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Theorem jensenlem2 22517
Description: Lemma for jensen 22518. (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 17968 . . . . . . 7  |-  0  =  ( 0g ` fld )
2 cnrng 17966 . . . . . . . 8  |-fld  e.  Ring
3 rngabl 16800 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Abel )
42, 3mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. 
Abel )
5 jensen.4 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
6 jensenlem.2 . . . . . . . . 9  |-  ( ph  ->  ( B  u.  {
z } )  C_  A )
76unssad 3644 . . . . . . . 8  |-  ( ph  ->  B  C_  A )
8 ssfi 7647 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  B  e.  Fin )
95, 7, 8syl2anc 661 . . . . . . 7  |-  ( ph  ->  B  e.  Fin )
10 resubdrg 18166 . . . . . . . . 9  |-  ( RR  e.  (SubRing ` fld )  /\ RRfld  e.  DivRing )
1110simpli 458 . . . . . . . 8  |-  RR  e.  (SubRing ` fld )
12 subrgsubg 16997 . . . . . . . 8  |-  ( RR  e.  (SubRing ` fld )  ->  RR  e.  (SubGrp ` fld ) )
1311, 12mp1i 12 . . . . . . 7  |-  ( ph  ->  RR  e.  (SubGrp ` fld )
)
14 remulcl 9481 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
1514adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
16 jensen.5 . . . . . . . . . 10  |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )
17 0re 9500 . . . . . . . . . . 11  |-  0  e.  RR
18 pnfxr 11206 . . . . . . . . . . 11  |- +oo  e.  RR*
19 icossre 11490 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
2017, 18, 19mp2an 672 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
21 fss 5678 . . . . . . . . . 10  |-  ( ( T : A --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  T : A --> RR )
2216, 20, 21sylancl 662 . . . . . . . . 9  |-  ( ph  ->  T : A --> RR )
23 jensen.6 . . . . . . . . . 10  |-  ( ph  ->  X : A --> D )
24 jensen.1 . . . . . . . . . 10  |-  ( ph  ->  D  C_  RR )
25 fss 5678 . . . . . . . . . 10  |-  ( ( X : A --> D  /\  D  C_  RR )  ->  X : A --> RR )
2623, 24, 25syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  X : A --> RR )
27 inidm 3670 . . . . . . . . 9  |-  ( A  i^i  A )  =  A
2815, 22, 26, 5, 5, 27off 6447 . . . . . . . 8  |-  ( ph  ->  ( T  oF  x.  X ) : A --> RR )
29 fssres 5689 . . . . . . . 8  |-  ( ( ( T  oF  x.  X ) : A --> RR  /\  B  C_  A )  ->  (
( T  oF  x.  X )  |`  B ) : B --> RR )
3028, 7, 29syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B ) : B --> RR )
31 c0ex 9494 . . . . . . . . 9  |-  0  e.  _V
3231a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  _V )
3330, 9, 32fdmfifsupp 7744 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B ) finSupp  0 )
341, 4, 9, 13, 30, 33gsumsubgcl 16530 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  e.  RR )
3534recnd 9526 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  e.  CC )
36 ax-resscn 9453 . . . . . . . 8  |-  RR  C_  CC
3720, 36sstri 3476 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
386unssbd 3645 . . . . . . . . 9  |-  ( ph  ->  { z }  C_  A )
39 vex 3081 . . . . . . . . . 10  |-  z  e. 
