MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  jensenlem2 Structured version   Visualization version   Unicode version

Theorem jensenlem2 23992
Description: Lemma for jensen 23993. (Contributed by Mario Carneiro, 21-Jun-2015.)
Hypotheses
Ref Expression
jensen.1  |-  ( ph  ->  D  C_  RR )
jensen.2  |-  ( ph  ->  F : D --> RR )
jensen.3  |-  ( (
ph  /\  ( a  e.  D  /\  b  e.  D ) )  -> 
( a [,] b
)  C_  D )
jensen.4  |-  ( ph  ->  A  e.  Fin )
jensen.5  |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )
jensen.6  |-  ( ph  ->  X : A --> D )
jensen.7  |-  ( ph  ->  0  <  (fld  gsumg  T ) )
jensen.8  |-  ( (
ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) )  <_  ( (
t  x.  ( F `
 x ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) ) )
jensenlem.1  |-  ( ph  ->  -.  z  e.  B
)
jensenlem.2  |-  ( ph  ->  ( B  u.  {
z } )  C_  A )
jensenlem.s  |-  S  =  (fld 
gsumg  ( T  |`  B ) )
jensenlem.l  |-  L  =  (fld 
gsumg  ( T  |`  ( B  u.  { z } ) ) )
jensenlem.3  |-  ( ph  ->  S  e.  RR+ )
jensenlem.4  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D
)
jensenlem.5  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  <_ 
( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )
Assertion
Ref Expression
jensenlem2  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  e.  D  /\  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L ) )  <_  ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  /  L ) ) )
Distinct variable groups:    a, b,
t, x, y, A    D, a, b, t, x, y    ph, a, b, t, x, y    F, a, b, t, x, y    T, a, b, t, x, y    X, a, b, t, x, y    z, a, B, b, t, x, y    t, L, x, y    S, a, b, t, x, y
Allowed substitution hints:    ph( z)    A( z)    D( z)    S( z)    T( z)    F( z)    L( z, a, b)    X( z)

Proof of Theorem jensenlem2
StepHypRef Expression
1 cnfld0 19069 . . . . . . 7  |-  0  =  ( 0g ` fld )
2 cnring 19067 . . . . . . . 8  |-fld  e.  Ring
3 ringabl 17888 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Abel )
42, 3mp1i 13 . . . . . . 7  |-  ( ph  ->fld  e. 
Abel )
5 jensen.4 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
6 jensenlem.2 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  {
z } )  C_  A )
76unssad 3602 . . . . . . . 8  |-  ( ph  ->  B  C_  A )
8 ssfi 7810 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  B  e.  Fin )
95, 7, 8syl2anc 673 . . . . . . 7  |-  ( ph  ->  B  e.  Fin )
10 resubdrg 19253 . . . . . . . . 9  |-  ( RR  e.  (SubRing ` fld )  /\ RRfld  e.  DivRing )
1110simpli 465 . . . . . . . 8  |-  RR  e.  (SubRing ` fld )
12 subrgsubg 18092 . . . . . . . 8  |-  ( RR  e.  (SubRing ` fld )  ->  RR  e.  (SubGrp ` fld ) )
1311, 12mp1i 13 . . . . . . 7  |-  ( ph  ->  RR  e.  (SubGrp ` fld )
)
14 remulcl 9642 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
1514adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
16 jensen.5 . . . . . . . . . 10  |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )
17 rge0ssre 11766 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
18 fss 5749 . . . . . . . . . 10  |-  ( ( T : A --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  T : A --> RR )
1916, 17, 18sylancl 675 . . . . . . . . 9  |-  ( ph  ->  T : A --> RR )
20 jensen.6 . . . . . . . . . 10  |-  ( ph  ->  X : A --> D )
21 jensen.1 . . . . . . . . . 10  |-  ( ph  ->  D  C_  RR )
2220, 21fssd 5750 . . . . . . . . 9  |-  ( ph  ->  X : A --> RR )
23 inidm 3632 . . . . . . . . 9  |-  ( A  i^i  A )  =  A
2415, 19, 22, 5, 5, 23off 6565 . . . . . . . 8  |-  ( ph  ->  ( T  oF  x.  X ) : A --> RR )
2524, 7fssresd 5762 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B ) : B --> RR )
26 c0ex 9655 . . . . . . . . 9  |-  0  e.  _V
2726a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  _V )
2825, 9, 27fdmfifsupp 7911 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B ) finSupp  0 )
291, 4, 9, 13, 25, 28gsumsubgcl 17631 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  e.  RR )
3029recnd 9687 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  e.  CC )
31 ax-resscn 9614 . . . . . . . 8  |-  RR  C_  CC
3217, 31sstri 3427 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
336unssbd 3603 . . . . . . . . 9  |-  ( ph  ->  { z }  C_  A )
34 vex 3034 . . . . . . . . . 10  |-  z  e. 
