Metamath Proof Explorer


Theorem cmvth

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) Avoid ax-mulf . (Revised by GG, 16-Mar-2025)

Ref Expression
Hypotheses 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 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 ) ) )

Proof

Step Hyp Ref Expression
1 cmvth.a
 |-  ( ph -> A e. RR )
2 cmvth.b
 |-  ( ph -> B e. RR )
3 cmvth.lt
 |-  ( ph -> A < B )
4 cmvth.f
 |-  ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
5 cmvth.g
 |-  ( ph -> G e. ( ( A [,] B ) -cn-> RR ) )
6 cmvth.df
 |-  ( ph -> dom ( RR _D F ) = ( A (,) B ) )
7 cmvth.dg
 |-  ( ph -> dom ( RR _D G ) = ( A (,) B ) )
8 eqid
 |-  ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
9 8 subcn
 |-  - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) )
10 cncff
 |-  ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
11 4 10 syl
 |-  ( ph -> F : ( A [,] B ) --> RR )
12 1 rexrd
 |-  ( ph -> A e. RR* )
13 2 rexrd
 |-  ( ph -> B e. RR* )
14 1 2 3 ltled
 |-  ( ph -> A <_ B )
15 ubicc2
 |-  ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
16 12 13 14 15 syl3anc
 |-  ( ph -> B e. ( A [,] B ) )
17 11 16 ffvelcdmd
 |-  ( ph -> ( F ` B ) e. RR )
18 lbicc2
 |-  ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
19 12 13 14 18 syl3anc
 |-  ( ph -> A e. ( A [,] B ) )
20 11 19 ffvelcdmd
 |-  ( ph -> ( F ` A ) e. RR )
21 17 20 resubcld
 |-  ( ph -> ( ( F ` B ) - ( F ` A ) ) e. RR )
22 21 recnd
 |-  ( ph -> ( ( F ` B ) - ( F ` A ) ) e. CC )
23 22 adantr
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
24 cncff
 |-  ( G e. ( ( A [,] B ) -cn-> RR ) -> G : ( A [,] B ) --> RR )
25 5 24 syl
 |-  ( ph -> G : ( A [,] B ) --> RR )
26 25 ffvelcdmda
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( G ` z ) e. RR )
27 26 recnd
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( G ` z ) e. CC )
28 ovmpot
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. CC /\ ( G ` z ) e. CC ) -> ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) )
29 23 27 28 syl2anc
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) )
30 29 eqeq2d
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( w = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) <-> w = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) )
31 30 pm5.32da
 |-  ( ph -> ( ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) <-> ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) ) )
32 31 opabbidv
 |-  ( ph -> { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) } = { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) } )
33 df-mpt
 |-  ( z e. ( A [,] B ) |-> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) = { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) }
34 32 33 eqtr4di
 |-  ( ph -> { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) } = ( z e. ( A [,] B ) |-> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) )
35 df-mpt
 |-  ( z e. ( A [,] B ) |-> ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) = { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) }
36 8 mpomulcn
 |-  ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) )
37 1 2 iccssred
 |-  ( ph -> ( A [,] B ) C_ RR )
38 ax-resscn
 |-  RR C_ CC
39 37 38 sstrdi
 |-  ( ph -> ( A [,] B ) C_ CC )
40 38 a1i
 |-  ( ph -> RR C_ CC )
41 cncfmptc
 |-  ( ( ( ( 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 ) )
42 21 39 40 41 syl3anc
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( F ` B ) - ( F ` A ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
43 25 feqmptd
 |-  ( ph -> G = ( z e. ( A [,] B ) |-> ( G ` z ) ) )
44 43 5 eqeltrrd
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( G ` z ) ) e. ( ( A [,] B ) -cn-> RR ) )
45 simpl
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( ( F ` B ) - ( F ` A ) ) e. RR )
46 45 recnd
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
47 simpr
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( G ` z ) e. RR )
48 47 recnd
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( G ` z ) e. CC )
49 28 eqcomd
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. CC /\ ( G ` z ) e. CC ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) )
50 46 48 49 syl2anc
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) )
51 remulcl
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. RR )
52 50 51 eqeltrrd
