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.

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 5159	. . . 4 |- ( z e. ( A [,] B ) |-> z ) = ( _I |` ( A [,] B ) )
6		iccssre 11716	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
7	1, 2, 6	syl2anc 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 ) )
10	7, 8, 9	sylancl 668	. . . 4 |- ( ph -> ( z e. ( A [,] B ) |-> z ) e. ( ( A [,] B ) -cn-> RR ) )
11	5, 10	syl5eqelr 2534	. . 3 |- ( ph -> ( _I |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
12		mvth.d	. . 3 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
13	5	oveq2i 6301	. . . . . 6 |- ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( RR _D ( _I |` ( A [,] B ) ) )
14		reelprrecn 9631	. . . . . . . 8 |- RR e. { RR , CC }
15	14	a1i 11	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
16		simpr 463	. . . . . . . 8 |- ( ( ph /\ z e. RR ) -> z e. RR )
17	16	recnd 9669	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> z e. CC )
18		1red 9658	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> 1 e. RR )
19	15	dvmptid 22911	. . . . . . 7 |- ( ph -> ( RR _D ( z e. RR |-> z ) ) = ( z e. RR |-> 1 ) )
20		eqid 2451	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21821	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22		iccntr 21839	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
23	1, 2, 22	syl2anc 667	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
24	15, 17, 18, 19, 7, 21, 20, 23	dvmptres2 22916	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( z e. ( A (,) B ) |-> 1 ) )
25	13, 24	syl5eqr 2499	. . . . 5 |- ( ph -> ( RR _D ( _I |` ( A [,] B ) ) ) = ( z e. ( A (,) B ) |-> 1 ) )
26	25	dmeqd 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 )
29	27, 28	dmmpti 5707	. . . 4 |- dom ( z e. ( A (,) B ) |-> 1 ) = ( A (,) B )
30	26, 29	syl6eq 2501	. . 3 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = ( A (,) B ) )
31	1, 2, 3, 4, 11, 12, 30	cmvth 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 ) ) )
32	1	rexrd 9690	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
33	2	rexrd 9690	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
34	1, 2, 3	ltled 9783	. . . . . . . . . . 11 |- ( ph -> A <_ B )
35		ubicc2 11749	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
36	32, 33, 34, 35	syl3anc 1268	. . . . . . . . . 10 |- ( ph -> B e. ( A [,] B ) )
37		fvresi 6090	. . . . . . . . . 10 |- ( B e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
38	36, 37	syl 17	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
39		lbicc2 11748	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
40	32, 33, 34, 39	syl3anc 1268	. . . . . . . . . 10 |- ( ph -> A e. ( A [,] B ) )
41		fvresi 6090	. . . . . . . . . 10 |- ( A e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` A ) = A )
42	40, 41	syl 17	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` A ) = A )
43	38, 42	oveq12d 6308	. . . . . . . 8 |- ( ph -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
44	43	adantr 467	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
45	44	oveq1d 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 ) ) )
46	25	fveq1d 5867	. . . . . . . . 9 |- ( ph -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = ( ( z e. ( A (,) B ) |-> 1 ) ` x ) )
47		eqidd 2452	. . . . . . . . . 10 |- ( z = x -> 1 = 1 )
48	47, 28, 27	fvmpt3i 5953	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> ( ( z e. ( A (,) B ) |-> 1 ) ` x ) = 1 )
49	46, 48	sylan9eq 2505	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = 1 )
50	49	oveq2d 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 )
52	4, 51	syl 17	. . . . . . . . . . . 12 |- ( ph -> F : ( A [,] B ) --> RR )
53	52, 36	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ph -> ( F ` B ) e. RR )
54	52, 40	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ph -> ( F ` A ) e. RR )
55	53, 54	resubcld 10047	. . . . . . . . . 10 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
56	55	recnd 9669	. . . . . . . . 9 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. CC )
57	56	adantr 467	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
58	57	mulid1d 9660	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. 1 ) = ( ( F ` B ) - ( F ` A ) ) )
59	50, 58	eqtrd 2485	. . . . . 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 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 ) ) ) )
61	2, 1	resubcld 10047	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
62	61	recnd 9669	. . . . . . 7 |- ( ph -> ( B - A ) e. CC )
63	62	adantr 467	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) e. CC )
64		dvf 22862	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
65	12	feq2d 5715	. . . . . . . 8 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
66	64, 65	mpbii 215	. . . . . . 7 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
67	66	ffvelrnda 6022	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
68	1, 2	posdifd 10200	. . . . . . . . 9 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
69	3, 68	mpbid 214	. . . . . . . 8 |- ( ph -> 0 < ( B - A ) )
70	69	gt0ne0d 10178	. . . . . . 7 |- ( ph -> ( B - A ) =/= 0 )
71	70	adantr 467	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) =/= 0 )
72	57, 63, 67, 71	divmuld 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 ) ) ) )
73	60, 72	bitr4d 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 ) )
76	73, 74, 75	3bitr4g 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 ) ) ) )
77	76	rexbidva 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 ) ) ) )
78	31, 77	mpbid 214	1 |- ( ph -> E. x e. ( A (,) B ) ( ( RR _D F ) ` x ) = ( ( ( F ` B ) - ( F ` A ) ) / ( B - A ) ) )

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  ℂ_fldccnfld 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