_V
3534snss 4087 . . . . . . . . 9  |-  ( z  e.  A  <->  { z }  C_  A )
3633, 35sylibr 217 . . . . . . . 8  |-  ( ph  ->  z  e.  A )
3716, 36ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( T `  z
)  e.  ( 0 [,) +oo ) )
3832, 37sseldi 3416 . . . . . 6  |-  ( ph  ->  ( T `  z
)  e.  CC )
3920, 36ffvelrnd 6038 . . . . . . . 8  |-  ( ph  ->  ( X `  z
)  e.  D )
4021, 39sseldd 3419 . . . . . . 7  |-  ( ph  ->  ( X `  z
)  e.  RR )
4140recnd 9687 . . . . . 6  |-  ( ph  ->  ( X `  z
)  e.  CC )
4238, 41mulcld 9681 . . . . 5  |-  ( ph  ->  ( ( T `  z )  x.  ( X `  z )
)  e.  CC )
43 jensen.2 . . . . . . . 8  |-  ( ph  ->  F : D --> RR )
44 jensen.3 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  D  /\  b  e.  D ) )  -> 
( a [,] b
)  C_  D )
45 jensen.7 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  T ) )
46 jensen.8 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  D  /\  y  e.  D  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) )  <_  ( (
t  x.  ( F `
 x ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) ) )
47 jensenlem.1 . . . . . . . 8  |-  ( ph  ->  -.  z  e.  B
)
48 jensenlem.s . . . . . . . 8  |-  S  =  (fld 
gsumg  ( T  |`  B ) )
49 jensenlem.l . . . . . . . 8  |-  L  =  (fld 
gsumg  ( T  |`  ( B  u.  { z } ) ) )
5021, 43, 44, 5, 16, 20, 45, 46, 47, 6, 48, 49jensenlem1 23991 . . . . . . 7  |-  ( ph  ->  L  =  ( S  +  ( T `  z ) ) )
51 jensenlem.3 . . . . . . . . 9  |-  ( ph  ->  S  e.  RR+ )
5251rpred 11364 . . . . . . . 8  |-  ( ph  ->  S  e.  RR )
53 elrege0 11764 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  <->  ( ( T `
 z )  e.  RR  /\  0  <_ 
( T `  z
) ) )
5453simplbi 467 . . . . . . . . 9  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  ->  ( T `
 z )  e.  RR )
5537, 54syl 17 . . . . . . . 8  |-  ( ph  ->  ( T `  z
)  e.  RR )
5652, 55readdcld 9688 . . . . . . 7  |-  ( ph  ->  ( S  +  ( T `  z ) )  e.  RR )
5750, 56eqeltrd 2549 . . . . . 6  |-  ( ph  ->  L  e.  RR )
5857recnd 9687 . . . . 5  |-  ( ph  ->  L  e.  CC )
59 0red 9662 . . . . . . 7  |-  ( ph  ->  0  e.  RR )
6051rpgt0d 11367 . . . . . . 7  |-  ( ph  ->  0  <  S )
6153simprbi 471 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  ->  0  <_ 
( T `  z
) )
6237, 61syl 17 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( T `  z ) )
6352, 55addge01d 10222 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  ( T `  z )  <->  S  <_  ( S  +  ( T `  z ) ) ) )
6462, 63mpbid 215 . . . . . . . 8  |-  ( ph  ->  S  <_  ( S  +  ( T `  z ) ) )
6564, 50breqtrrd 4422 . . . . . . 7  |-  ( ph  ->  S  <_  L )
6659, 52, 57, 60, 65ltletrd 9812 . . . . . 6  |-  ( ph  ->  0  <  L )
6766gt0ne0d 10199 . . . . 5  |-  ( ph  ->  L  =/=  0 )
6830, 42, 58, 67divdird 10443 . . . 4  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  +  ( ( T `  z )  x.  ( X `  z )
) )  /  L
)  =  ( ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  L )  +  ( ( ( T `  z )  x.  ( X `  z )
)  /  L ) ) )
69 cnfldbas 19051 . . . . . . 7  |-  CC  =  ( Base ` fld )
70 cnfldadd 19052 . . . . . . 7  |-  +  =  ( +g  ` fld )
71 ringcmn 17889 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
722, 71mp1i 13 . . . . . . 7  |-  ( ph  ->fld  e. CMnd
)
737sselda 3418 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
7416ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( T `  x )  e.  ( 0 [,) +oo ) )
7573, 74syldan 478 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( T `  x )  e.  ( 0 [,) +oo ) )
7632, 75sseldi 3416 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( T `  x )  e.  CC )
7721adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  D  C_  RR )
7820ffvelrnda 6037 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( X `  x )  e.  D )
7973, 78syldan 478 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  D )
8077, 79sseldd 3419 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  RR )