_V
4039snss 4110 . . . . . . . . 9  |-  ( z  e.  A  <->  { z }  C_  A )
4138, 40sylibr 212 . . . . . . . 8  |-  ( ph  ->  z  e.  A )
4216, 41ffvelrnd 5956 . . . . . . 7  |-  ( ph  ->  ( T `  z
)  e.  ( 0 [,) +oo ) )
4337, 42sseldi 3465 . . . . . 6  |-  ( ph  ->  ( T `  z
)  e.  CC )
4423, 41ffvelrnd 5956 . . . . . . . 8  |-  ( ph  ->  ( X `  z
)  e.  D )
4524, 44sseldd 3468 . . . . . . 7  |-  ( ph  ->  ( X `  z
)  e.  RR )
4645recnd 9526 . . . . . 6  |-  ( ph  ->  ( X `  z
)  e.  CC )
4743, 46mulcld 9520 . . . . 5  |-  ( ph  ->  ( ( T `  z )  x.  ( X `  z )
)  e.  CC )
48 jensen.2 . . . . . . . 8  |-  ( ph  ->  F : D --> RR )
49 jensen.3 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  D  /\  b  e.  D ) )  -> 
( a [,] b
)  C_  D )
50 jensen.7 . . . . . . . 8  |-  ( ph  ->  0  <  (fld  gsumg  T ) )
51 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
) ) ) )
52 jensenlem.1 . . . . . . . 8  |-  ( ph  ->  -.  z  e.  B
)
53 jensenlem.s . . . . . . . 8  |-  S  =  (fld 
gsumg  ( T  |`  B ) )
54 jensenlem.l . . . . . . . 8  |-  L  =  (fld 
gsumg  ( T  |`  ( B  u.  { z } ) ) )
5524, 48, 49, 5, 16, 23, 50, 51, 52, 6, 53, 54jensenlem1 22516 . . . . . . 7  |-  ( ph  ->  L  =  ( S  +  ( T `  z ) ) )
56 jensenlem.3 . . . . . . . . 9  |-  ( ph  ->  S  e.  RR+ )
5756rpred 11141 . . . . . . . 8  |-  ( ph  ->  S  e.  RR )
58 elrege0 11512 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  <->  ( ( T `
 z )  e.  RR  /\  0  <_ 
( T `  z
) ) )
5958simplbi 460 . . . . . . . . 9  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  ->  ( T `
 z )  e.  RR )
6042, 59syl 16 . . . . . . . 8  |-  ( ph  ->  ( T `  z
)  e.  RR )
6157, 60readdcld 9527 . . . . . . 7  |-  ( ph  ->  ( S  +  ( T `  z ) )  e.  RR )
6255, 61eqeltrd 2542 . . . . . 6  |-  ( ph  ->  L  e.  RR )
6362recnd 9526 . . . . 5  |-  ( ph  ->  L  e.  CC )
64 0red 9501 . . . . . . 7  |-  ( ph  ->  0  e.  RR )
6556rpgt0d 11144 . . . . . . 7  |-  ( ph  ->  0  <  S )
6658simprbi 464 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  ->  0  <_ 
( T `  z
) )
6742, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( T `  z ) )
6857, 60addge01d 10041 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  ( T `  z )  <->  S  <_  ( S  +  ( T `  z ) ) ) )
6967, 68mpbid 210 . . . . . . . 8  |-  ( ph  ->  S  <_  ( S  +  ( T `  z ) ) )
7069, 55breqtrrd 4429 . . . . . . 7  |-  ( ph  ->  S  <_  L )
7164, 57, 62, 65, 70ltletrd 9645 . . . . . 6  |-  ( ph  ->  0  <  L )
7271gt0ne0d 10018 . . . . 5  |-  ( ph  ->  L  =/=  0 )
7335, 47, 63, 72divdird 10259 . . . 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 ) ) )
74 cnfldbas 17950 . . . . . . 7  |-  CC  =  ( Base ` fld )
75 cnfldadd 17951 . . . . . . 7  |-  +  =  ( +g  ` fld )
76 rngcmn 16801 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
772, 76mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. CMnd
)
787sselda 3467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
7916ffvelrnda 5955 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( T `  x )  e.  ( 0 [,) +oo ) )
8078, 79syldan 470 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( T `  x )  e.  ( 0 [,) +oo ) )
8137, 80sseldi 3465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( T `  x )  e.  CC )
8224adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  D  C_  RR )
8323ffvelrnda 5955 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( X `  x )  e.  D )
8478, 83syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  D )
8582, 84sseldd 3468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  RR )
8685recnd 9526 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  CC )
8781, 86mulcld 9520 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( T `  x
)  x.  ( X `
 x ) )  e.  CC )
88 fveq2 5802 . . . . . . . 8  |-  ( x  =  z  ->  ( T `  x )  =  ( T `  z ) )
89 fveq2 5802 . . . . . . . 8  |-  ( x  =  z  ->  ( X `  x )  =  ( X `  z ) )
9088, 89oveq12d 6221 . . . . . . 7  |-  ( x  =  z  ->  (
( T `  x
)  x.  ( X `
 x ) )  =  ( ( T `
 z )  x.  ( X `  z
) ) )
9174, 75, 77, 9, 87, 41, 52, 47, 90gsumunsn 16578 . . . . . 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 ) ) ) )
9216feqmptd 5856 . . . . . . . . . 10  |-  ( ph  ->  T  =  ( x  e.  A  |->  ( T `
 x ) ) )
9323feqmptd 5856 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( x  e.  A  |->  ( X `
 x ) ) )
