Theorem mvth 22125

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 5326	. . . 4 |- ( z e. ( A [,] B ) |-> z ) = ( _I |` ( A [,] B ) )
6		iccssre 11602	. . . . . 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 9545	. . . . 5 |- RR C_ CC
9		cncfmptid 21148	. . . . 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 2560	. . 3 |- ( ph -> ( _I |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
12		mvth.d	. . 3 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
13	5	oveq2i 6293	. . . . . 6 |- ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( RR _D ( _I |` ( A [,] B ) ) )
14		reelprrecn 9580	. . . . . . . 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 9618	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> z e. CC )
18		1red 9607	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> 1 e. RR )
19	15	dvmptid 22092	. . . . . . 7 |- ( ph -> ( RR _D ( z e. RR |-> z ) ) = ( z e. RR |-> 1 ) )
20		eqid 2467	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21040	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22		iccntr 21058	. . . . . . . 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 22097	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( z e. ( A (,) B ) |-> 1 ) )
25	13, 24	syl5eqr 2522	. . . . 5 |- ( ph -> ( RR _D ( _I |` ( A [,] B ) ) ) = ( z e. ( A (,) B ) |-> 1 ) )
26	25	dmeqd 5203	. . . 4 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = dom ( z e. ( A (,) B ) |-> 1 ) )
27		1ex 9587	. . . . 5 |- 1 e. _V
28		eqid 2467	. . . . 5 |- ( z e. ( A (,) B ) |-> 1 ) = ( z e. ( A (,) B ) |-> 1 )
29	27, 28	dmmpti 5708	. . . 4 |- dom ( z e. ( A (,) B ) |-> 1 ) = ( A (,) B )
30	26, 29	syl6eq 2524	. . 3 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = ( A (,) B ) )
31	1, 2, 3, 4, 11, 12, 30	cmvth 22124	. 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 9639	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
33	2	rexrd 9639	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
34	1, 2, 3	ltled 9728	. . . . . . . . . . 11 |- ( ph -> A <_ B )
35		ubicc2 11633	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
36	32, 33, 34, 35	syl3anc 1228	. . . . . . . . . 10 |- ( ph -> B e. ( A [,] B ) )
37		fvresi 6085	. . . . . . . . . 10 |- ( B e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
38	36, 37	syl 16	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
39		lbicc2 11632	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
40	32, 33, 34, 39	syl3anc 1228	. . . . . . . . . 10 |- ( ph -> A e. ( A [,] B ) )
41		fvresi 6085	. . . . . . . . . 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 6300	. . . . . . . 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 6297	. . . . . 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 5866	. . . . . . . . 9 |- ( ph -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = ( ( z e. ( A (,) B ) |-> 1 ) ` x ) )
47		eqidd 2468	. . . . . . . . . 10 |- ( z = x -> 1 = 1 )
48	47, 28, 27	fvmpt3i 5952	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> ( ( z e. ( A (,) B ) |-> 1 ) ` x ) = 1 )
49	46, 48	sylan9eq 2528	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = 1 )
50	49	oveq2d 6298	. . . . . . 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 21129	. . . . . . . . . . . . 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 6020	. . . . . . . . . . 11 |- ( ph -> ( F ` B ) e. RR )
54	52, 40	ffvelrnd 6020	. . . . . . . . . . 11 |- ( ph -> ( F ` A ) e. RR )
55	53, 54	resubcld 9983	. . . . . . . . . 10 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
56	55	recnd 9618	. . . . . . . . 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 9609	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. 1 ) = ( ( F ` B ) - ( F ` A ) ) )
59	50, 58	eqtrd 2508	. . . . . 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 2489	. . . . 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 9983	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
62	61	recnd 9618	. . . . . . 7 |- ( ph -> ( B - A ) e. CC )
63	62	adantr 465	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) e. CC )
64		dvf 22043	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
65	12	feq2d 5716	. . . . . . . 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 6019	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
68	1, 2	posdifd 10135	. . . . . . . . 9 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
69	3, 68	mpbid 210	. . . . . . . 8 |- ( ph -> 0 < ( B - A ) )
70	69	gt0ne0d 10113	. . . . . . 7 |- ( ph -> ( B - A ) =/= 0 )
71	70	adantr 465	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) =/= 0 )
72	57, 63, 67, 71	divmuld 10338	. . . . 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 2476	. . . 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 2476	. . . 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 2970	. 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 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   C_ wss 3476   {cpr 4029   class class class wbr 4447   |-> cmpt 4505   _I cid 4790   dom cdm 4999   ran crn 5000   |` cres 5001   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   x. cmul 9493   RR*cxr 9623   < clt 9624   <_ cle 9625   - cmin 9801   / cdiv 10202   (,)cioo 11525   [,]cicc 11528   TopOpenctopn 14670   topGenctg 14686  ℂ_fldccnfld 18188   intcnt 19281   -cn->ccncf 21112   _D cdv 21999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-cmp 19650  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003

This theorem is referenced by:  dvlip  22126  c1liplem1  22129  dvgt0lem1  22135  dvcvx  22153  dvbdfbdioolem1  31258