8180recnd 9687 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  CC )
8276, 81mulcld 9681 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( T `  x
)  x.  ( X `
 x ) )  e.  CC )
83 fveq2 5879 . . . . . . . 8  |-  ( x  =  z  ->  ( T `  x )  =  ( T `  z ) )
84 fveq2 5879 . . . . . . . 8  |-  ( x  =  z  ->  ( X `  x )  =  ( X `  z ) )
8583, 84oveq12d 6326 . . . . . . 7  |-  ( x  =  z  ->  (
( T `  x
)  x.  ( X `
 x ) )  =  ( ( T `
 z )  x.  ( X `  z
) ) )
8669, 70, 72, 9, 82, 36, 47, 42, 85gsumunsn 17670 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) )  =  ( (fld  gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( X `  x ) ) ) )  +  ( ( T `  z )  x.  ( X `  z ) ) ) )
8716feqmptd 5932 . . . . . . . . . 10  |-  ( ph  ->  T  =  ( x  e.  A  |->  ( T `
 x ) ) )
8820feqmptd 5932 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( x  e.  A  |->  ( X `
 x ) ) )
895, 74, 78, 87, 88offval2 6567 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  X )  =  ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) ) )
9089reseq1d 5110 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x ) ) )  |`  ( B  u.  {
z } ) ) )
916resmptd 5162 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( X `  x
) ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) )
9290, 91eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) )  =  ( x  e.  ( B  u.  {
z } )  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9392oveq2d 6324 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  =  (fld  gsumg  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) ) )
9489reseq1d 5110 . . . . . . . . 9  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) )  |`  B ) )
957resmptd 5162 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( X `  x
) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9694, 95eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9796oveq2d 6324 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  =  (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( X `  x )
) ) ) )
9897oveq1d 6323 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  +  ( ( T `  z )  x.  ( X `  z )
) )  =  ( (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( X `  x )
) ) )  +  ( ( T `  z )  x.  ( X `  z )
) ) )
9986, 93, 983eqtr4d 2515 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  +  ( ( T `  z )  x.  ( X `  z ) ) ) )
10099oveq1d 6323 . . . 4  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  =  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  +  ( ( T `  z )  x.  ( X `  z ) ) )  /  L ) )
10152recnd 9687 . . . . . 6  |-  ( ph  ->  S  e.  CC )
10251rpne0d 11369 . . . . . 6  |-  ( ph  ->  S  =/=  0 )
10330, 101, 58, 102, 67dmdcand 10434 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  =  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  L ) )
10458, 101, 58, 67divsubdird 10444 . . . . . . . 8  |-  ( ph  ->  ( ( L  -  S )  /  L
)  =  ( ( L  /  L )  -  ( S  /  L ) ) )
10550oveq1d 6323 . . . . . . . . . 10  |-  ( ph  ->  ( L  -  S
)  =  ( ( S  +  ( T `
 z ) )  -  S ) )
106101, 38pncan2d 10007 . . . . . . . . . 10  |-  ( ph  ->  ( ( S  +  ( T `  z ) )  -  S )  =  ( T `  z ) )
107105, 106eqtrd 2505 . . . . . . . . 9  |-  ( ph  ->  ( L  -  S
)  =  ( T `
 z ) )
108107oveq1d 6323 . . . . . . . 8  |-  ( ph  ->  ( ( L  -  S )  /  L
)  =  ( ( T `  z )  /  L ) )
10958, 67dividd 10403 . . . . . . . . 9  |-  ( ph  ->  ( L  /  L
)  =  1 )
110109oveq1d 6323 . . . . . . . 8  |-  ( ph  ->  ( ( L  /  L )  -  ( S  /  L ) )  =  ( 1  -  ( S  /  L
) ) )
111104, 108, 1103eqtr3rd 2514 . . . . . . 7  |-  ( ph  ->  ( 1  -  ( S  /  L ) )  =  ( ( T `
 z )  /  L ) )
112111oveq1d 6323 . . . . . 6  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
)  =  ( ( ( T `  z
)  /  L )  x.  ( X `  z ) ) )
11338, 41, 58, 67div23d 10442 . . . . . 6  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( X `  z
) )  /  L
)  =  ( ( ( T `  z