945, 79, 83, 92, 93offval2 6449 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  X )  =  ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) ) )
9594reseq1d 5220 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x ) ) )  |`  ( B  u.  {
z } ) ) )
96 resmpt 5267 . . . . . . . . 9  |-  ( ( B  u.  { z } )  C_  A  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( X `  x
) ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) )
976, 96syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( X `  x
) ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) )
9895, 97eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) )  =  ( x  e.  ( B  u.  {
z } )  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9998oveq2d 6219 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  =  (fld  gsumg  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) ) )
10094reseq1d 5220 . . . . . . . . 9  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) )  |`  B ) )
101 resmpt 5267 . . . . . . . . . 10  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) )  |`  B )  =  ( x  e.  B  |->  ( ( T `
 x )  x.  ( X `  x
) ) ) )
1027, 101syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( X `  x
) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
103100, 102eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
104103oveq2d 6219 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  =  (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( X `  x )
) ) ) )
105104oveq1d 6218 . . . . . 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 )
) ) )
10691, 99, 1053eqtr4d 2505 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  +  ( ( T `  z )  x.  ( X `  z ) ) ) )
107106oveq1d 6218 . . . 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 ) )
10857recnd 9526 . . . . . 6  |-  ( ph  ->  S  e.  CC )
10956rpne0d 11146 . . . . . 6  |-  ( ph  ->  S  =/=  0 )
11035, 108, 63, 109, 72dmdcand 10250 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  =  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  L ) )
11163, 108, 63, 72divsubdird 10260 . . . . . . . 8  |-  ( ph  ->  ( ( L  -  S )  /  L
)  =  ( ( L  /  L )  -  ( S  /  L ) ) )
11255oveq1d 6218 . . . . . . . . . 10  |-  ( ph  ->  ( L  -  S
)  =  ( ( S  +  ( T `
 z ) )  -  S ) )
113108, 43pncan2d 9835 . . . . . . . . . 10  |-  ( ph  ->  ( ( S  +  ( T `  z ) )  -  S )  =  ( T `  z ) )
114112, 113eqtrd 2495 . . . . . . . . 9  |-  ( ph  ->  ( L  -  S
)  =  ( T `
 z ) )
115114oveq1d 6218 . . . . . . . 8  |-  ( ph  ->  ( ( L  -  S )  /  L
)  =  ( ( T `  z )  /  L ) )
11663, 72dividd 10219 . . . . . . . . 9  |-  ( ph  ->  ( L  /  L
)  =  1 )
117116oveq1d 6218 . . . . . . . 8  |-  ( ph  ->  ( ( L  /  L )  -  ( S  /  L ) )  =  ( 1  -  ( S  /  L
) ) )
118111, 115, 1173eqtr3rd 2504 . . . . . . 7  |-  ( ph  ->  ( 1  -  ( S  /  L ) )  =  ( ( T `
 z )  /  L ) )
119118oveq1d 6218 . . . . . 6  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
)  =  ( ( ( T `  z
)  /  L )  x.  ( X `  z ) ) )
12043, 46, 63, 72div23d 10258 . . . . . 6  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( X `  z
) )  /  L
)  =  ( ( ( T `  z
)  /  L )  x.  ( X `  z ) ) )
121119, 120eqtr4d 2498 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
)  =  ( ( ( T `  z
)  x.  ( X `
 z ) )  /  L ) )
122110, 121oveq12d 6221 . . . 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 ) ) )
12373, 107, 1223eqtr4d 2505 . . 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 )
) ) )
124 jensenlem.4 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D
)
12557, 62, 72redivcld 10273 . . . . . 6  |-  ( ph  ->  ( S  /  L
)  e.  RR )
12656rpge0d 11145 . . . . . . 7  |-  ( ph  ->  0  <_  S )
127 divge0 10312 . . . . . . 7  |-  ( ( ( S  e.  RR  /\  0  <_  S )  /\  ( L  e.  RR  /\  0  <  L ) )  ->  0  <_  ( S  /  L ) )
12857, 126, 62, 71, 127syl22anc 1220 . . . . . 6  |-  ( ph  ->  0  <_  ( S  /  L ) )
12963mulid1d 9517 . . . . . . . 8  |-  ( ph  ->  ( L  x.  1 )  =  L )
13070, 129breqtrrd 4429 . . . . . . 7  |-  ( ph  ->  S  <_  ( L  x.  1 ) )
131 1red 9515 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
132 ledivmul 10319 . . . . . . . 8  |-  ( ( S  e.  RR  /\  1  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) )  -> 
( ( S  /  L )  <_  1  <->  S  <_  ( L  x.  1 ) ) )
13357, 131, 62, 71, 132syl112anc 1223 . . . . . . 7  |-  ( ph  ->  ( ( S  /  L )  <_  1  <->  S  <_  ( L  x.  1 ) ) )
