Theorem cmvth 23754

Description: Cauchy's Mean Value Theorem. If F , G are real continuous functions on [ A , B ] differentiable on ( A , B ), then there is some x e. ( A , B ) such that F' ( x ) / G' ( x ) = ( F ( A ) - F ( B ) ) / ( G ( A ) - G ( B ) ). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.)

Hypotheses

Ref	Expression
cmvth.a	|- ( ph -> A e. RR )
cmvth.b	|- ( ph -> B e. RR )
cmvth.lt	|- ( ph -> A < B )
cmvth.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
cmvth.g	|- ( ph -> G e. ( ( A [,] B ) -cn-> RR ) )
cmvth.df	|- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
cmvth.dg	|- ( ph -> dom ( RR _D G ) = ( A (,) B ) )

Assertion

Ref	Expression
cmvth	|- ( ph -> E. x e. ( A (,) B ) ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) )

Distinct variable groups:   x, A   x, B   x, F   x, G   ph, x

Proof of Theorem cmvth

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmvth.a	. . 3 |- ( ph -> A e. RR )
2		cmvth.b	. . 3 |- ( ph -> B e. RR )
3		cmvth.lt	. . 3 |- ( ph -> A < B )
4		eqid 2622	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
5	4	subcn 22669	. . . 4 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
6	4	mulcn 22670	. . . . 5 |- x. e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
7		cmvth.f	. . . . . . . . 9 |- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
8		cncff 22696	. . . . . . . . 9 |- ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
9	7, 8	syl 17	. . . . . . . 8 |- ( ph -> F : ( A [,] B ) --> RR )
10	1	rexrd 10089	. . . . . . . . 9 |- ( ph -> A e. RR* )
11	2	rexrd 10089	. . . . . . . . 9 |- ( ph -> B e. RR* )
12	1, 2, 3	ltled 10185	. . . . . . . . 9 |- ( ph -> A <_ B )
13		ubicc2 12289	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
14	10, 11, 12, 13	syl3anc 1326	. . . . . . . 8 |- ( ph -> B e. ( A [,] B ) )
15	9, 14	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( F ` B ) e. RR )
16		lbicc2 12288	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
17	10, 11, 12, 16	syl3anc 1326	. . . . . . . 8 |- ( ph -> A e. ( A [,] B ) )
18	9, 17	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( F ` A ) e. RR )
19	15, 18	resubcld 10458	. . . . . 6 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
20		iccssre 12255	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
21	1, 2, 20	syl2anc 693	. . . . . . 7 |- ( ph -> ( A [,] B ) C_ RR )
22		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
23	21, 22	syl6ss 3615	. . . . . 6 |- ( ph -> ( A [,] B ) C_ CC )
24	22	a1i 11	. . . . . 6 |- ( ph -> RR C_ CC )
25		cncfmptc 22714	. . . . . 6 |- ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( A [,] B ) C_ CC /\ RR C_ CC ) -> ( z e. ( A [,] B ) |-> ( ( F ` B ) - ( F ` A ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
26	19, 23, 24, 25	syl3anc 1326	. . . . 5 |- ( ph -> ( z e. ( A [,] B ) |-> ( ( F ` B ) - ( F ` A ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
27		cmvth.g	. . . . . . . 8 |- ( ph -> G e. ( ( A [,] B ) -cn-> RR ) )
28		cncff 22696	. . . . . . . 8 |- ( G e. ( ( A [,] B ) -cn-> RR ) -> G : ( A [,] B ) --> RR )
29	27, 28	syl 17	. . . . . . 7 |- ( ph -> G : ( A [,] B ) --> RR )
30	29	feqmptd 6249	. . . . . 6 |- ( ph -> G = ( z e. ( A [,] B ) |-> ( G ` z ) ) )
31	30, 27	eqeltrrd 2702	. . . . 5 |- ( ph -> ( z e. ( A [,] B ) |-> ( G ` z ) ) e. ( ( A [,] B ) -cn-> RR ) )
32		remulcl 10021	. . . . 5 |- ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. RR )
33	4, 6, 26, 31, 22, 32	cncfmpt2ss 22718	. . . 4 |- ( ph -> ( z e. ( A [,] B ) |-> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
34	29, 14	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( G ` B ) e. RR )
35	29, 17	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( G ` A ) e. RR )
36	34, 35	resubcld 10458	. . . . . 6 |- ( ph -> ( ( G ` B ) - ( G ` A ) ) e. RR )
37		cncfmptc 22714	. . . . . 6 |- ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( A [,] B ) C_ CC /\ RR C_ CC ) -> ( z e. ( A [,] B ) |-> ( ( G ` B ) - ( G ` A ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
38	36, 23, 24, 37	syl3anc 1326	. . . . 5 |- ( ph -> ( z e. ( A [,] B ) |-> ( ( G ` B ) - ( G ` A ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
39	9	feqmptd 6249	. . . . . 6 |- ( ph -> F = ( z e. ( A [,] B ) |-> ( F ` z ) ) )
40	39, 7	eqeltrrd 2702	. . . . 5 |- ( ph -> ( z e. ( A [,] B ) |-> ( F ` z ) ) e. ( ( A [,] B ) -cn-> RR ) )
41		remulcl 10021	. . . . 5 |- ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. RR )
42	4, 6, 38, 40, 22, 41	cncfmpt2ss 22718	. . . 4 |- ( ph -> ( z e. ( A [,] B ) |-> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