)  /  L )  x.  ( X `  z ) ) )
114112, 113eqtr4d 2508 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
)  =  ( ( ( T `  z
)  x.  ( X `
 z ) )  /  L ) )
115103, 114oveq12d 6326 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) )  =  ( ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  L )  +  ( ( ( T `  z )  x.  ( X `  z )
)  /  L ) ) )
11668, 100, 1153eqtr4d 2515 . . 3  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  =  ( ( ( S  /  L )  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )
117 jensenlem.4 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D
)
11852, 57, 67redivcld 10457 . . . . . 6  |-  ( ph  ->  ( S  /  L
)  e.  RR )
11951rpge0d 11368 . . . . . . 7  |-  ( ph  ->  0  <_  S )
120 divge0 10496 . . . . . . 7  |-  ( ( ( S  e.  RR  /\  0  <_  S )  /\  ( L  e.  RR  /\  0  <  L ) )  ->  0  <_  ( S  /  L ) )
12152, 119, 57, 66, 120syl22anc 1293 . . . . . 6  |-  ( ph  ->  0  <_  ( S  /  L ) )
12258mulid1d 9678 . . . . . . . 8  |-  ( ph  ->  ( L  x.  1 )  =  L )
12365, 122breqtrrd 4422 . . . . . . 7  |-  ( ph  ->  S  <_  ( L  x.  1 ) )
124 1red 9676 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
125 ledivmul 10503 . . . . . . . 8  |-  ( ( S  e.  RR  /\  1  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) )  -> 
( ( S  /  L )  <_  1  <->  S  <_  ( L  x.  1 ) ) )
12652, 124, 57, 66, 125syl112anc 1296 . . . . . . 7  |-  ( ph  ->  ( ( S  /  L )  <_  1  <->  S  <_  ( L  x.  1 ) ) )
127123, 126mpbird 240 . . . . . 6  |-  ( ph  ->  ( S  /  L
)  <_  1 )
128 0re 9661 . . . . . . 7  |-  0  e.  RR
129 1re 9660 . . . . . . 7  |-  1  e.  RR
130128, 129elicc2i 11725 . . . . . 6  |-  ( ( S  /  L )  e.  ( 0 [,] 1 )  <->  ( ( S  /  L )  e.  RR  /\  0  <_ 
( S  /  L
)  /\  ( S  /  L )  <_  1
) )
131118, 121, 127, 130syl3anbrc 1214 . . . . 5  |-  ( ph  ->  ( S  /  L
)  e.  ( 0 [,] 1 ) )
132117, 39, 1313jca 1210 . . . 4  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D  /\  ( X `  z
)  e.  D  /\  ( S  /  L
)  e.  ( 0 [,] 1 ) ) )
13321, 44cvxcl 23989 . . . 4  |-  ( (
ph  /\  ( (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D  /\  ( X `  z
)  e.  D  /\  ( S  /  L
)  e.  ( 0 [,] 1 ) ) )  ->  ( (
( S  /  L
)  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( X `  z ) ) )  e.  D )
134132, 133mpdan 681 . . 3  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) )  e.  D
)
135116, 134eqeltrd 2549 . 2  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  e.  D )
13643, 134ffvelrnd 6038 . . . 4  |-  ( ph  ->  ( F `  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  e.  RR )
13743, 117ffvelrnd 6038 . . . . . 6  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  e.  RR )
138118, 137remulcld 9689 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  e.  RR )
13943, 39ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( F `  ( X `  z )
)  e.  RR )
14055, 139remulcld 9689 . . . . . 6  |-  ( ph  ->  ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  e.  RR )
141140, 57, 67redivcld 10457 . . . . 5  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  e.  RR )
142138, 141readdcld 9688 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )  e.  RR )
143 fco 5751 . . . . . . . . . . 11  |-  ( ( F : D --> RR  /\  X : A --> D )  ->  ( F  o.  X ) : A --> RR )
14443, 20, 143syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  X
) : A --> RR )
14515, 19, 144, 5, 5, 23off 6565 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  ( F  o.  X ) ) : A --> RR )
146145, 7fssresd 5762 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) : B --> RR )
147146, 9, 27fdmfifsupp 7911 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) finSupp  0 )
1481, 4, 9, 13, 146, 147gsumsubgcl 17631 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  e.  RR )
149148, 52, 102redivcld 10457 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S )  e.  RR )