134130, 133mpbird 232 . . . . . 6  |-  ( ph  ->  ( S  /  L
)  <_  1 )
135 1re 9499 . . . . . . 7  |-  1  e.  RR
13617, 135elicc2i 11475 . . . . . 6  |-  ( ( S  /  L )  e.  ( 0 [,] 1 )  <->  ( ( S  /  L )  e.  RR  /\  0  <_ 
( S  /  L
)  /\  ( S  /  L )  <_  1
) )
137125, 128, 134, 136syl3anbrc 1172 . . . . 5  |-  ( ph  ->  ( S  /  L
)  e.  ( 0 [,] 1 ) )
138124, 44, 1373jca 1168 . . . 4  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D  /\  ( X `  z
)  e.  D  /\  ( S  /  L
)  e.  ( 0 [,] 1 ) ) )
13924, 49cvxcl 22514 . . . 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 )
140138, 139mpdan 668 . . 3  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) )  e.  D
)
141123, 140eqeltrd 2542 . 2  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  e.  D )
14248, 140ffvelrnd 5956 . . . 4  |-  ( ph  ->  ( F `  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  e.  RR )
14348, 124ffvelrnd 5956 . . . . . 6  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  e.  RR )
144125, 143remulcld 9528 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  e.  RR )
14548, 44ffvelrnd 5956 . . . . . . 7  |-  ( ph  ->  ( F `  ( X `  z )
)  e.  RR )
14660, 145remulcld 9528 . . . . . 6  |-  ( ph  ->  ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  e.  RR )
147146, 62, 72redivcld 10273 . . . . 5  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  e.  RR )
148144, 147readdcld 9527 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  +  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )  e.  RR )
149 fco 5679 . . . . . . . . . . 11  |-  ( ( F : D --> RR  /\  X : A --> D )  ->  ( F  o.  X ) : A --> RR )
15048, 23, 149syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  X
) : A --> RR )
15115, 22, 150, 5, 5, 27off 6447 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  ( F  o.  X ) ) : A --> RR )
152 fssres 5689 . . . . . . . . 9  |-  ( ( ( T  oF  x.  ( F  o.  X ) ) : A --> RR  /\  B  C_  A )  ->  (
( T  oF  x.  ( F  o.  X ) )  |`  B ) : B --> RR )
153151, 7, 152syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) : B --> RR )
154153, 9, 32fdmfifsupp 7744 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) finSupp  0 )
1551, 4, 9, 13, 153, 154gsumsubgcl 16530 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  e.  RR )
156155, 57, 109redivcld 10273 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S )  e.  RR )
157125, 156remulcld 9528 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  e.  RR )
158 resubcl 9787 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( S  /  L
)  e.  RR )  ->  ( 1  -  ( S  /  L
) )  e.  RR )
159135, 125, 158sylancr 663 . . . . . 6  |-  ( ph  ->  ( 1  -  ( S  /  L ) )  e.  RR )
160159, 145remulcld 9528 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  e.  RR )
161157, 160readdcld 9527 . . . 4  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) ) )  e.  RR )
162 oveq2 6211 . . . . . . . . . . . 12  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
t  x.  x )  =  ( t  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
163162oveq1d 6218 . . . . . . . . . . 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
) ) )
164163fveq2d 5806 . . . . . . . . . 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
) ) ) )
165 fveq2 5802 . . . . . . . . . . . 12  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  ( F `  x )  =  ( F `  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
166165oveq2d 6219 . . . . . . . . . . 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 ) ) ) )
167166oveq1d 6218 . . . . . . . . . 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
) ) ) )
168164, 167breq12d 4416 . . . . . . . . 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
) ) ) ) )
169168imbi2d 316 . . . . . . . 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
) ) ) ) ) )
170 oveq2 6211 . . . . . . . . . . . 12  |-  ( y  =  ( X `  z )  ->  (
( 1  -  t
)  x.  y )  =  ( ( 1  -  t )  x.  ( X `  z
) ) )
171170oveq2d 6219 . . . . . . . . . . 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 )
) ) )
172171fveq2d 5806 . . . . . . . . . 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 )
) ) ) )
173 fveq2 5802 . . . . . . . . . . . 12  |-  ( y  =  ( X `  z )  ->  ( F `  y )  =  ( F `  ( X `  z ) ) )
174173oveq2d 6219 . . . . . . . . . . 11  |-  ( y  =  ( X `  z )  ->  (
