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Theorem mvth 19829
Description: The Mean Value Theorem. If  F is a real continuous function on  [ A ,  B ] which is differentiable on  ( A ,  B
), then there is some  x  e.  ( A ,  B ) such that  ( RR  _D  F
) `  x is equal to the average slope over  [ A ,  B ]. (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
mvth.a  |-  ( ph  ->  A  e.  RR )
mvth.b  |-  ( ph  ->  B  e.  RR )
mvth.lt  |-  ( ph  ->  A  <  B )
mvth.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
mvth.d  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
Assertion
Ref Expression
mvth  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR  _D  F ) `  x
)  =  ( ( ( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) ) )
Distinct variable groups:    x, A    x, B    x, F    ph, x

Proof of Theorem mvth
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mvth.a . . 3  |-  ( ph  ->  A  e.  RR )
2 mvth.b . . 3  |-  ( ph  ->  B  e.  RR )
3 mvth.lt . . 3  |-  ( ph  ->  A  <  B )
4 mvth.f . . 3  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
5 mptresid 5154 . . . 4  |-  ( z  e.  ( A [,] B )  |->  z )  =  (  _I  |`  ( A [,] B ) )
6 iccssre 10948 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
71, 2, 6syl2anc 643 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ax-resscn 9003 . . . . 5  |-  RR  C_  CC
9 cncfmptid 18895 . . . . 5  |-  ( ( ( A [,] B
)  C_  RR  /\  RR  C_  CC )  ->  (
z  e.  ( A [,] B )  |->  z )  e.  ( ( A [,] B )
-cn-> RR ) )
107, 8, 9sylancl 644 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  z )  e.  ( ( A [,] B
) -cn-> RR ) )
115, 10syl5eqelr 2489 . . 3  |-  ( ph  ->  (  _I  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
12 mvth.d . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
135oveq2i 6051 . . . . . 6  |-  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  z ) )  =  ( RR  _D  (  _I  |`  ( A [,] B ) ) )
14 reex 9037 . . . . . . . . 9  |-  RR  e.  _V
1514prid1 3872 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
1615a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
17 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  RR )
1817recnd 9070 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  CC )
19 1re 9046 . . . . . . . 8  |-  1  e.  RR
2019a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  1  e.  RR )
2116dvmptid 19796 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  RR  |->  z ) )  =  ( z  e.  RR  |->  1 ) )
22 eqid 2404 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2322tgioo2 18787 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
24 iccntr 18805 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
251, 2, 24syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2616, 18, 20, 21, 7, 23, 22, 25dvmptres2 19801 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  z ) )  =  ( z  e.  ( A (,) B )  |->  1 ) )
2713, 26syl5eqr 2450 . . . . 5  |-  ( ph  ->  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( z  e.  ( A (,) B
)  |->  1 ) )
2827dmeqd 5031 . . . 4  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  dom  (
z  e.  ( A (,) B )  |->  1 ) )
29 1ex 9042 . . . . 5  |-  1  e.  _V
30 eqid 2404 . . . . 5  |-  ( z  e.  ( A (,) B )  |->  1 )  =  ( z  e.  ( A (,) B
)  |->  1 )
3129, 30dmmpti 5533 . . . 4  |-  dom  (
z  e.  ( A (,) B )  |->  1 )  =  ( A (,) B )
3228, 31syl6eq 2452 . . 3  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( A (,) B ) )
331, 2, 3, 4, 11, 12, 32cmvth 19828 . 2  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  (  _I  |`  ( A [,] B
) ) ) `  x ) )  =  ( ( ( (  _I  |`  ( A [,] B ) ) `  B )  -  (
(  _I  |`  ( A [,] B ) ) `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) ) )
341rexrd 9090 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
352rexrd 9090 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
361, 2, 3ltled 9177 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
37 ubicc2 10970 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
3834, 35, 36, 37syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( A [,] B ) )
39 fvresi 5883 . . . . . . . . . 10  |-  ( B  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 B )  =  B )
4038, 39syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 B )  =  B )
41 lbicc2 10969 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4234, 35, 36, 41syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( A [,] B ) )
43 fvresi 5883 . . . . . . . . . 10  |-  ( A  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4442, 43syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4540, 44oveq12d 6058 . . . . . . . 8  |-  ( ph  ->  ( ( (  _I  |`  ( A [,] B
) ) `  B
)  -  ( (  _I  |`  ( A [,] B ) ) `  A ) )  =  ( B  -  A
) )
4645adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
(  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  =  ( B  -  A ) )