150118, 149remulcld 9689 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  e.  RR )
151 resubcl 9958 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( S  /  L
)  e.  RR )  ->  ( 1  -  ( S  /  L
) )  e.  RR )
152129, 118, 151sylancr 676 . . . . . 6  |-  ( ph  ->  ( 1  -  ( S  /  L ) )  e.  RR )
153152, 139remulcld 9689 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  e.  RR )
154150, 153readdcld 9688 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) )  e.  RR )
155 oveq2 6316 . . . . . . . . . . . 12  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
t  x.  x )  =  ( t  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
156155oveq1d 6323 . . . . . . . . . . 11  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) ) )
157156fveq2d 5883 . . . . . . . . . 10  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  ( F `  ( (
t  x.  x )  +  ( ( 1  -  t )  x.  y ) ) )  =  ( F `  ( ( t  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) ) ) )
158 fveq2 5879 . . . . . . . . . . . 12  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  ( F `  x )  =  ( F `  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
159158oveq2d 6324 . . . . . . . . . . 11  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
t  x.  ( F `
 x ) )  =  ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) ) )
160159oveq1d 6323 . . . . . . . . . 10  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
( t  x.  ( F `  x )
)  +  ( ( 1  -  t )  x.  ( F `  y ) ) )  =  ( ( t  x.  ( F `  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) ) )
161157, 160breq12d 4408 . . . . . . . . 9  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
( F `  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) ) )  <_  ( (
t  x.  ( F `
 x ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) )  <->  ( F `  ( ( t  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) ) ) )
162161imbi2d 323 . . . . . . . 8  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
( ph  ->  ( F `
 ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) ) )  <_ 
( ( t  x.  ( F `  x
) )  +  ( ( 1  -  t
)  x.  ( F `
 y ) ) ) )  <->  ( ph  ->  ( F `  (
( t  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) ) ) ) )
163 oveq2 6316 . . . . . . . . . . . 12  |-  ( y  =  ( X `  z )  ->  (
( 1  -  t
)  x.  y )  =  ( ( 1  -  t )  x.  ( X `  z
) ) )
164163oveq2d 6324 . . . . . . . . . . 11  |-  ( y  =  ( X `  z )  ->  (
( t  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) )  =  ( ( t  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) ) )
165164fveq2d 5883 . . . . . . . . . 10  |-  ( y  =  ( X `  z )  ->  ( F `  ( (
t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y ) ) )  =  ( F `
 ( ( t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) ) ) )
166 fveq2 5879 . . . . . . . . . . . 12  |-  ( y  =  ( X `  z )  ->  ( F `  y )  =  ( F `  ( X `  z ) ) )
167166oveq2d 6324 . . . . . . . . . . 11  |-  ( y  =  ( X `  z )  ->  (
( 1  -  t
)  x.  ( F `
 y ) )  =  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) )
168167oveq2d 6324 . . . . . . . . . 10  |-  ( y  =  ( X `  z )  ->  (
( t  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) )  =  ( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) ) )
169165, 168breq12d 4408 . . . . . . . . 9  |-  ( y  =  ( X `  z )  ->  (
( F `  (
( t  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) )  <->  ( F `  ( ( t  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) ) ) )
170169imbi2d 323 . . . . . . . 8  |-  ( y  =  ( X `  z )  ->  (
( ph  ->  ( F `
 ( ( t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  y
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  y
) ) ) )  <-> 
( ph  ->  ( F `
 ( ( t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) ) ) ) )
171 oveq1 6315 . . . . . . . . . . . 12  |-  ( t  =  ( S  /  L )  ->  (
t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  =  ( ( S  /  L )  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
