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.

Step	Hyp	Ref	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 )
7	1, 2, 6	syl2anc 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 ) )
10	7, 8, 9	sylancl 644	. . . 4 |- ( ph -> ( z e. ( A [,] B ) |-> z ) e. ( ( A [,] B ) -cn-> RR ) )
11	5, 10	syl5eqelr 2489	. . 3 |- ( ph -> ( _I |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
12		mvth.d	. . 3 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
13	5	oveq2i 6051	. . . . . 6 |- ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( RR _D ( _I |` ( A [,] B ) ) )
14		reex 9037	. . . . . . . . 9 |- RR e. _V
15	14	prid1 3872	. . . . . . . 8 |- RR e. { RR , CC }
16	15	a1i 11	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
17		simpr 448	. . . . . . . 8 |- ( ( ph /\ z e. RR ) -> z e. RR )
18	17	recnd 9070	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> z e. CC )
19		1re 9046	. . . . . . . 8 |- 1 e. RR
20	19	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> 1 e. RR )
21	16	dvmptid 19796	. . . . . . 7 |- ( ph -> ( RR _D ( z e. RR |-> z ) ) = ( z e. RR |-> 1 ) )
22		eqid 2404	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
23	22	tgioo2 18787	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
24		iccntr 18805	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
25	1, 2, 24	syl2anc 643	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
26	16, 18, 20, 21, 7, 23, 22, 25	dvmptres2 19801	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( z e. ( A (,) B ) |-> 1 ) )
27	13, 26	syl5eqr 2450	. . . . 5 |- ( ph -> ( RR _D ( _I |` ( A [,] B ) ) ) = ( z e. ( A (,) B ) |-> 1 ) )
28	27	dmeqd 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 )
31	29, 30	dmmpti 5533	. . . 4 |- dom ( z e. ( A (,) B ) |-> 1 ) = ( A (,) B )
32	28, 31	syl6eq 2452	. . 3 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = ( A (,) B ) )
33	1, 2, 3, 4, 11, 12, 32	cmvth 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 ) ) )
34	1	rexrd 9090	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
35	2	rexrd 9090	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
36	1, 2, 3	ltled 9177	. . . . . . . . . . 11 |- ( ph -> A <_ B )
37		ubicc2 10970	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
38	34, 35, 36, 37	syl3anc 1184	. . . . . . . . . 10 |- ( ph -> B e. ( A [,] B ) )
39		fvresi 5883	. . . . . . . . . 10 |- ( B e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
40	38, 39	syl 16	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
41		lbicc2 10969	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
42	34, 35, 36, 41	syl3anc 1184	. . . . . . . . . 10 |- ( ph -> A e. ( A [,] B ) )
43		fvresi 5883	. . . . . . . . . 10 |- ( A e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` A ) = A )
44	42, 43	syl 16	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` A ) = A )
45	40, 44	oveq12d 6058	. . . . . . . 8 |- ( ph -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
46	45	adantr 452	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
47	46	oveq1d 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 ) ) )
48	27	fveq1d 5689	. . . . . . . . 9 |- ( ph -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = ( ( z e. ( A (,) B ) |-> 1 ) ` x ) )
49		eqidd 2405	. . . . . . . . . 10 |- ( z = x -> 1 = 1 )
50	49, 30, 29	fvmpt3i 5768	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> ( ( z e. ( A (,) B ) |-> 1 ) ` x ) = 1 )
51	48, 50	sylan9eq 2456	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = 1 )
52	51	oveq2d 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 )
54	4, 53	syl 16	. . . . . . . . . . . 12 |- ( ph -> F : ( A [,] B ) --> RR )
55	54, 38	ffvelrnd 5830	. . . . . . . . . . 11 |- ( ph -> ( F ` B ) e. RR )
56	54, 42	ffvelrnd 5830	. . . . . . . . . . 11 |- ( ph -> ( F ` A ) e. RR )
57	55, 56	resubcld 9421	. . . . . . . . . 10 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
58	57	recnd 9070	. . . . . . . . 9 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. CC )
59	58	adantr 452	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
60	59	mulid1d 9061	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. 1 ) = ( ( F ` B ) - ( F ` A ) ) )
61	52, 60	eqtrd 2436	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) ) = ( ( F ` B ) - ( F ` A ) ) )
62	47, 61	eqeq12d 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 ) ) ) )
63	2, 1	resubcld 9421	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
64	63	recnd 9070	. . . . . . 7 |- ( ph -> ( B - A ) e. CC )
65	64	adantr 452	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) e. CC )
66		dvf 19747	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
67	12	feq2d 5540	. . . . . . . 8 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
68	66, 67	mpbii 203	. . . . . . 7 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
69	68	ffvelrnda 5829	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
70	1, 2	posdifd 9569	. . . . . . . . 9 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
71	3, 70	mpbid 202	. . . . . . . 8 |- ( ph -> 0 < ( B - A ) )
72	71	gt0ne0d 9547	. . . . . . 7 |- ( ph -> ( B - A ) =/= 0 )
73	72	adantr 452	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) =/= 0 )
74	59, 65, 69, 73	divmuld 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 ) ) ) )
75	62, 74	bitr4d 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 ) )
78	75, 76, 77	3bitr4g 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 ) ) ) )
79	78	rexbidva 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 ) ) ) )
80	33, 79	mpbid 202	1 |- ( ph -> E. x e. ( A (,) B ) ( ( RR _D F ) ` x ) = ( ( ( F ` B ) - ( F ` A ) ) / ( B - A ) ) )

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  ℂ_fldccnfld 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