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) e. RR /\ ( G ` z ) e. RR ) -> ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) e. RR )
53 8 36 42 44 38 52 cncfmpt2ss
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
54 35 53 eqeltrrid
 |-  ( ph -> { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( F ` B ) - ( F ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( G ` z ) ) ) } e. ( ( A [,] B ) -cn-> RR ) )
55 34 54 eqeltrrd
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
56 25 16 ffvelcdmd
 |-  ( ph -> ( G ` B ) e. RR )
57 25 19 ffvelcdmd
 |-  ( ph -> ( G ` A ) e. RR )
58 56 57 resubcld
 |-  ( ph -> ( ( G ` B ) - ( G ` A ) ) e. RR )
59 58 adantr
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( G ` B ) - ( G ` A ) ) e. RR )
60 59 recnd
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( G ` B ) - ( G ` A ) ) e. CC )
61 11 ffvelcdmda
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( F ` z ) e. RR )
62 61 recnd
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( F ` z ) e. CC )
63 ovmpot
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. CC /\ ( F ` z ) e. CC ) -> ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) )
64 60 62 63 syl2anc
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) )
65 64 eqeq2d
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( w = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) <-> w = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) )
66 65 pm5.32da
 |-  ( ph -> ( ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) <-> ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) )
67 66 opabbidv
 |-  ( ph -> { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) } = { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) } )
68 df-mpt
 |-  ( z e. ( A [,] B ) |-> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) = { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) }
69 67 68 eqtr4di
 |-  ( ph -> { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) } = ( z e. ( A [,] B ) |-> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) )
70 df-mpt
 |-  ( z e. ( A [,] B ) |-> ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) = { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) }
71 cncfmptc
 |-  ( ( ( ( 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 ) )
72 58 39 40 71 syl3anc
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( G ` B ) - ( G ` A ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
73 11 feqmptd
 |-  ( ph -> F = ( z e. ( A [,] B ) |-> ( F ` z ) ) )
74 73 4 eqeltrrd
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( F ` z ) ) e. ( ( A [,] B ) -cn-> RR ) )
75 simpl
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( ( G ` B ) - ( G ` A ) ) e. RR )
76 75 recnd
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( ( G ` B ) - ( G ` A ) ) e. CC )
77 simpr
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( F ` z ) e. RR )
78 77 recnd
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( F ` z ) e. CC )
79 63 eqcomd
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. CC /\ ( F ` z ) e. CC ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) )
80 76 78 79 syl2anc
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) )
81 remulcl
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. RR )
82 80 81 eqeltrrd
 |-  ( ( ( ( G ` B ) - ( G ` A ) ) e. RR /\ ( F ` z ) e. RR ) -> ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) e. RR )
83 8 36 72 74 38 82 cncfmpt2ss
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
84 70 83 eqeltrrid
 |-  ( ph -> { <. z , w >. | ( z e. ( A [,] B ) /\ w = ( ( ( G ` B ) - ( G ` A ) ) ( u e. CC , v e. CC |-> ( u x. v ) ) ( F ` z ) ) ) } e. ( ( A [,] B ) -cn-> RR ) )
85 69 84 eqeltrrd
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
86 resubcl
 |-  ( ( ( ( ( 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 )
87 8 9 55 85 38 86 cncfmpt2ss
 |-  ( ph -> ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) e. ( ( A [,] B ) -cn-> RR ) )
88 23 27 mulcld
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. CC )
89 59 61 remulcld
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. RR )
90 89 recnd
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. CC )
91 88 90 subcld
 |-  ( ( ph /\ z e. ( A [,] B ) ) -> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. CC )
92 8 tgioo2
 |-  ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR )
93 iccntr
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