43		resubcl 10345	. . . 4 |- ( ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. RR /\ ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. RR ) -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. RR )
44	4, 5, 33, 42, 22, 43	cncfmpt2ss 22718	. . 3 |- ( ph -> ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
45	19	recnd 10068	. . . . . . . . . 10 |- ( ph -> ( ( F ` B ) - ( F ` A ) ) e. CC )
46	45	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
47	29	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( G ` z ) e. RR )
48	47	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( G ` z ) e. CC )
49	46, 48	mulcld 10060	. . . . . . . 8 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. CC )
50	36	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( G ` B ) - ( G ` A ) ) e. RR )
51	9	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( F ` z ) e. RR )
52	50, 51	remulcld 10070	. . . . . . . . 9 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. RR )
53	52	recnd 10068	. . . . . . . 8 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. CC )
54	49, 53	subcld 10392	. . . . . . 7 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. CC )
55	4	tgioo2 22606	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
56		iccntr 22624	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
57	1, 2, 56	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
58	24, 21, 54, 55, 4, 57	dvmptntr 23734	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) = ( RR _D ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) )
59		reelprrecn 10028	. . . . . . . 8 |- RR e. { RR , CC }
60	59	a1i 11	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
61		ioossicc 12259	. . . . . . . . 9 |- ( A (,) B ) C_ ( A [,] B )
62	61	sseli 3599	. . . . . . . 8 |- ( z e. ( A (,) B ) -> z e. ( A [,] B ) )
63	62, 49	sylan2 491	. . . . . . 7 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. CC )
64		ovex 6678	. . . . . . . 8 |- ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) e. _V
65	64	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) e. _V )
66	62, 48	sylan2 491	. . . . . . . 8 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( G ` z ) e. CC )
67		fvexd 6203	. . . . . . . 8 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( ( RR _D G ) ` z ) e. _V )
68	30	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( RR _D G ) = ( RR _D ( z e. ( A [,] B ) |-> ( G ` z ) ) ) )
69		dvf 23671	. . . . . . . . . . 11 |- ( RR _D G ) : dom ( RR _D G ) --> CC
70		cmvth.dg	. . . . . . . . . . . 12 |- ( ph -> dom ( RR _D G ) = ( A (,) B ) )
71	70	feq2d 6031	. . . . . . . . . . 11 |- ( ph -> ( ( RR _D G ) : dom ( RR _D G ) --> CC <-> ( RR _D G ) : ( A (,) B ) --> CC ) )
72	69, 71	mpbii 223	. . . . . . . . . 10 |- ( ph -> ( RR _D G ) : ( A (,) B ) --> CC )
73	72	feqmptd 6249	. . . . . . . . 9 |- ( ph -> ( RR _D G ) = ( z e. ( A (,) B ) |-> ( ( RR _D G ) ` z ) ) )
74	24, 21, 48, 55, 4, 57	dvmptntr 23734	. . . . . . . . 9 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> ( G ` z ) ) ) = ( RR _D ( z e. ( A (,) B ) |-> ( G ` z ) ) ) )
75	68, 73, 74	3eqtr3rd 2665	. . . . . . . 8 |- ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( G ` z ) ) ) = ( z e. ( A (,) B ) |-> ( ( RR _D G ) ` z ) ) )
76	60, 66, 67, 75, 45	dvmptcmul 23727	. . . . . . 7 |- ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) ) = ( z e. ( A (,) B ) |-> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) ) )
77	62, 53	sylan2 491	. . . . . . 7 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. CC )
78		ovex 6678	. . . . . . . 8 |- ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) e. _V
79	78	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) e. _V )
80	51	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ z e. ( A [,] B ) ) -> ( F ` z ) e. CC )
81	62, 80	sylan2 491	. . . . . . . 8 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( F ` z ) e. CC )
82		fvexd 6203	. . . . . . . 8 |- ( ( ph /\ z e. ( A (,) B ) ) -> ( ( RR _D F ) ` z ) e. _V )
83	39	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( RR _D F ) = ( RR _D ( z e. ( A [,] B ) |-> ( F ` z ) ) ) )
84		dvf 23671	. . . . . . . . . . 11 |- ( RR _D F ) : dom ( RR _D F ) --> CC
85		cmvth.df	. . . . . . . . . . . 12 |- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
86	85	feq2d 6031	. . . . . . . . . . 11 |- ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
87	84, 86	mpbii 223	. . . . . . . . . 10 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
88	87	feqmptd 6249	. . . . . . . . 9 |- ( ph -> ( RR _D F ) = ( z e. ( A (,) B ) |-> ( ( RR _D F ) ` z ) ) )