( 1  -  t
)  x.  ( F `
 y ) )  =  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) )
175174oveq2d 6219 . . . . . . . . . 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 )
) ) ) )
176172, 175breq12d 4416 . . . . . . . . 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 )
) ) ) ) )
177176imbi2d 316 . . . . . . . 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 )
) ) ) ) ) )
178 oveq1 6210 . . . . . . . . . . . 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 ) ) )
179 oveq2 6211 . . . . . . . . . . . . 13  |-  ( t  =  ( S  /  L )  ->  (
1  -  t )  =  ( 1  -  ( S  /  L
) ) )
180179oveq1d 6218 . . . . . . . . . . . 12  |-  ( t  =  ( S  /  L )  ->  (
( 1  -  t
)  x.  ( X `
 z ) )  =  ( ( 1  -  ( S  /  L ) )  x.  ( X `  z
) ) )
181178, 180oveq12d 6221 . . . . . . . . . . 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 )
) ) )
182181fveq2d 5806 . . . . . . . . . 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 )
) ) ) )
183 oveq1 6210 . . . . . . . . . . 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 ) ) ) )
184179oveq1d 6218 . . . . . . . . . . 11  |-  ( t  =  ( S  /  L )  ->  (
( 1  -  t
)  x.  ( F `
 ( X `  z ) ) )  =  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) )
185183, 184oveq12d 6221 . . . . . . . . . 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 )
) ) ) )
186182, 185breq12d 4416 . . . . . . . . 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 )
) ) ) ) )
187186imbi2d 316 . . . . . . . 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 )
) ) ) ) ) )
18851expcom 435 . . . . . . . 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 )
) ) ) )
189169, 177, 187, 188vtocl3ga 3146 . . . . . . 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 )
) ) ) ) )
190124, 44, 137, 189syl3anc 1219 . . . . . 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 )
) ) ) ) )
191190pm2.43i 47 . . . . 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 )
) ) ) )
192118oveq1d 6218 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  =  ( ( ( T `  z )  /  L )  x.  ( F `  ( X `  z )
) ) )
193145recnd 9526 . . . . . . . 8  |-  ( ph  ->  ( F `  ( X `  z )
)  e.  CC )
19443, 193, 63, 72div23d 10258 . . . . . . 7  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  =  ( ( ( T `  z
)  /  L )  x.  ( F `  ( X `  z ) ) ) )
195192, 194eqtr4d 2498 . . . . . 6  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  =  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )
196195oveq2d 6219 . . . . 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
) ) )
197191, 196breqtrd 4427 . . . 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
) ) )
198194, 192eqtr4d 2498 . . . . . 6  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  =  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z ) ) ) )
199198oveq2d 6219 . . . . 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 )
) ) ) )
200 jensenlem.5 . . . . . . 7  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  <_ 
( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )
20157, 62, 65, 71divgt0d 10382 . . . . . . . 8  |-  ( ph  ->  0  <  ( S  /  L ) )
202 lemul2 10296 . . . . . . . 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 ) ) ) )
203143, 156, 125, 201, 202syl112anc 1223 . . . . . . 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 ) ) ) )
204200, 203mpbid 210 . . . . . 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 ) ) )
205144, 157, 160, 204leadd1dd 10067 . . . . 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 ) ) ) ) )
206199, 205eqbrtrd 4423 . . . 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 ) ) ) ) )
207142, 148, 161, 197, 206letrd 9642 . . 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 ) ) ) ) )
208123fveq2d 5806 . . 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 )
) ) ) )
209155recnd 9526 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  e.  CC )
210146recnd 9526 . . . . 5  |-  ( ph  ->  ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  e.  CC )
211209, 210, 63, 72divdird 10259 . . . 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
) ) )
21220, 79sseldi 3465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( T `  x )  e.  RR )
21348ffvelrnda 5955 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  x )  e.  D
)  ->  ( F `  ( X `  x
) )  e.  RR )
