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 ) ) ) |