172 oveq2 6316 . . . . . . . . . . . . 13  |-  ( t  =  ( S  /  L )  ->  (
1  -  t )  =  ( 1  -  ( S  /  L
) ) )
173172oveq1d 6323 . . . . . . . . . . . 12  |-  ( t  =  ( S  /  L )  ->  (
( 1  -  t
)  x.  ( X `
 z ) )  =  ( ( 1  -  ( S  /  L ) )  x.  ( X `  z
) ) )
174171, 173oveq12d 6326 . . . . . . . . . . 11  |-  ( t  =  ( S  /  L )  ->  (
( t  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) )  =  ( ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )
175174fveq2d 5883 . . . . . . . . . 10  |-  ( t  =  ( S  /  L )  ->  ( F `  ( (
t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z ) ) ) )  =  ( F `
 ( ( ( S  /  L )  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) ) )
176 oveq1 6315 . . . . . . . . . . 11  |-  ( t  =  ( S  /  L )  ->  (
t  x.  ( F `
 ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  =  ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) ) )
177172oveq1d 6323 . . . . . . . . . . 11  |-  ( t  =  ( S  /  L )  ->  (
( 1  -  t
)  x.  ( F `
 ( X `  z ) ) )  =  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) )
178176, 177oveq12d 6326 . . . . . . . . . 10  |-  ( t  =  ( S  /  L )  ->  (
( t  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) )  =  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) )
179175, 178breq12d 4408 . . . . . . . . 9  |-  ( t  =  ( S  /  L )  ->  (
( F `  (
( t  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) )  <->  ( F `  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  <_ 
( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) ) )
180179imbi2d 323 . . . . . . . 8  |-  ( t  =  ( S  /  L )  ->  (
( ph  ->  ( F `
 ( ( t  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  t )  x.  ( X `  z )
) ) )  <_ 
( ( t  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) ) )  <-> 
( ph  ->  ( F `
 ( ( ( S  /  L )  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  <_ 
( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) ) ) )
18146expcom 442 . . . . . . . 8  |-  ( ( x  e.  D  /\  y  e.  D  /\  t  e.  ( 0 [,] 1 ) )  ->  ( ph  ->  ( F `  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y ) ) )  <_  ( ( t  x.  ( F `  x ) )  +  ( ( 1  -  t )  x.  ( F `  y )
) ) ) )
182162, 170, 180, 181vtocl3ga 3103 . . . . . . 7  |-  ( ( ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D  /\  ( X `  z
)  e.  D  /\  ( S  /  L
)  e.  ( 0 [,] 1 ) )  ->  ( ph  ->  ( F `  ( ( ( S  /  L
)  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( X `  z ) ) ) )  <_  ( (
( S  /  L
)  x.  ( F `
 ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) ) )
183117, 39, 131, 182syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( ph  ->  ( F `  ( (
( S  /  L
)  x.  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( X `  z ) ) ) )  <_  ( (
( S  /  L
)  x.  ( F `
 ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) ) )
184183pm2.43i 48 . . . . 5  |-  ( ph  ->  ( F `  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  <_ 
( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) )
185111oveq1d 6323 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  =  ( ( ( T `  z )  /  L )  x.  ( F `  ( X `  z )
) ) )
186139recnd 9687 . . . . . . . 8  |-  ( ph  ->  ( F `  ( X `  z )
)  e.  CC )
18738, 186, 58, 67div23d 10442 . . . . . . 7  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  =  ( ( ( T `  z
)  /  L )  x.  ( F `  ( X `  z ) ) ) )
188185, 187eqtr4d 2508 . . . . . 6  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  =  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )
189188oveq2d 6324 . . . . 5  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) )  =  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) ) )
190184, 189breqtrd 4420 . . . 4  |-  ( ph  ->  ( F `  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  <_ 
( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) ) )
