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Theorem dveq0 22129
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 21125 . . . 4  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
31, 2syl 16 . . 3  |-  ( ph  ->  F : ( A [,] B ) --> CC )
4 ffn 5722 . . 3  |-  ( F : ( A [,] B ) --> CC  ->  F  Fn  ( A [,] B ) )
53, 4syl 16 . 2  |-  ( ph  ->  F  Fn  ( A [,] B ) )
6 fvex 5867 . . 3  |-  ( F `
 A )  e. 
_V
7 fnconstg 5764 . . 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 6107 . . . 4  |-  ( x  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  {
( F `  A
) } ) `  x )  =  ( F `  A ) )
109adantl 466 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( (
( A [,] B
)  X.  { ( F `  A ) } ) `  x
)  =  ( F `
 A ) )
113adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F :
( A [,] B
) --> CC )
12 dveq0.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1312adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
1413rexrd 9632 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR* )
15 dveq0.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
1716rexrd 9632 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
18 elicc2 11578 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1912, 15, 18syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
2019biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
2120simp1d 1003 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
2220simp2d 1004 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
2320simp3d 1005 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
2413, 21, 16, 22, 23letrd 9727 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  B )
25 lbicc2 11625 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2614, 17, 24, 25syl3anc 1223 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  ( A [,] B ) )
2711, 26ffvelrnd 6013 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
283ffvelrnda 6012 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
2927, 28subcld 9919 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  e.  CC )
30 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
3126, 30jca 532 . . . . . . . 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 5196 . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  F )  =  dom  ( ( A (,) B )  X.  {
0 } ) )
34 c0ex 9579 . . . . . . . . . . . 12  |-  0  e.  _V
3534snnz 4138 . . . . . . . . . . 11  |-  { 0 }  =/=  (/)
36 dmxp 5212 . . . . . . . . . . 11  |-  ( { 0 }  =/=  (/)  ->  dom  ( ( A (,) B )  X.  {
0 } )  =  ( A (,) B
) )
3735, 36ax-mp 5 . . . . . . . . . 10  |-  dom  (
( A (,) B
)  X.  { 0 } )  =  ( A (,) B )
3833, 37syl6eq 2517 . . . . . . . . 9  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
39 0red 9586 . . . . . . . . 9  |-  ( ph  ->  0  e.  RR )
4032fveq1d 5859 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( RR  _D  F ) `  y
)  =  ( ( ( A (,) B
)  X.  { 0 } ) `  y
) )
4134fvconst2 6107 . . . . . . . . . . . 12  |-  ( y  e.  ( A (,) B )  ->  (
( ( A (,) B )  X.  {
0 } ) `  y )  =  0 )
4240, 41sylan9eq 2521 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  y )  =  0 )
4342abs00bd 13074 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  =  0 )
44 0le0 10614 . . . . . . . . . 10  |-  0  <_  0
4543, 44syl6eqbr 4477 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  y
) )  <_  0
)
4612, 15, 1, 38, 39, 45dvlip 22122 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  ( A [,] B
)  /\  x  e.  ( A [,] B ) ) )  ->  ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
4731, 46syldan 470 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  (
0  x.  ( abs `  ( A  -  x
) ) ) )
4813recnd 9611 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  CC )
4921recnd 9611 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  CC )
5048, 49subcld 9919 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A  -  x )  e.  CC )
5150abscld 13216 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  RR )
5251recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( A  -  x
) )  e.  CC )
5352mul02d 9766 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 0  x.  ( abs `  ( A  -  x )
) )  =  0 )
5447, 53breqtrd 4464 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  <_  0
)
5529absge0d 13224 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  0  <_  ( abs `  ( ( F `  A )  -  ( F `  x ) ) ) )
5629abscld 13216 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  e.  RR )
57 0re 9585 . . . . . . 7  |-  0  e.  RR
58 letri3 9659 . . . . . . 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 ) ) ) ) ) )
5956, 57, 58sylancl 662 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( abs `  ( ( F `
 A )  -  ( F `  x ) ) )  =  0  <-> 
( ( abs `  (
( F `  A
)  -  ( F `
 x ) ) )  <_  0  /\  0  <_  ( abs `  (
( F `  A
)  -  ( F `
 x ) ) ) ) ) )
6054, 55, 59mpbir2and 915 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( abs `  ( ( F `  A )  -  ( F `  x )
) )  =  0 )
6129, 60abs00d 13226 . . . 4  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  A )  -  ( F `  x ) )  =  0 )
6227, 28, 61subeq0d 9927 . . 3  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  A )  =  ( F `  x ) )
6310, 62eqtr2d 2502 . 2  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  =  ( ( ( A [,] B )  X.  {
( F `  A
) } ) `  x ) )
645, 8, 63eqfnfvd 5969 1  |-  ( ph  ->  F  =  ( ( A [,] B )  X.  { ( F `
 A ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106   (/)c0 3778   {csn 4020   class class class wbr 4440    X. cxp 4990   dom cdm 4992    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486   RR*cxr 9616    <_ cle 9618    - cmin 9794   (,)cioo 11518   [,]cicc 11521   abscabs 13017   -cn->ccncf 21108    _D cdv 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-cmp 19646  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999
This theorem is referenced by:  ftc2  22173  ftc2nc  29663
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