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Theorem jensenlem2 23515
Description: Lemma for jensen 23516. (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 18637 . . . . . . 7  |-  0  =  ( 0g ` fld )
2 cnring 18635 . . . . . . . 8  |-fld  e.  Ring
3 ringabl 17423 . . . . . . . 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 3667 . . . . . . . 8  |-  ( ph  ->  B  C_  A )
8 ssfi 7733 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  B  e.  Fin )
95, 7, 8syl2anc 659 . . . . . . 7  |-  ( ph  ->  B  e.  Fin )
10 resubdrg 18817 . . . . . . . . 9  |-  ( RR  e.  (SubRing ` fld )  /\ RRfld  e.  DivRing )
1110simpli 456 . . . . . . . 8  |-  RR  e.  (SubRing ` fld )
12 subrgsubg 17630 . . . . . . . 8  |-  ( RR  e.  (SubRing ` fld )  ->  RR  e.  (SubGrp ` fld ) )
1311, 12mp1i 12 . . . . . . 7  |-  ( ph  ->  RR  e.  (SubGrp ` fld )
)
14 remulcl 9566 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
1514adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
16 jensen.5 . . . . . . . . . 10  |-  ( ph  ->  T : A --> ( 0 [,) +oo ) )
17 rge0ssre 11631 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
18 fss 5721 . . . . . . . . . 10  |-  ( ( T : A --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  T : A --> RR )
1916, 17, 18sylancl 660 . . . . . . . . 9  |-  ( ph  ->  T : A --> RR )
20 jensen.6 . . . . . . . . . 10  |-  ( ph  ->  X : A --> D )
21 jensen.1 . . . . . . . . . 10  |-  ( ph  ->  D  C_  RR )
2220, 21fssd 5722 . . . . . . . . 9  |-  ( ph  ->  X : A --> RR )
23 inidm 3693 . . . . . . . . 9  |-  ( A  i^i  A )  =  A
2415, 19, 22, 5, 5, 23off 6527 . . . . . . . 8  |-  ( ph  ->  ( T  oF  x.  X ) : A --> RR )
2524, 7fssresd 5734 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B ) : B --> RR )
26 c0ex 9579 . . . . . . . . 9  |-  0  e.  _V
2726a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  _V )
2825, 9, 27fdmfifsupp 7831 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B ) finSupp  0 )
291, 4, 9, 13, 25, 28gsumsubgcl 17131 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  e.  RR )
3029recnd 9611 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  e.  CC )
31 ax-resscn 9538 . . . . . . . 8  |-  RR  C_  CC
3217, 31sstri 3498 . . . . . . 7  |-  ( 0 [,) +oo )  C_  CC
336unssbd 3668 . . . . . . . . 9  |-  ( ph  ->  { z }  C_  A )
34 vex 3109 . . . . . . . . . 10  |-  z  e. 