191187, 185eqtr4d 2508 . . . . . 6  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  =  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z ) ) ) )
192191oveq2d 6324 . . . . 5  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )  =  ( ( ( S  /  L )  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) ) )
193 jensenlem.5 . . . . . . 7  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  <_ 
( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )
19452, 57, 60, 66divgt0d 10564 . . . . . . . 8  |-  ( ph  ->  0  <  ( S  /  L ) )
195 lemul2 10480 . . . . . . . 8  |-  ( ( ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  e.  RR  /\  ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X
) )  |`  B ) )  /  S )  e.  RR  /\  (
( S  /  L
)  e.  RR  /\  0  <  ( S  /  L ) ) )  ->  ( ( F `
 ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  <_ 
( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S )  <->  ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  <_  ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) ) ) )
196137, 149, 118, 194, 195syl112anc 1296 . . . . . . 7  |-  ( ph  ->  ( ( F `  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  <_ 
( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S )  <->  ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  <_  ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) ) ) )
197193, 196mpbid 215 . . . . . 6  |-  ( ph  ->  ( ( S  /  L )  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  <_  ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) ) )
198138, 150, 153, 197leadd1dd 10248 . . . . 5  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) )  <_ 
( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) ) )
199192, 198eqbrtrd 4416 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )  <_  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) ) )
200136, 142, 154, 190, 199letrd 9809 . . 3  |-  ( ph  ->  ( F `  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  <_ 
( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) ) )
201116fveq2d 5883 . . 3  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  { z } ) ) )  /  L ) )  =  ( F `  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) ) )
202148recnd 9687 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  e.  CC )
203140recnd 9687 . . . . 5  |-  ( ph  ->  ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  e.  CC )
204202, 203, 58, 67divdird 10443 . . . 4  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  +  ( ( T `  z )  x.  ( F `  ( X `  z ) ) ) )  /  L )  =  ( ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X
) )  |`  B ) )  /  L )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) ) )
20517, 74sseldi 3416 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( T `  x )  e.  RR )
20643ffvelrnda 6037 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  x )  e.  D
)  ->  ( F `  ( X `  x
) )  e.  RR )
20778, 206syldan 478 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  ( X `  x ) )  e.  RR )
208205, 207remulcld 9689 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  RR )
209208recnd 9687 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  CC )
21073, 209syldan 478 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  CC )
21184fveq2d 5883 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  ( X `  x ) )  =  ( F `  ( X `  z )
) )
21283, 211oveq12d 6326 . . . . . . 7  |-  ( x  =  z  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  =  ( ( T `
 z )  x.  ( F `  ( X `  z )
) ) )
21369, 70, 72, 9, 210, 36, 47, 203, 212gsumunsn 17670 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )  =  ( (fld  gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )  +  ( ( T `  z
)  x.  ( F `
 ( X `  z ) ) ) ) )
21443feqmptd 5932 . . . . . . . . . . 11  |-  ( ph  ->  F  =  ( y  e.  D  |->  ( F `
 y ) ) )
