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Theorem mvth 22577
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 ]. This is Metamath 100 proof #75. (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 5269 . . . 4  |-  ( z  e.  ( A [,] B )  |->  z )  =  (  _I  |`  ( A [,] B ) )
6 iccssre 11577 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
71, 2, 6syl2anc 659 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ax-resscn 9499 . . . . 5  |-  RR  C_  CC
9 cncfmptid 21600 . . . . 5  |-  ( ( ( A [,] B
)  C_  RR  /\  RR  C_  CC )  ->  (
z  e.  ( A [,] B )  |->  z )  e.  ( ( A [,] B )
-cn-> RR ) )
107, 8, 9sylancl 660 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  z )  e.  ( ( A [,] B
) -cn-> RR ) )
115, 10syl5eqelr 2495 . . 3  |-  ( ph  ->  (  _I  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
12 mvth.d . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
135oveq2i 6245 . . . . . 6  |-  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  z ) )  =  ( RR  _D  (  _I  |`  ( A [,] B ) ) )
14 reelprrecn 9534 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
1514a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
16 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  RR )
1716recnd 9572 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  CC )
18 1red 9561 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  1  e.  RR )
1915dvmptid 22544 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  RR  |->  z ) )  =  ( z  e.  RR  |->  1 ) )
20 eqid 2402 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21492 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
22 iccntr 21510 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
231, 2, 22syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2415, 17, 18, 19, 7, 21, 20, 23dvmptres2 22549 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  z ) )  =  ( z  e.  ( A (,) B )  |->  1 ) )
2513, 24syl5eqr 2457 . . . . 5  |-  ( ph  ->  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( z  e.  ( A (,) B
)  |->  1 ) )
2625dmeqd 5147 . . . 4  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  dom  (
z  e.  ( A (,) B )  |->  1 ) )
27 1ex 9541 . . . . 5  |-  1  e.  _V
28 eqid 2402 . . . . 5  |-  ( z  e.  ( A (,) B )  |->  1 )  =  ( z  e.  ( A (,) B
)  |->  1 )
2927, 28dmmpti 5649 . . . 4  |-  dom  (
z  e.  ( A (,) B )  |->  1 )  =  ( A (,) B )
3026, 29syl6eq 2459 . . 3  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( A (,) B ) )
311, 2, 3, 4, 11, 12, 30cmvth 22576 . 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 ) ) )
321rexrd 9593 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
332rexrd 9593 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
341, 2, 3ltled 9685 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
35 ubicc2 11608 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
3632, 33, 34, 35syl3anc 1230 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( A [,] B ) )
37 fvresi 6033 . . . . . . . . . 10  |-  ( B  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 B )  =  B )
3836, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 B )  =  B )
39 lbicc2 11607 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4032, 33, 34, 39syl3anc 1230 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( A [,] B ) )
41 fvresi 6033 . . . . . . . . . 10  |-  ( A  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4240, 41syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4338, 42oveq12d 6252 . . . . . . . 8  |-  ( ph  ->  ( ( (  _I  |`  ( A [,] B
) ) `  B
)  -  ( (  _I  |`  ( A [,] B ) ) `  A ) )  =  ( B  -  A
) )
4443adantr 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
(  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  =  ( B  -  A ) )
4544oveq1d 6249 . . . . . 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 )
) )
4625fveq1d 5807 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  ( ( z  e.  ( A (,) B )  |->  1 ) `
 x ) )
47 eqidd 2403 . . . . . . . . . 10  |-  ( z  =  x  ->  1  =  1 )
4847, 28, 27fvmpt3i 5893 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  (
( z  e.  ( A (,) B ) 
|->  1 ) `  x
)  =  1 )
4946, 48sylan9eq 2463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  1 )
5049oveq2d 6250 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( ( F `  B
)  -  ( F `
 A ) )  x.  1 ) )
51 cncff 21581 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