21483, 213syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  ( X `  x ) )  e.  RR )
215212, 214remulcld 9528 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  RR )
216215recnd 9526 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  CC )
21778, 216syldan 470 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  CC )
21889fveq2d 5806 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  ( X `  x ) )  =  ( F `  ( X `  z )
) )
21988, 218oveq12d 6221 . . . . . . 7  |-  ( x  =  z  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  =  ( ( T `
 z )  x.  ( F `  ( X `  z )
) ) )
22074, 75, 77, 9, 217, 41, 52, 210, 219gsumunsn 16578 . . . . . 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 ) ) ) ) )
22148feqmptd 5856 . . . . . . . . . . 11  |-  ( ph  ->  F  =  ( y  e.  D  |->  ( F `
 y ) ) )
222 fveq2 5802 . . . . . . . . . . 11  |-  ( y  =  ( X `  x )  ->  ( F `  y )  =  ( F `  ( X `  x ) ) )
22383, 93, 221, 222fmptco 5988 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  X
)  =  ( x  e.  A  |->  ( F `
 ( X `  x ) ) ) )
2245, 79, 214, 92, 223offval2 6449 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  ( F  o.  X ) )  =  ( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
225224reseq1d 5220 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) )  =  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  ( B  u.  { z } ) ) )
226 resmpt 5267 . . . . . . . . 9  |-  ( ( B  u.  { z } )  C_  A  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
2276, 226syl 16 . . . . . . . 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 ) ) ) ) )
228225, 227eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
229228oveq2d 6219 . . . . . 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 ) ) ) ) ) )
230224reseq1d 5220 . . . . . . . . 9  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) )  |`  B )
)
231 resmpt 5267 . . . . . . . . . 10  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) ) )
2327, 231syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( F `
 ( X `  x ) ) ) ) )
233230, 232eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( F `
 ( X `  x ) ) ) ) )
234233oveq2d 6219 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  =  (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) ) )
235234oveq1d 6218 . . . . . 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 )
) ) ) )
236220, 229, 2353eqtr4d 2505 . . . . 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 ) ) ) ) )
237236oveq1d 6218 . . . 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 ) )
238209, 108, 63, 109, 72dmdcand 10250 . . . . 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 ) )
239238, 195oveq12d 6221 . . . 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 ) ) )
240211, 237, 2393eqtr4d 2505 . . 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 ) ) ) ) )
241207, 208, 2403brtr4d 4433 . 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 ) )
242141, 241jca 532 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3437    C_ wss 3439   {csn 3988   class class class wbr 4403    |-> cmpt 4461    |` cres 4953    o. ccom 4955   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   Fincfn 7423   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401   +oocpnf 9529   RR*cxr 9531    < clt 9532    <_ cle 9533    - cmin 9709    / cdiv 10107   RR+crp 11105   [,)cico 11416   [,]cicc 11417    gsumg cgsu 14501  SubGrpcsubg 15797  CMndccmn 16401   Abelcabel 16402   Ringcrg 16771   DivRingcdr 16958  SubRingcsubrg 16987  ℂfldccnfld 17946  RRfldcrefld 18162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-addf 9475  ax-mulf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-tpos 6858  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-rp 11106  df-ico 11420  df-icc 11421  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-starv 14375  df-tset 14379  df-ple 14380  df-ds 14382  df-unif 14383  df-0g 14502  df-gsum 14503  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-submnd 15587  df-grp 15667  df-minusg 15668  df-mulg 15670  df-subg 15800  df-cntz 15957  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-cring 16774  df-oppr 16841  df-dvdsr 16859  df-unit 16860  df-invr 16890  df-dvr 16901  df-drng 16960  df-subrg 16989  df-cnfld 17947  df-refld 18163
This theorem is referenced by:  jensen  22518
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