_V
3534snss 4140 . . . . . . . . 9  |-  ( z  e.  A  <->  { z }  C_  A )
3633, 35sylibr 212 . . . . . . . 8  |-  ( ph  ->  z  e.  A )
3716, 36ffvelrnd 6008 . . . . . . 7  |-  ( ph  ->  ( T `  z
)  e.  ( 0 [,) +oo ) )
3832, 37sseldi 3487 . . . . . 6  |-  ( ph  ->  ( T `  z
)  e.  CC )
3920, 36ffvelrnd 6008 . . . . . . . 8  |-  ( ph  ->  ( X `  z
)  e.  D )
4021, 39sseldd 3490 . . . . . . 7  |-  ( ph  ->  ( X `  z
)  e.  RR )
4140recnd 9611 . . . . . 6  |-  ( ph  ->  ( X `  z
)  e.  CC )
4238, 41mulcld 9605 . . . . 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 23514 . . . . . . 7  |-  ( ph  ->  L  =  ( S  +  ( T `  z ) ) )
51 jensenlem.3 . . . . . . . . 9  |-  ( ph  ->  S  e.  RR+ )
5251rpred 11259 . . . . . . . 8  |-  ( ph  ->  S  e.  RR )
53 elrege0 11630 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  <->  ( ( T `
 z )  e.  RR  /\  0  <_ 
( T `  z
) ) )
5453simplbi 458 . . . . . . . . 9  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  ->  ( T `
 z )  e.  RR )
5537, 54syl 16 . . . . . . . 8  |-  ( ph  ->  ( T `  z
)  e.  RR )
5652, 55readdcld 9612 . . . . . . 7  |-  ( ph  ->  ( S  +  ( T `  z ) )  e.  RR )
5750, 56eqeltrd 2542 . . . . . 6  |-  ( ph  ->  L  e.  RR )
5857recnd 9611 . . . . 5  |-  ( ph  ->  L  e.  CC )
59 0red 9586 . . . . . . 7  |-  ( ph  ->  0  e.  RR )
6051rpgt0d 11262 . . . . . . 7  |-  ( ph  ->  0  <  S )
6153simprbi 462 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ( 0 [,) +oo )  ->  0  <_ 
( T `  z
) )
6237, 61syl 16 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( T `  z ) )
6352, 55addge01d 10136 . . . . . . . . 9  |-  ( ph  ->  ( 0  <_  ( T `  z )  <->  S  <_  ( S  +  ( T `  z ) ) ) )
6462, 63mpbid 210 . . . . . . . 8  |-  ( ph  ->  S  <_  ( S  +  ( T `  z ) ) )
6564, 50breqtrrd 4465 . . . . . . 7  |-  ( ph  ->  S  <_  L )
6659, 52, 57, 60, 65ltletrd 9731 . . . . . 6  |-  ( ph  ->  0  <  L )
6766gt0ne0d 10113 . . . . 5  |-  ( ph  ->  L  =/=  0 )
6830, 42, 58, 67divdird 10354 . . . 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 18619 . . . . . . 7  |-  CC  =  ( Base ` fld )
70 cnfldadd 18620 . . . . . . 7  |-  +  =  ( +g  ` fld )
71 ringcmn 17424 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e. CMnd )
722, 71mp1i 12 . . . . . . 7  |-  ( ph  ->fld  e. CMnd
)
737sselda 3489 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  A )
7416ffvelrnda 6007 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( T `  x )  e.  ( 0 [,) +oo ) )
7573, 74syldan 468 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( T `  x )  e.  ( 0 [,) +oo ) )
7632, 75sseldi 3487 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( T `  x )  e.  CC )
7721adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  D  C_  RR )
7820ffvelrnda 6007 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( X `  x )  e.  D )
7973, 78syldan 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  D )
8077, 79sseldd 3490 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  RR )
8180recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  ( X `  x )  e.  CC )
8276, 81mulcld 9605 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( T `  x
)  x.  ( X `
 x ) )  e.  CC )
83 fveq2 5848 . . . . . . . 8  |-  ( x  =  z  ->  ( T `  x )  =  ( T `  z ) )
84 fveq2 5848 . . . . . . . 8  |-  ( x  =  z  ->  ( X `  x )  =  ( X `  z ) )
8583, 84oveq12d 6288 . . . . . . 7  |-  ( x  =  z  ->  (
( T `  x
)  x.  ( X `
 x ) )  =  ( ( T `
 z )  x.  ( X `  z
) ) )
8669, 70, 72, 9, 82, 36, 47, 42, 85gsumunsn 17182 . . . . . 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 5901 . . . . . . . . . 10  |-  ( ph  ->  T  =  ( x  e.  A  |->  ( T `
 x ) ) )
8820feqmptd 5901 . . . . . . . . . 10  |-  ( ph  ->  X  =  ( x  e.  A  |->  ( X `
 x ) ) )
895, 74, 78, 87, 88offval2 6529 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  X )  =  ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) ) )
