Theorem mvth 21596

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.

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 5267	. . . 4 |- ( z e. ( A [,] B ) |-> z ) = ( _I |` ( A [,] B ) )
6		iccssre 11487	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
7	1, 2, 6	syl2anc 661	. . . . 5 |- ( ph -> ( A [,] B ) C_ RR )
8		ax-resscn 9449	. . . . 5 |- RR C_ CC
9		cncfmptid 20619	. . . . 5 |- ( ( ( A [,] B ) C_ RR /\ RR C_ CC ) -> ( z e. ( A [,] B ) |-> z ) e. ( ( A [,] B ) -cn-> RR ) )
10	7, 8, 9	sylancl 662	. . . 4 |- ( ph -> ( z e. ( A [,] B ) |-> z ) e. ( ( A [,] B ) -cn-> RR ) )
11	5, 10	syl5eqelr 2547	. . 3 |- ( ph -> ( _I |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
12		mvth.d	. . 3 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
13	5	oveq2i 6210	. . . . . 6 |- ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( RR _D ( _I |` ( A [,] B ) ) )
14		reelprrecn 9484	. . . . . . . 8 |- RR e. { RR , CC }
15	14	a1i 11	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
16		simpr 461	. . . . . . . 8 |- ( ( ph /\ z e. RR ) -> z e. RR )
17	16	recnd 9522	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> z e. CC )
18		1red 9511	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> 1 e. RR )
19	15	dvmptid 21563	. . . . . . 7 |- ( ph -> ( RR _D ( z e. RR |-> z ) ) = ( z e. RR |-> 1 ) )
20		eqid 2454	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 20511	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22		iccntr 20529	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
23	1, 2, 22	syl2anc 661	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
24	15, 17, 18, 19, 7, 21, 20, 23	dvmptres2 21568	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( z e. ( A (,) B ) |-> 1 ) )
25	13, 24	syl5eqr 2509	. . . . 5 |- ( ph -> ( RR _D ( _I |` ( A [,] B ) ) ) = ( z e. ( A (,) B ) |-> 1 ) )
26	25	dmeqd 5149	. . . 4 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = dom ( z e. ( A (,) B ) |-> 1 ) )
27		1ex 9491	. . . . 5 |- 1 e. _V
28		eqid 2454	. . . . 5 |- ( z e. ( A (,) B ) |-> 1 ) = ( z e. ( A (,) B ) |-> 1 )
29	27, 28	dmmpti 5647	. . . 4 |- dom ( z e. ( A (,) B ) |-> 1 ) = ( A (,) B )
30	26, 29	syl6eq 2511	. . 3 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = ( A (,) B ) )
31	1, 2, 3, 4, 11, 12, 30	cmvth 21595	. 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 ) ) )
32	1	rexrd 9543	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
33	2	rexrd 9543	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
34	1, 2, 3	ltled 9632	. . . . . . . . . . 11 |- ( ph -> A <_ B )
35		ubicc2 11518	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
36	32, 33, 34, 35	syl3anc 1219	. . . . . . . . . 10 |- ( ph -> B e. ( A [,] B ) )
37		fvresi 6012	. . . . . . . . . 10 |- ( B e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
38	36, 37	syl 16	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
39		lbicc2 11517	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
40	32, 33, 34, 39	syl3anc 1219	. . . . . . . . . 10 |- ( ph -> A e. ( A [,] B ) )
41		fvresi 6012	. . . . . . . . . 10 |- ( A e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` A ) = A )
42	40, 41	syl 16	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` A ) = A )
43	38, 42	oveq12d 6217	. . . . . . . 8 |- ( ph -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
44	43	adantr 465	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
45	44	oveq1d 6214	. . . . . 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 ) ) )
46	25	fveq1d 5800	. . . . . . . . 9 |- ( ph -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = ( ( z e. ( A (,) B ) |-> 1 ) ` x ) )
47		eqidd 2455	. . . . . . . . . 10 |- ( z = x -> 1 = 1 )
48	47, 28, 27	fvmpt3i 5886	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> ( ( z e. ( A (,) B ) |-> 1 ) ` x ) = 1 )
49	46, 48	sylan9eq 2515	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = 1 )
50	49	oveq2d 6215	. . . . . . 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 20600	. . . . . . . . . . . . 13 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
52	4, 51	syl 16	. . . . . . . . . . . 12 |- ( ph -> F : ( A [,] B ) --> RR )
53	52, 36	ffvelrnd 5952	. . . . . . . . . . 11 |- ( ph -> ( F ` B ) e. RR )
54	52, 40	ffvelrnd 5952	. . . . . . . . . . 11 |- ( ph -> ( F ` A ) e. RR )
55	53, 54	resubcld 9886	. . . . . . . . . 10 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
56	55	recnd 9522	. . . . . . . . 9 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. CC )
57	56	adantr 465	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
58	57	mulid1d 9513	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. 1 ) = ( ( F ` B ) - ( F ` A ) ) )
59	50, 58	eqtrd 2495	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) ) = ( ( F ` B ) - ( F ` A ) ) )
60	45, 59	eqeq12d 2476	. . . . 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 ) ) ) )
61	2, 1	resubcld 9886	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
62	61	recnd 9522	. . . . . . 7 |- ( ph -> ( B - A ) e. CC )
63	62	adantr 465	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) e. CC )
64		dvf 21514	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
65	12	feq2d 5654	. . . . . . . 8 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
66	64, 65	mpbii 211	. . . . . . 7 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
67	66	ffvelrnda 5951	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
68	1, 2	posdifd 10036	. . . . . . . . 9 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
69	3, 68	mpbid 210	. . . . . . . 8 |- ( ph -> 0 < ( B - A ) )
70	69	gt0ne0d 10014	. . . . . . 7 |- ( ph -> ( B - A ) =/= 0 )
71	70	adantr 465	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) =/= 0 )
72	57, 63, 67, 71	divmuld 10239	. . . . 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 ) ) ) )
73	60, 72	bitr4d 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 2463	. . . 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 2463	. . . 4 |- ( ( ( RR _D F ) ` x ) = ( ( ( F ` B ) - ( F ` A ) ) / ( B - A ) ) <-> ( ( ( F ` B ) - ( F ` A ) ) / ( B - A ) ) = ( ( RR _D F ) ` x ) )
76	73, 74, 75	3bitr4g 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 ) ) ) )
77	76	rexbidva 2861	. 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 ) ) ) )
78	31, 77	mpbid 210	1 |- ( ph -> E. x e. ( A (,) B ) ( ( RR _D F ) ` x ) = ( ( ( F ` B ) - ( F ` A ) ) / ( B - A ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2647   E.wrex 2799   C_ wss 3435   {cpr 3986   class class class wbr 4399   |-> cmpt 4457   _I cid 4738   dom cdm 4947   ran crn 4948   |` cres 4949   -->wf 5521   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393   x. cmul 9397   RR*cxr 9527   < clt 9528   <_ cle 9529   - cmin 9705   / cdiv 10103   (,)cioo 11410   [,]cicc 11413   TopOpenctopn 14478   topGenctg 14494  ℂ_fldccnfld 17942   intcnt 18752   -cn->ccncf 20583   _D cdv 21470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  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 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-cmp 19121  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474

This theorem is referenced by:  dvlip  21597  c1liplem1  21600  dvgt0lem1  21606  dvcvx  21624