 11912 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  RR )
5756recnd 8856 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  CC )
5857mul02d 9005 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 0  x.  ( abs `  ( A  -  x )
) )  =  0 )
5952, 58breqtrd 4048 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  0
)
6031absge0d 11920 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  0  <_  ( abs `  ( ( F `  A )  -  ( F `  x ) ) ) )
6131abscld 11912 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  e.  RR )
62 letri3 8902 . . . . . . 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 645 . . . . . 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 890 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  =  0 )
6531, 64abs00d 11922 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  =  0 )
6628, 30, 65subeq0d 9160 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  =  ( F `  x ) )
6710, 66eqtr2d 2317 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  ( ( ( A [,] B )  X.  {
( F `  A
) } ) `  x ) )
685, 8, 67eqfnfvd 5586 1  |-  ( ph  ->  F  =  ( ( A [,] B )  X.  { ( F `
 A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   _Vcvv 2789   (/)c0 3456   {csn 3641   class class class wbr 4024    X. cxp 4686   dom cdm 4688    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732    x. cmul 8737   RR*cxr 8861    <_ cle 8863    - cmin 9032   (,)cioo 10650   [,]cicc 10653   abscabs 11713   -cn->ccncf 18374    _D cdv 19207
This theorem is referenced by:  ftc2  19385
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-cmp 17108  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211
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