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Theorem mvth 22944
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 5159 . . . 4  |-  ( z  e.  ( A [,] B )  |->  z )  =  (  _I  |`  ( A [,] B ) )
6 iccssre 11716 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
71, 2, 6syl2anc 667 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ax-resscn 9596 . . . . 5  |-  RR  C_  CC
9 cncfmptid 21944 . . . . 5  |-  ( ( ( A [,] B
)  C_  RR  /\  RR  C_  CC )  ->  (
z  e.  ( A [,] B )  |->  z )  e.  ( ( A [,] B )
-cn-> RR ) )
107, 8, 9sylancl 668 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  z )  e.  ( ( A [,] B
) -cn-> RR ) )
115, 10syl5eqelr 2534 . . 3  |-  ( ph  ->  (  _I  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
12 mvth.d . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
135oveq2i 6301 . . . . . 6  |-  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  z ) )  =  ( RR  _D  (  _I  |`  ( A [,] B ) ) )
14 reelprrecn 9631 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
1514a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
16 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  RR )
1716recnd 9669 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  CC )
18 1red 9658 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  1  e.  RR )
1915dvmptid 22911 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  RR  |->  z ) )  =  ( z  e.  RR  |->  1 ) )
20 eqid 2451 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21821 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
22 iccntr 21839 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
231, 2, 22syl2anc 667 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2415, 17, 18, 19, 7, 21, 20, 23dvmptres2 22916 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  z ) )  =  ( z  e.  ( A (,) B )  |->  1 ) )
2513, 24syl5eqr 2499 . . . . 5  |-  ( ph  ->  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( z  e.  ( A (,) B
)  |->  1 ) )
2625dmeqd 5037 . . . 4  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  dom  (
z  e.  ( A (,) B )  |->  1 ) )
27 1ex 9638 . . . . 5  |-  1  e.  _V
28 eqid 2451 . . . . 5  |-  ( z  e.  ( A (,) B )  |->  1 )  =  ( z  e.  ( A (,) B
)  |->  1 )
2927, 28dmmpti 5707 . . . 4  |-  dom  (
z  e.  ( A (,) B )  |->  1 )  =  ( A (,) B )
3026, 29syl6eq 2501 . . 3  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( A (,) B ) )
311, 2, 3, 4, 11, 12, 30cmvth 22943 . 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 9690 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
332rexrd 9690 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
341, 2, 3ltled 9783 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
35 ubicc2 11749 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
3632, 33, 34, 35syl3anc 1268 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( A [,] B ) )
37 fvresi 6090 . . . . . . . . . 10  |-  ( B  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 B )  =  B )
3836, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 B )  =  B )
39 lbicc2 11748 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4032, 33, 34, 39syl3anc 1268 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( A [,] B ) )
41 fvresi 6090 . . . . . . . . . 10  |-  ( A  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4240, 41syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4338, 42oveq12d 6308 . . . . . . . 8  |-  ( ph  ->  ( ( (  _I  |`  ( A [,] B
) ) `  B
)  -  ( (  _I  |`  ( A [,] B ) ) `  A ) )  =  ( B  -  A
) )
4443adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
(  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  =  ( B  -  A ) )
4544oveq1d 6305 . . . . . 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 5867 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  ( ( z  e.  ( A (,) B )  |->  1 ) `
 x ) )
47 eqidd 2452 . . . . . . . . . 10  |-  ( z  =  x  ->  1  =  1 )
4847, 28, 27fvmpt3i 5953 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  (
( z  e.  ( A (,) B ) 
|->  1 ) `  x
)  =  1 )
4946, 48sylan9eq 2505 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  1 )
5049oveq2d 6306 . . . . . . 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 21925 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
524, 51syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5352, 36ffvelrnd 6023 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  B
)  e.  RR )
5452, 40ffvelrnd 6023 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  A
)  e.  RR )
5553, 54resubcld 10047 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
5655recnd 9669 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
5756adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
5857mulid1d 9660 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  1 )  =  ( ( F `  B )  -  ( F `  A )
) )
5950, 58eqtrd 2485 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
6045, 59eqeq12d 2466 . . . . 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 10047 . . . . . . . 8  |-  ( ph  ->  ( B  -  A
)  e.  RR )
6261recnd 9669 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
6362adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  e.  CC )
64 dvf 22862 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
6512feq2d 5715 . . . . . . . 8  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
6664, 65mpbii 215 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
6766ffvelrnda 6022 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
681, 2posdifd 10200 . . . . . . . . 9  |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
693, 68mpbid 214 . . . . . . . 8  |-  ( ph  ->  0  <  ( B  -  A ) )
7069gt0ne0d 10178 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
7170adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  =/=  0
)
7257, 63, 67, 71divmuld 10405 . . . . 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 260 . . . 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 2458 . . . 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 2458 . . . 4  |-  ( ( ( RR  _D  F
) `  x )  =  ( ( ( F `  B )  -  ( F `  A ) )  / 
( B  -  A
) )  <->  ( (
( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) )  =  ( ( RR  _D  F ) `  x
) )
7673, 74, 753bitr4g 292 . . 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 2898 . 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 214 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 371    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738    C_ wss 3404   {cpr 3970   class class class wbr 4402    |-> cmpt 4461    _I cid 4744   dom cdm 4834   ran crn 4835    |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   (,)cioo 11635   [,]cicc 11638   TopOpenctopn 15320   topGenctg 15336  ℂfldccnfld 18970   intcnt 20032   -cn->ccncf 21908    _D cdv 22818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822
This theorem is referenced by:  dvlip  22945  c1liplem1  22948  dvgt0lem1  22954  dvcvx  22972  dvbdfbdioolem1  37800
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