4746oveq1d 6055 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( (  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  =  ( ( B  -  A )  x.  (
( RR  _D  F
) `  x )
) )
4827fveq1d 5689 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  ( ( z  e.  ( A (,) B )  |->  1 ) `
 x ) )
49 eqidd 2405 . . . . . . . . . 10  |-  ( z  =  x  ->  1  =  1 )
5049, 30, 29fvmpt3i 5768 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  (
( z  e.  ( A (,) B ) 
|->  1 ) `  x
)  =  1 )
5148, 50sylan9eq 2456 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  1 )
5251oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( ( F `  B
)  -  ( F `
 A ) )  x.  1 ) )
53 cncff 18876 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
544, 53syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5554, 38ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  B
)  e.  RR )
5654, 42ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  A
)  e.  RR )
5755, 56resubcld 9421 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
5857recnd 9070 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
5958adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
6059mulid1d 9061 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  1 )  =  ( ( F `  B )  -  ( F `  A )
) )
6152, 60eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
6247, 61eqeq12d 2418 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( ( (  _I  |`  ( A [,] B
) ) `  B
)  -  ( (  _I  |`  ( A [,] B ) ) `  A ) )  x.  ( ( RR  _D  F ) `  x
) )  =  ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  (  _I  |`  ( A [,] B
) ) ) `  x ) )  <->  ( ( B  -  A )  x.  ( ( RR  _D  F ) `  x
) )  =  ( ( F `  B
)  -  ( F `
 A ) ) ) )
632, 1resubcld 9421 . . . . . . . 8  |-  ( ph  ->  ( B  -  A
)  e.  RR )
6463recnd 9070 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
6564adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  e.  CC )
66 dvf 19747 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
6712feq2d 5540 . . . . . . . 8  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
6866, 67mpbii 203 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
6968ffvelrnda 5829 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
701, 2posdifd 9569 . . . . . . . . 9  |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
713, 70mpbid 202 . . . . . . . 8  |-  ( ph  ->  0  <  ( B  -  A ) )
7271gt0ne0d 9547 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
7372adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  =/=  0
)
7459, 65, 69, 73divmuld 9768 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  /  ( B  -  A ) )  =  ( ( RR 
_D  F ) `  x )  <->  ( ( B  -  A )  x.  ( ( RR  _D  F ) `  x
) )  =  ( ( F `  B
)  -  ( F `
 A ) ) ) )
7562, 74bitr4d 248 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( ( (  _I  |`  ( A [,] B
) ) `  B
)  -  ( (  _I  |`  ( A [,] B ) ) `  A ) )  x.  ( ( RR  _D  F ) `  x
) )  =  ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  (  _I  |`  ( A [,] B
) ) ) `  x ) )  <->  ( (
( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) )  =  ( ( RR  _D  F ) `  x
) ) )
76 eqcom 2406 . . . 4  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  (  _I  |`  ( A [,] B
) ) ) `  x ) )  =  ( ( ( (  _I  |`  ( A [,] B ) ) `  B )  -  (
(  _I  |`  ( A [,] B ) ) `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  <->  ( (
( (  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  =  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  (  _I  |`  ( A [,] B
) ) ) `  x ) ) )
77 eqcom 2406 . . . 4  |-  ( ( ( RR  _D  F
) `  x )  =  ( ( ( F `  B )  -  ( F `  A ) )  / 
( B  -  A
) )  <->  ( (
( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) )  =  ( ( RR  _D  F ) `  x
) )
7875, 76, 773bitr4g 280 . . 3  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  (  _I  |`  ( A [,] B
) ) ) `  x ) )  =  ( ( ( (  _I  |`  ( A [,] B ) ) `  B )  -  (
(  _I  |`  ( A [,] B ) ) `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  <->  ( ( RR  _D  F ) `  x )  =  ( ( ( F `  B )  -  ( F `  A )
)  /  ( B  -  A ) ) ) )
7978rexbidva 2683 . 2  |-  ( ph  ->  ( E. x  e.  ( A (,) B
) ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( ( (  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  <->  E. x  e.  ( A (,) B
) ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  B )  -  ( F `  A )
)  /  ( B  -  A ) ) ) )
8033, 79mpbid 202 1  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR  _D  F ) `  x
)  =  ( ( ( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226    _I cid 4453   dom cdm 4837   ran crn 4838    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   (,)cioo 10872   [,]cicc 10875   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvlip  19830  c1liplem1  19833  dvgt0lem1  19839  dvcvx  19857
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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