215 fveq2 5879 . . . . . . . . . . 11  |-  ( y  =  ( X `  x )  ->  ( F `  y )  =  ( F `  ( X `  x ) ) )
21678, 88, 214, 215fmptco 6072 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  X
)  =  ( x  e.  A  |->  ( F `
 ( X `  x ) ) ) )
2175, 74, 207, 87, 216offval2 6567 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  ( F  o.  X ) )  =  ( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
218217reseq1d 5110 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) )  =  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  ( B  u.  { z } ) ) )
2196resmptd 5162 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
220218, 219eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
221220oveq2d 6324 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  =  (fld 
gsumg  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) ) )
222217reseq1d 5110 . . . . . . . . 9  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) )  |`  B )
)
2237resmptd 5162 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( F `
 ( X `  x ) ) ) ) )
224222, 223eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( F `
 ( X `  x ) ) ) ) )
225224oveq2d 6324 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  =  (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) ) )
226225oveq1d 6323 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  +  ( ( T `  z )  x.  ( F `  ( X `  z ) ) ) )  =  ( (fld  gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )  +  ( ( T `
 z )  x.  ( F `  ( X `  z )
) ) ) )
227213, 221, 2263eqtr4d 2515 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  =  ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  +  ( ( T `  z )  x.  ( F `  ( X `  z ) ) ) ) )
228227oveq1d 6323 . . . 4  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  /  L )  =  ( ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  +  ( ( T `  z )  x.  ( F `  ( X `  z ) ) ) )  /  L ) )
229202, 101, 58, 102, 67dmdcand 10434 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  =  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  L ) )
230229, 188oveq12d 6326 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) )  =  ( ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  L )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L ) ) )
231204, 228, 2303eqtr4d 2515 . . 3  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  /  L )  =  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) ) )
232200, 201, 2313brtr4d 4426 . 2  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  { z } ) ) )  /  L ) )  <_  ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  /  L ) )
233135, 232jca 541 1  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  e.  D  /\  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L ) )  <_  ( (fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) ) )  /  L ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388    C_ wss 3390   {csn 3959   class class class wbr 4395    |-> cmpt 4454    |` cres 4841    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   RR+crp 11325   [,)cico 11662   [,]cicc 11663    gsumg cgsu 15417  SubGrpcsubg 16889  CMndccmn 17508   Abelcabl 17509   Ringcrg 17858   DivRingcdr 18053  SubRingcsubrg 18082  ℂfldccnfld 19047  RRfldcrefld 19249
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-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-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  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-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  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-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-0g 15418  df-gsum 15419  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-subg 16892  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-subrg 18084  df-cnfld 19048  df-refld 19250
This theorem is referenced by:  jensen  23993
  Copyright terms: Public domain W3C validator