524, 51syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5352, 36ffvelrnd 5966 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  B
)  e.  RR )
5452, 40ffvelrnd 5966 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  A
)  e.  RR )
5553, 54resubcld 9948 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
5655recnd 9572 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
5756adantr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
5857mulid1d 9563 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  1 )  =  ( ( F `  B )  -  ( F `  A )
) )
5950, 58eqtrd 2443 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
6045, 59eqeq12d 2424 . . . . 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 ) ) ) )
612, 1resubcld 9948 . . . . . . . 8  |-  ( ph  ->  ( B  -  A
)  e.  RR )
6261recnd 9572 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
6362adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  e.  CC )
64 dvf 22495 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
6512feq2d 5657 . . . . . . . 8  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
6664, 65mpbii 211 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
6766ffvelrnda 5965 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
681, 2posdifd 10099 . . . . . . . . 9  |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
693, 68mpbid 210 . . . . . . . 8  |-  ( ph  ->  0  <  ( B  -  A ) )
7069gt0ne0d 10077 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
7170adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  =/=  0
)
7257, 63, 67, 71divmuld 10303 . . . . 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 ) ) ) )
7360, 72bitr4d 256 . . . 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
) ) )
74 eqcom 2411 . . . 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 ) ) )
75 eqcom 2411 . . . 4  |-  ( ( ( RR  _D  F
) `  x )  =  ( ( ( F `  B )  -  ( F `  A ) )  / 
( B  -  A
) )  <->  ( (
( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) )  =  ( ( RR  _D  F ) `  x
) )
7673, 74, 753bitr4g 288 . . 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 ) ) ) )
7776rexbidva 2914 . 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 ) ) ) )
7831, 77mpbid 210 1  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR  _D  F ) `  x
)  =  ( ( ( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754    C_ wss 3413   {cpr 3973   class class class wbr 4394    |-> cmpt 4452    _I cid 4732   dom cdm 4942   ran crn 4943    |` cres 4944   -->wf 5521   ` cfv 5525  (class class class)co 6234   CCcc 9440   RRcr 9441   0cc0 9442   1c1 9443    x. cmul 9447   RR*cxr 9577    < clt 9578    <_ cle 9579    - cmin 9761    / cdiv 10167   (,)cioo 11500   [,]cicc 11503   TopOpenctopn 14928   topGenctg 14944  ℂfldccnfld 18632   intcnt 19702   -cn->ccncf 21564    _D cdv 22451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521  ax-mulf 9522
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-supp 6857  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-ixp 7428  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fsupp 7784  df-fi 7825  df-sup 7855  df-oi 7889  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-q 11146  df-rp 11184  df-xneg 11289  df-xadd 11290  df-xmul 11291  df-ioo 11504  df-ico 11506  df-icc 11507  df-fz 11644  df-fzo 11768  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-mulr 14815  df-starv 14816  df-sca 14817  df-vsca 14818  df-ip 14819  df-tset 14820  df-ple 14821  df-ds 14823  df-unif 14824  df-hom 14825  df-cco 14826  df-rest 14929  df-topn 14930  df-0g 14948  df-gsum 14949  df-topgen 14950  df-pt 14951  df-prds 14954  df-xrs 15008  df-qtop 15013  df-imas 15014  df-xps 15016  df-mre 15092  df-mrc 15093  df-acs 15095  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-mulg 16276  df-cntz 16571  df-cmn 17016  df-psmet 18623  df-xmet 18624  df-met 18625  df-bl 18626  df-mopn 18627  df-fbas 18628  df-fg 18629  df-cnfld 18633  df-top 19583  df-bases 19585  df-topon 19586  df-topsp 19587  df-cld 19704  df-ntr 19705  df-cls 19706  df-nei 19784  df-lp 19822  df-perf 19823  df-cn 19913  df-cnp 19914  df-haus 20001  df-cmp 20072  df-tx 20247  df-hmeo 20440  df-fil 20531  df-fm 20623  df-flim 20624  df-flf 20625  df-xms 21007  df-ms 21008  df-tms 21009  df-cncf 21566  df-limc 22454  df-dv 22455
This theorem is referenced by:  dvlip  22578  c1liplem1  22581  dvgt0lem1  22587  dvcvx  22605  dvbdfbdioolem1  37075
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