89	24, 21, 80, 55, 4, 57	dvmptntr 23734	. . . . . . . . 9 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> ( F ` z ) ) ) = ( RR _D ( z e. ( A (,) B ) |-> ( F ` z ) ) ) )
90	83, 88, 89	3eqtr3rd 2665	. . . . . . . 8 |- ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( F ` z ) ) ) = ( z e. ( A (,) B ) |-> ( ( RR _D F ) ` z ) ) )
91	36	recnd 10068	. . . . . . . 8 |- ( ph -> ( ( G ` B ) - ( G ` A ) ) e. CC )
92	60, 81, 82, 90, 91	dvmptcmul 23727	. . . . . . 7 |- ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) = ( z e. ( A (,) B ) |-> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) )
93	60, 63, 65, 76, 77, 79, 92	dvmptsub 23730	. . . . . 6 |- ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) = ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) )
94	58, 93	eqtrd 2656	. . . . 5 |- ( ph -> ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) = ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) )
95	94	dmeqd 5326	. . . 4 |- ( ph -> dom ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) = dom ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) )
96		ovex 6678	. . . . 5 |- ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) e. _V
97		eqid 2622	. . . . 5 |- ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) = ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) )
98	96, 97	dmmpti 6023	. . . 4 |- dom ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) = ( A (,) B )
99	95, 98	syl6eq 2672	. . 3 |- ( ph -> dom ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) = ( A (,) B ) )
100	15	recnd 10068	. . . . . . . 8 |- ( ph -> ( F ` B ) e. CC )
101	35	recnd 10068	. . . . . . . 8 |- ( ph -> ( G ` A ) e. CC )
102	100, 101	mulcld 10060	. . . . . . 7 |- ( ph -> ( ( F ` B ) x. ( G ` A ) ) e. CC )
103	18	recnd 10068	. . . . . . . 8 |- ( ph -> ( F ` A ) e. CC )
104	34	recnd 10068	. . . . . . . 8 |- ( ph -> ( G ` B ) e. CC )
105	103, 104	mulcld 10060	. . . . . . 7 |- ( ph -> ( ( F ` A ) x. ( G ` B ) ) e. CC )
106	103, 101	mulcld 10060	. . . . . . 7 |- ( ph -> ( ( F ` A ) x. ( G ` A ) ) e. CC )
107	102, 105, 106	nnncan2d 10427	. . . . . 6 |- ( ph -> ( ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` A ) ) ) - ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) ) = ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` B ) ) ) )
108	100, 104	mulcld 10060	. . . . . . 7 |- ( ph -> ( ( F ` B ) x. ( G ` B ) ) e. CC )
109	108, 105, 102	nnncan1d 10426	. . . . . 6 |- ( ph -> ( ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` B ) ) ) - ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) ) = ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` B ) ) ) )
110	107, 109	eqtr4d 2659	. . . . 5 |- ( ph -> ( ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` A ) ) ) - ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) ) = ( ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` B ) ) ) - ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) ) )
111	100, 103, 101	subdird 10487	. . . . . 6 |- ( ph -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) = ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` A ) ) ) )
112	91, 103	mulcomd 10061	. . . . . . 7 |- ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) = ( ( F ` A ) x. ( ( G ` B ) - ( G ` A ) ) ) )
113	103, 104, 101	subdid 10486	. . . . . . 7 |- ( ph -> ( ( F ` A ) x. ( ( G ` B ) - ( G ` A ) ) ) = ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) )
114	112, 113	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) = ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) )
115	111, 114	oveq12d 6668	. . . . 5 |- ( ph -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) ) = ( ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` A ) ) ) - ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) ) )
116	100, 103, 104	subdird 10487	. . . . . 6 |- ( ph -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) = ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` B ) ) ) )
117	91, 100	mulcomd 10061	. . . . . . 7 |- ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) = ( ( F ` B ) x. ( ( G ` B ) - ( G ` A ) ) ) )
118	100, 104, 101	subdid 10486	. . . . . . 7 |- ( ph -> ( ( F ` B ) x. ( ( G ` B ) - ( G ` A ) ) ) = ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) )
119	117, 118	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) = ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) )
120	116, 119	oveq12d 6668	. . . . 5 |- ( ph -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) ) = ( ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` B ) ) ) - ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) ) )
