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.

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 5165	. . . 4 |- ( z e. ( A [,] B ) |-> z ) = ( _I |` ( A [,] B ) )
6		iccssre 11741	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
7	1, 2, 6	syl2anc 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 ) )
10	7, 8, 9	sylancl 675	. . . 4 |- ( ph -> ( z e. ( A [,] B ) |-> z ) e. ( ( A [,] B ) -cn-> RR ) )
11	5, 10	syl5eqelr 2554	. . 3 |- ( ph -> ( _I |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
12		mvth.d	. . 3 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
13	5	oveq2i 6319	. . . . . 6 |- ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( RR _D ( _I |` ( A [,] B ) ) )
14		reelprrecn 9649	. . . . . . . 8 |- RR e. { RR , CC }
15	14	a1i 11	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
16		simpr 468	. . . . . . . 8 |- ( ( ph /\ z e. RR ) -> z e. RR )
17	16	recnd 9687	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> z e. CC )
18		1red 9676	. . . . . . 7 |- ( ( ph /\ z e. RR ) -> 1 e. RR )
19	15	dvmptid 22990	. . . . . . 7 |- ( ph -> ( RR _D ( z e. RR |-> z ) ) = ( z e. RR |-> 1 ) )
20		eqid 2471	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21899	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22		iccntr 21917	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
23	1, 2, 22	syl2anc 673	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
24	15, 17, 18, 19, 7, 21, 20, 23	dvmptres2 22995	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> z ) ) = ( z e. ( A (,) B ) |-> 1 ) )
25	13, 24	syl5eqr 2519	. . . . 5 |- ( ph -> ( RR _D ( _I |` ( A [,] B ) ) ) = ( z e. ( A (,) B ) |-> 1 ) )
26	25	dmeqd 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 )
29	27, 28	dmmpti 5717	. . . 4 |- dom ( z e. ( A (,) B ) |-> 1 ) = ( A (,) B )
30	26, 29	syl6eq 2521	. . 3 |- ( ph -> dom ( RR _D ( _I |` ( A [,] B ) ) ) = ( A (,) B ) )
31	1, 2, 3, 4, 11, 12, 30	cmvth 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 ) ) )
32	1	rexrd 9708	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
33	2	rexrd 9708	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
34	1, 2, 3	ltled 9800	. . . . . . . . . . 11 |- ( ph -> A <_ B )
35		ubicc2 11775	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
36	32, 33, 34, 35	syl3anc 1292	. . . . . . . . . 10 |- ( ph -> B e. ( A [,] B ) )
37		fvresi 6106	. . . . . . . . . 10 |- ( B e. ( A [,] B ) -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
38	36, 37	syl 17	. . . . . . . . 9 |- ( ph -> ( ( _I |` ( A [,] B ) ) ` B ) = B )
39		lbicc2 11774	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
40	32, 33, 34, 39	syl3anc 1292	. . . . . . . . . 10 |- ( ph -> A e. ( A [,] B ) )
41		fvresi 6106	. . . . . . . . . 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 6326	. . . . . . . 8 |- ( ph -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
44	43	adantr 472	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( _I |` ( A [,] B ) ) ` B ) - ( ( _I |` ( A [,] B ) ) ` A ) ) = ( B - A ) )
45	44	oveq1d 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 ) ) )
46	25	fveq1d 5881	. . . . . . . . 9 |- ( ph -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = ( ( z e. ( A (,) B ) |-> 1 ) ` x ) )
47		eqidd 2472	. . . . . . . . . 10 |- ( z = x -> 1 = 1 )
48	47, 28, 27	fvmpt3i 5968	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> ( ( z e. ( A (,) B ) |-> 1 ) ` x ) = 1 )
49	46, 48	sylan9eq 2525	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( _I |` ( A [,] B ) ) ) ` x ) = 1 )
50	49	oveq2d 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 )
52	4, 51	syl 17	. . . . . . . . . . . 12 |- ( ph -> F : ( A [,] B ) --> RR )
53	52, 36	ffvelrnd 6038	. . . . . . . . . . 11 |- ( ph -> ( F ` B ) e. RR )
54	52, 40	ffvelrnd 6038	. . . . . . . . . . 11 |- ( ph -> ( F ` A ) e. RR )
55	53, 54	resubcld 10068	. . . . . . . . . 10 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
56	55	recnd 9687	. . . . . . . . 9 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. CC )
57	56	adantr 472	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
58	57	mulid1d 9678	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. 1 ) = ( ( F ` B ) - ( F ` A ) ) )
59	50, 58	eqtrd 2505	. . . . . 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 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 ) ) ) )
61	2, 1	resubcld 10068	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
62	61	recnd 9687	. . . . . . 7 |- ( ph -> ( B - A ) e. CC )
63	62	adantr 472	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) e. CC )
64		dvf 22941	. . . . . . . 8 |- ( RR _D F ) : dom ( RR _D F ) --> CC
65	12	feq2d 5725	. . . . . . . 8 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
66	64, 65	mpbii 216	. . . . . . 7 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
67	66	ffvelrnda 6037	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
68	1, 2	posdifd 10221	. . . . . . . . 9 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
69	3, 68	mpbid 215	. . . . . . . 8 |- ( ph -> 0 < ( B - A ) )
70	69	gt0ne0d 10199	. . . . . . 7 |- ( ph -> ( B - A ) =/= 0 )
71	70	adantr 472	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( B - A ) =/= 0 )
72	57, 63, 67, 71	divmuld 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 ) ) ) )
73	60, 72	bitr4d 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 ) )
76	73, 74, 75	3bitr4g 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 ) ) ) )
77	76	rexbidva 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 ) ) ) )
78	31, 77	mpbid 215	1 |- ( ph -> E. x e. ( A (,) B ) ( ( RR _D F ) ` x ) = ( ( ( F ` B ) - ( F ` A ) ) / ( B - A ) ) )

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  ℂ_fldccnfld 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