94 1 2 93 syl2anc
 |-  ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
95 40 37 91 92 8 94 dvmptntr
 |-  ( 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 ) ) ) ) ) )
96 reelprrecn
 |-  RR e. { RR , CC }
97 96 a1i
 |-  ( ph -> RR e. { RR , CC } )
98 ioossicc
 |-  ( A (,) B ) C_ ( A [,] B )
99 98 sseli
 |-  ( z e. ( A (,) B ) -> z e. ( A [,] B ) )
100 99 88 sylan2
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) e. CC )
101 ovexd
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) e. _V )
102 99 27 sylan2
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( G ` z ) e. CC )
103 fvexd
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( ( RR _D G ) ` z ) e. _V )
104 43 oveq2d
 |-  ( ph -> ( RR _D G ) = ( RR _D ( z e. ( A [,] B ) |-> ( G ` z ) ) ) )
105 dvf
 |-  ( RR _D G ) : dom ( RR _D G ) --> CC
106 7 feq2d
 |-  ( ph -> ( ( RR _D G ) : dom ( RR _D G ) --> CC <-> ( RR _D G ) : ( A (,) B ) --> CC ) )
107 105 106 mpbii
 |-  ( ph -> ( RR _D G ) : ( A (,) B ) --> CC )
108 107 feqmptd
 |-  ( ph -> ( RR _D G ) = ( z e. ( A (,) B ) |-> ( ( RR _D G ) ` z ) ) )
109 40 37 27 92 8 94 dvmptntr
 |-  ( ph -> ( RR _D ( z e. ( A [,] B ) |-> ( G ` z ) ) ) = ( RR _D ( z e. ( A (,) B ) |-> ( G ` z ) ) ) )
110 104 108 109 3eqtr3rd
 |-  ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( G ` z ) ) ) = ( z e. ( A (,) B ) |-> ( ( RR _D G ) ` z ) ) )
111 97 102 103 110 22 dvmptcmul
 |-  ( 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 ) ) ) )
112 99 90 sylan2
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) e. CC )
113 ovexd
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) e. _V )
114 99 62 sylan2
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( F ` z ) e. CC )
115 fvexd
 |-  ( ( ph /\ z e. ( A (,) B ) ) -> ( ( RR _D F ) ` z ) e. _V )
116 73 oveq2d
 |-  ( ph -> ( RR _D F ) = ( RR _D ( z e. ( A [,] B ) |-> ( F ` z ) ) ) )
117 dvf
 |-  ( RR _D F ) : dom ( RR _D F ) --> CC
118 6 feq2d
 |-  ( ph -> ( ( RR _D F ) : dom ( RR _D F ) --> CC <-> ( RR _D F ) : ( A (,) B ) --> CC ) )
119 117 118 mpbii
 |-  ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
120 119 feqmptd
 |-  ( ph -> ( RR _D F ) = ( z e. ( A (,) B ) |-> ( ( RR _D F ) ` z ) ) )
121 40 37 62 92 8 94 dvmptntr
 |-  ( ph -> ( RR _D ( z e. ( A [,] B ) |-> ( F ` z ) ) ) = ( RR _D ( z e. ( A (,) B ) |-> ( F ` z ) ) ) )
122 116 120 121 3eqtr3rd
 |-  ( ph -> ( RR _D ( z e. ( A (,) B ) |-> ( F ` z ) ) ) = ( z e. ( A (,) B ) |-> ( ( RR _D F ) ` z ) ) )
123 58 recnd
 |-  ( ph -> ( ( G ` B ) - ( G ` A ) ) e. CC )
124 97 114 115 122 123 dvmptcmul
 |-  ( 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 ) ) ) )
125 97 100 101 111 112 113 124 dvmptsub
 |-  ( 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 ) ) ) ) )
126 95 125 eqtrd
 |-  ( 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 ) ) ) ) )
127 126 dmeqd
 |-  ( 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 ) ) ) ) )
128 ovex
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) ) e. _V
129 eqid
 |-  ( 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 ) ) ) )
130 128 129 dmmpti
 |-  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 )
131 127 130 eqtrdi
 |-  ( ph -> dom ( RR _D ( z e. ( A [,] B ) |-> ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) ) ) = ( A (,) B ) )
132 17 recnd
 |-  ( ph -> ( F ` B ) e. CC )
133 57 recnd
 |-  ( ph -> ( G ` A ) e. CC )
134 132 133 mulcld
 |-  ( ph -> ( ( F ` B ) x. ( G ` A ) ) e. CC )
135 20 recnd
 |-  ( ph -> ( F ` A ) e. CC )
136 56 recnd
 |-  ( ph -> ( G ` B ) e. CC )
137 135 136 mulcld
 |-  ( ph -> ( ( F ` A ) x. ( G ` B ) ) e. CC )
138 135 133 mulcld
 |-  ( ph -> ( ( F ` A ) x. ( G ` A ) ) e. CC )
139 134 137 138 nnncan2d
 |-  ( 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 ) ) ) )
140 132 136 mulcld
 |-  ( ph -> ( ( F ` B ) x. ( G ` B ) ) e. CC )
141 140 137 134 nnncan1d
 |-  ( 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 ) ) ) )
142 139 141 eqtr4d
 |-  ( 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 ) ) ) ) )
143 132 135 133 subdird
 |-  ( ph -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) = ( ( ( F ` B ) x. ( G ` A ) ) - ( ( F ` A ) x. ( G ` A ) ) ) )
144 123 135 mulcomd
 |-  ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) = ( ( F ` A ) x. ( ( G ` B ) - ( G ` A ) ) ) )
145 135 136 133 subdid
 |-  ( ph -> ( ( F ` A ) x. ( ( G ` B ) - ( G ` A ) ) ) = ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) )