121	110, 115, 120	3eqtr4d 2666	. . . 4 |- ( ph -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) ) )
122		fveq2 6191	. . . . . . . 8 |- ( z = A -> ( G ` z ) = ( G ` A ) )
123	122	oveq2d 6666	. . . . . . 7 |- ( z = A -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) )
124		fveq2 6191	. . . . . . . 8 |- ( z = A -> ( F ` z ) = ( F ` A ) )
125	124	oveq2d 6666	. . . . . . 7 |- ( z = A -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) )
126	123, 125	oveq12d 6668	. . . . . 6 |- ( z = A -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) ) )
127		eqid 2622	. . . . . 6 |- ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) = ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) )
128		ovex 6678	. . . . . 6 |- ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. _V
129	126, 127, 128	fvmpt3i 6287	. . . . 5 |- ( A e. ( A [,] B ) -> ( ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ` A ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) ) )
130	17, 129	syl 17	. . . 4 |- ( ph -> ( ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ` A ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) ) )
131		fveq2 6191	. . . . . . . 8 |- ( z = B -> ( G ` z ) = ( G ` B ) )
132	131	oveq2d 6666	. . . . . . 7 |- ( z = B -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) )
133		fveq2 6191	. . . . . . . 8 |- ( z = B -> ( F ` z ) = ( F ` B ) )
134	133	oveq2d 6666	. . . . . . 7 |- ( z = B -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) )
135	132, 134	oveq12d 6668	. . . . . 6 |- ( z = B -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) ) )
136	135, 127, 128	fvmpt3i 6287	. . . . 5 |- ( B e. ( A [,] B ) -> ( ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ` B ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) ) )
137	14, 136	syl 17	. . . 4 |- ( ph -> ( ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ` B ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) ) )
138	121, 130, 137	3eqtr4d 2666	. . 3 |- ( ph -> ( ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ` A ) = ( ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ` B ) )
139	1, 2, 3, 44, 99, 138	rolle 23753	. 2 |- ( ph -> E. x e. ( A (,) B ) ( ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) ` x ) = 0 )
140	94	fveq1d 6193	. . . . . 6 |- ( ph -> ( ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) ` x ) = ( ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) ` x ) )
141		fveq2 6191	. . . . . . . . 9 |- ( z = x -> ( ( RR _D G ) ` z ) = ( ( RR _D G ) ` x ) )
142	141	oveq2d 6666	. . . . . . . 8 |- ( z = x -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) )
143		fveq2 6191	. . . . . . . . 9 |- ( z = x -> ( ( RR _D F ) ` z ) = ( ( RR _D F ) ` x ) )
144	143	oveq2d 6666	. . . . . . . 8 |- ( z = x -> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) )
145	142, 144	oveq12d 6668	. . . . . . 7 |- ( z = x -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) )
146	145, 97, 96	fvmpt3i 6287	. . . . . 6 |- ( x e. ( A (,) B ) -> ( ( z e. ( A (,) B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) ) ` x ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) )
147	140, 146	sylan9eq 2676	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) ` x ) = ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) )
148	147	eqeq1d 2624	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) ` x ) = 0 <-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) = 0 ) )
149	45	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
150	72	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D G ) ` x ) e. CC )
151	149, 150	mulcld 10060	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) e. CC )
152	91	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( G ` B ) - ( G ` A ) ) e. CC )
153	87	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
154	152, 153	mulcld 10060	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) e. CC )
155	151, 154	subeq0ad 10402	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) = 0 <-> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) )
156	148, 155	bitrd 268	. . 3 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) ` x ) = 0 <-> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) )
157	156	rexbidva 3049	. 2 |- ( ph -> ( E. x e. ( A (,) B ) ( ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) ` x ) = 0 <-> E. x e. ( A (,) B ) ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) ) )
158	139, 157	mpbid 222	1 |- ( ph -> E. x e. ( A (,) B ) ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  mvth  23755  lhop1lem  23776