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Theorem mvth 23023
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 5165 . . . 4  |-  ( z  e.  ( A [,] B )  |->  z )  =  (  _I  |`  ( A [,] B ) )
6 iccssre 11741 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
71, 2, 6syl2anc 673 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ax-resscn 9614 . . . . 5  |-  RR  C_  CC
9 cncfmptid 22022 . . . . 5  |-  ( ( ( A [,] B
)  C_  RR  /\  RR  C_  CC )  ->  (
z  e.  ( A [,] B )  |->  z )  e.  ( ( A [,] B )
-cn-> RR ) )
107, 8, 9sylancl 675 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  z )  e.  ( ( A [,] B
) -cn-> RR ) )
115, 10syl5eqelr 2554 . . 3  |-  ( ph  ->  (  _I  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
12 mvth.d . . 3  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
135oveq2i 6319 . . . . . 6  |-  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  z ) )  =  ( RR  _D  (  _I  |`  ( A [,] B ) ) )
14 reelprrecn 9649 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
1514a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
16 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  RR )
1716recnd 9687 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  z  e.  CC )
18 1red 9676 . . . . . . 7  |-  ( (
ph  /\  z  e.  RR )  ->  1  e.  RR )
1915dvmptid 22990 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  RR  |->  z ) )  =  ( z  e.  RR  |->  1 ) )
20 eqid 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2120tgioo2 21899 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
22 iccntr 21917 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
231, 2, 22syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2415, 17, 18, 19, 7, 21, 20, 23dvmptres2 22995 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  z ) )  =  ( z  e.  ( A (,) B )  |->  1 ) )
2513, 24syl5eqr 2519 . . . . 5  |-  ( ph  ->  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( z  e.  ( A (,) B
)  |->  1 ) )
2625dmeqd 5042 . . . 4  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  dom  (
z  e.  ( A (,) B )  |->  1 ) )
27 1ex 9656 . . . . 5  |-  1  e.  _V
28 eqid 2471 . . . . 5  |-  ( z  e.  ( A (,) B )  |->  1 )  =  ( z  e.  ( A (,) B
)  |->  1 )
2927, 28dmmpti 5717 . . . 4  |-  dom  (
z  e.  ( A (,) B )  |->  1 )  =  ( A (,) B )
3026, 29syl6eq 2521 . . 3  |-  ( ph  ->  dom  ( RR  _D  (  _I  |`  ( A [,] B ) ) )  =  ( A (,) B ) )
311, 2, 3, 4, 11, 12, 30cmvth 23022 . 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 9708 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR* )
332rexrd 9708 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR* )
341, 2, 3ltled 9800 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
35 ubicc2 11775 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
3632, 33, 34, 35syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ( A [,] B ) )
37 fvresi 6106 . . . . . . . . . 10  |-  ( B  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 B )  =  B )
3836, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 B )  =  B )
39 lbicc2 11774 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
4032, 33, 34, 39syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( A [,] B ) )
41 fvresi 6106 . . . . . . . . . 10  |-  ( A  e.  ( A [,] B )  ->  (
(  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4240, 41syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (  _I  |`  ( A [,] B ) ) `
 A )  =  A )
4338, 42oveq12d 6326 . . . . . . . 8  |-  ( ph  ->  ( ( (  _I  |`  ( A [,] B
) ) `  B
)  -  ( (  _I  |`  ( A [,] B ) ) `  A ) )  =  ( B  -  A
) )
4443adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
(  _I  |`  ( A [,] B ) ) `
 B )  -  ( (  _I  |`  ( A [,] B ) ) `
 A ) )  =  ( B  -  A ) )
4544oveq1d 6323 . . . . . 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 5881 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  ( ( z  e.  ( A (,) B )  |->  1 ) `
 x ) )
47 eqidd 2472 . . . . . . . . . 10  |-  ( z  =  x  ->  1  =  1 )
4847, 28, 27fvmpt3i 5968 . . . . . . . . 9  |-  ( x  e.  ( A (,) B )  ->  (
( z  e.  ( A (,) B ) 
|->  1 ) `  x
)  =  1 )
4946, 48sylan9eq 2525 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  (  _I  |`  ( A [,] B ) ) ) `  x )  =  1 )
5049oveq2d 6324 . . . . . . 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 22003 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
524, 51syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5352, 36ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  B
)  e.  RR )
5452, 40ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ph  ->  ( F `  A
)  e.  RR )
5553, 54resubcld 10068 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
5655recnd 9687 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
5756adantr 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
5857mulid1d 9678 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  1 )  =  ( ( F `  B )  -  ( F `  A )
) )
5950, 58eqtrd 2505 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  (  _I  |`  ( A [,] B ) ) ) `  x ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
6045, 59eqeq12d 2486 . . . . 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 10068 . . . . . . . 8  |-  ( ph  ->  ( B  -  A
)  e.  RR )
6261recnd 9687 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  CC )
6362adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  e.  CC )
64 dvf 22941 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
6512feq2d 5725 . . . . . . . 8  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
6664, 65mpbii 216 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
6766ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
681, 2posdifd 10221 . . . . . . . . 9  |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
693, 68mpbid 215 . . . . . . . 8  |-  ( ph  ->  0  <  ( B  -  A ) )
7069gt0ne0d 10199 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  =/=  0 )
7170adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( B  -  A )  =/=  0
)
7257, 63, 67, 71divmuld 10427 . . . . 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 264 . . . 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 2478 . . . 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 2478 . . . 4  |-  ( ( ( RR  _D  F
) `  x )  =  ( ( ( F `  B )  -  ( F `  A ) )  / 
( B  -  A
) )  <->  ( (
( F `  B
)  -  ( F `
 A ) )  /  ( B  -  A ) )  =  ( ( RR  _D  F ) `  x
) )
7673, 74, 753bitr4g 296 . . 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 2889 . 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 215 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 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    C_ wss 3390   {cpr 3961   class class class wbr 4395    |-> cmpt 4454    _I cid 4749   dom cdm 4839   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   (,)cioo 11660   [,]cicc 11663   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   intcnt 20109   -cn->ccncf 21986    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  dvlip  23024  c1liplem1  23027  dvgt0lem1  23033  dvcvx  23051  dvbdfbdioolem1  37897
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