146 144 145 eqtrd
 |-  ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) = ( ( ( F ` A ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` A ) ) ) )
147 143 146 oveq12d
 |-  ( 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 ) ) ) ) )
148 132 135 136 subdird
 |-  ( ph -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) = ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` A ) x. ( G ` B ) ) ) )
149 123 132 mulcomd
 |-  ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) = ( ( F ` B ) x. ( ( G ` B ) - ( G ` A ) ) ) )
150 132 136 133 subdid
 |-  ( ph -> ( ( F ` B ) x. ( ( G ` B ) - ( G ` A ) ) ) = ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) )
151 149 150 eqtrd
 |-  ( ph -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) = ( ( ( F ` B ) x. ( G ` B ) ) - ( ( F ` B ) x. ( G ` A ) ) ) )
152 148 151 oveq12d
 |-  ( 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 ) ) ) ) )
153 142 147 152 3eqtr4d
 |-  ( 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 ) ) ) )
154 fveq2
 |-  ( z = A -> ( G ` z ) = ( G ` A ) )
155 154 oveq2d
 |-  ( z = A -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` A ) ) )
156 fveq2
 |-  ( z = A -> ( F ` z ) = ( F ` A ) )
157 156 oveq2d
 |-  ( z = A -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` A ) ) )
158 155 157 oveq12d
 |-  ( 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 ) ) ) )
159 eqid
 |-  ( 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 ) ) ) )
160 ovex
 |-  ( ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) - ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) ) e. _V
161 158 159 160 fvmpt3i
 |-  ( 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 ) ) ) )
162 19 161 syl
 |-  ( 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 ) ) ) )
163 fveq2
 |-  ( z = B -> ( G ` z ) = ( G ` B ) )
164 163 oveq2d
 |-  ( z = B -> ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( G ` B ) ) )
165 fveq2
 |-  ( z = B -> ( F ` z ) = ( F ` B ) )
166 165 oveq2d
 |-  ( z = B -> ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( F ` B ) ) )
167 164 166 oveq12d
 |-  ( 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 ) ) ) )
168 167 159 160 fvmpt3i
 |-  ( 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 ) ) ) )
169 16 168 syl
 |-  ( 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 ) ) ) )
170 153 162 169 3eqtr4d
 |-  ( 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 ) )
171 1 2 3 87 131 170 rolle
 |-  ( 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 )
172 126 fveq1d
 |-  ( 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 ) )
173 fveq2
 |-  ( z = x -> ( ( RR _D G ) ` z ) = ( ( RR _D G ) ` x ) )
174 173 oveq2d
 |-  ( z = x -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` z ) ) = ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) )
175 fveq2
 |-  ( z = x -> ( ( RR _D F ) ` z ) = ( ( RR _D F ) ` x ) )
176 175 oveq2d
 |-  ( z = x -> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` z ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) )
177 174 176 oveq12d
 |-  ( 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 ) ) ) )
178 177 129 128 fvmpt3i
 |-  ( 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 ) ) ) )
179 172 178 sylan9eq
 |-  ( ( 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 ) ) ) )
180 179 eqeq1d
 |-  ( ( 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 ) )
181 22 adantr
 |-  ( ( ph /\ x e. ( A (,) B ) ) -> ( ( F ` B ) - ( F ` A ) ) e. CC )
182 107 ffvelcdmda
 |-  ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D G ) ` x ) e. CC )
183 181 182 mulcld
 |-  ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) e. CC )
184 123 adantr
 |-  ( ( ph /\ x e. ( A (,) B ) ) -> ( ( G ` B ) - ( G ` A ) ) e. CC )
185 119 ffvelcdmda
 |-  ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
186 184 185 mulcld
 |-  ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) e. CC )
187 183 186 subeq0ad
 |-  ( ( 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 ) ) ) )
188 180 187 bitrd
 |-  ( ( 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 ) ) ) )
189 188 rexbidva
 |-  ( 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 ) ) ) )
190 171 189 mpbid
 |-  ( ph -> E. x e. ( A (,) B ) ( ( ( F ` B ) - ( F ` A ) ) x. ( ( RR _D G ) ` x ) ) = ( ( ( G ` B ) - ( G ` A ) ) x. ( ( RR _D F ) ` x ) ) )