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Theorem mvth 19864
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 5186 . . . 4  |-  ( z  e.  ( A [,] B )  |->  z )  =  (  _I  |`  ( A [,] B ) )
6 iccssre 10981 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
71, 2, 6syl2anc 643 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ax-resscn 9036 . . . . 5  |-  RR  C_  CC
9 cncfmptid 18930 . . . . 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 2520 . . 3  |-  ( ph  ->  (  _I  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
12 mvth.d . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
135oveq2i 6083 . . . . . 6  |-  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  z ) )  =  ( RR  _D  (  _I  |`  ( A [,] B ) ) )
14 reex 9070 . . . . . . . . 9  |-  RR  e.  _V
1514prid1 3904 . . . . . . . 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 9103 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  CC )
19 1re 9079 . . . . . . . 8  |-  1  e.  RR
2019a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  1  e.  RR )
2116dvmptid 19831 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  RR  |->  z ) )  =  ( z  e.  RR  |->  1 ) )
22 eqid 2435 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2322tgioo2 18822 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
24 iccntr 18840 . . . . . . . 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 19836 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  z ) )  =  ( z  e.  ( A (,) B )  |->  1 ) )
2713, 26syl5eqr 2481 . . . . 5  |-  ( ph  ->  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( z  e.  ( A (,) B
)  |->  1 ) )
2827dmeqd 5063 . . . 4  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  dom  (
z  e.  ( A (,) B )  |->  1 ) )
29 1ex 9075 . . . . 5  |-  1  e.  _V
30 eqid 2435 . . . . 5  |-  ( z  e.  ( A (,) B )  |->  1 )  =  ( z  e.  ( A (,) B
)  |->  1 )
3129, 30dmmpti 5565 . . . 4  |-  dom  (
z  e.  ( A (,) B )  |->  1 )  =  ( A (,) B )
3228, 31syl6eq 2483 . . 3  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( A (,) B ) )
331, 2, 3, 4, 11, 12, 32cmvth 19863 . 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 9123 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
352rexrd 9123 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
361, 2, 3ltled 9210 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
37 ubicc2 11003 . . . . . . . . . . 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 5915 . . . . . . . . . 10  |-  ( B  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 B )  =  B )
4038, 39syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 B )  =  B )
41 lbicc2 11002 . . . . . . . . . . 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 5915 . . . . . . . . . 10  |-  ( A  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4442, 43syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4540, 44oveq12d 6090 . . . . . . . 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 6087 . . . . . 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 5721 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  ( ( z  e.  ( A (,) B )  |->  1 ) `
 x ) )
49 eqidd 2436 . . . . . . . . . 10  |-  ( z  =  x  ->  1  =  1 )
5049, 30, 29fvmpt3i 5800 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  (
( z  e.  ( A (,) B ) 
|->  1 ) `  x
)  =  1 )
5148, 50sylan9eq 2487 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  1 )
5251oveq2d 6088 . . . . . . 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 18911 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
544, 53syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5554, 38ffvelrnd 5862 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  B
)  e.  RR )
5654, 42ffvelrnd 5862 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  A
)  e.  RR )
5755, 56resubcld 9454 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
5857recnd 9103 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
5958adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
6059mulid1d 9094 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  1 )  =  ( ( F `  B )  -  ( F `  A )
) )
6152, 60eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
6247, 61eqeq12d 2449 . . . . 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 9454 . . . . . . . 8  |-  ( ph  ->  ( B  -  A
)  e.  RR )
6463recnd 9103 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
6564adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  e.  CC )
66 dvf 19782 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
6712feq2d 5572 . . . . . . . 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 5861 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
701, 2posdifd 9602 . . . . . . . . 9  |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
713, 70mpbid 202 . . . . . . . 8  |-  ( ph  ->  0  <  ( B  -  A ) )
7271gt0ne0d 9580 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
7372adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  =/=  0
)
7459, 65, 69, 73divmuld 9801 . . . . 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 2437 . . . 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 2437 . . . 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 2714 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    C_ wss 3312   {cpr 3807   class class class wbr 4204    e. cmpt 4258    _I cid 4485   dom cdm 4869   ran crn 4870    |` cres 4871   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    x. cmul 8984   RR*cxr 9108    < clt 9109    <_ cle 9110    - cmin 9280    / cdiv 9666   (,)cioo 10905   [,]cicc 10908   TopOpenctopn 13637   topGenctg 13653  ℂfldccnfld 16691   intcnt 17069   -cn->ccncf 18894    _D cdv 19738
This theorem is referenced by:  dvlip  19865  c1liplem1  19868  dvgt0lem1  19874  dvcvx  19892
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-cmp 17438  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742
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