9089reseq1d 5261 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x ) ) )  |`  ( B  u.  {
z } ) ) )
916resmptd 5313 . . . . . . . 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 2495 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) )  =  ( x  e.  ( B  u.  {
z } )  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9392oveq2d 6286 . . . . . 6  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  =  (fld  gsumg  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( X `  x ) ) ) ) )
9489reseq1d 5261 . . . . . . . . 9  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( X `  x )
) )  |`  B ) )
957resmptd 5313 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( X `  x
) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9694, 95eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  X )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( X `
 x ) ) ) )
9796oveq2d 6286 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  =  (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( X `  x )
) ) ) )
9897oveq1d 6285 . . . . . 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 2505 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  +  ( ( T `  z )  x.  ( X `  z ) ) ) )
10099oveq1d 6285 . . . 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 9611 . . . . . 6  |-  ( ph  ->  S  e.  CC )
10251rpne0d 11264 . . . . . 6  |-  ( ph  ->  S  =/=  0 )
10330, 101, 58, 102, 67dmdcand 10345 . . . . 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 10355 . . . . . . . 8  |-  ( ph  ->  ( ( L  -  S )  /  L
)  =  ( ( L  /  L )  -  ( S  /  L ) ) )
10550oveq1d 6285 . . . . . . . . . 10  |-  ( ph  ->  ( L  -  S
)  =  ( ( S  +  ( T `
 z ) )  -  S ) )
106101, 38pncan2d 9924 . . . . . . . . . 10  |-  ( ph  ->  ( ( S  +  ( T `  z ) )  -  S )  =  ( T `  z ) )
107105, 106eqtrd 2495 . . . . . . . . 9  |-  ( ph  ->  ( L  -  S
)  =  ( T `
 z ) )
108107oveq1d 6285 . . . . . . . 8  |-  ( ph  ->  ( ( L  -  S )  /  L
)  =  ( ( T `  z )  /  L ) )
10958, 67dividd 10314 . . . . . . . . 9  |-  ( ph  ->  ( L  /  L
)  =  1 )
110109oveq1d 6285 . . . . . . . 8  |-  ( ph  ->  ( ( L  /  L )  -  ( S  /  L ) )  =  ( 1  -  ( S  /  L
) ) )
111104, 108, 1103eqtr3rd 2504 . . . . . . 7  |-  ( ph  ->  ( 1  -  ( S  /  L ) )  =  ( ( T `
 z )  /  L ) )
112111oveq1d 6285 . . . . . 6  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
)  =  ( ( ( T `  z
)  /  L )  x.  ( X `  z ) ) )
11338, 41, 58, 67div23d 10353 . . . . . 6  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( X `  z
) )  /  L
)  =  ( ( ( T `  z
)  /  L )  x.  ( X `  z ) ) )
114112, 113eqtr4d 2498 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
)  =  ( ( ( T `  z
)  x.  ( X `
 z ) )  /  L ) )
115103, 114oveq12d 6288 . . . 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 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 )
) ) )
117 jensenlem.4 . . . . 5  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D
)
11852, 57, 67redivcld 10368 . . . . . 6  |-  ( ph  ->  ( S  /  L
)  e.  RR )
11951rpge0d 11263 . . . . . . 7  |-  ( ph  ->  0  <_  S )
120 divge0 10407 . . . . . . 7  |-  ( ( ( S  e.  RR  /\  0  <_  S )  /\  ( L  e.  RR  /\  0  <  L ) )  ->  0  <_  ( S  /  L ) )
12152, 119, 57, 66, 120syl22anc 1227 . . . . . 6  |-  ( ph  ->  0  <_  ( S  /  L ) )
12258mulid1d 9602 . . . . . . . 8  |-  ( ph  ->  ( L  x.  1 )  =  L )
12365, 122breqtrrd 4465 . . . . . . 7  |-  ( ph  ->  S  <_  ( L  x.  1 ) )
124 1red 9600 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
125 ledivmul 10414 . . . . . . . 8  |-  ( ( S  e.  RR  /\  1  e.  RR  /\  ( L  e.  RR  /\  0  <  L ) )  -> 
( ( S  /  L )  <_  1  <->  S  <_  ( L  x.  1 ) ) )
12652, 124, 57, 66, 125syl112anc 1230 . . . . . . 7  |-  ( ph  ->  ( ( S  /  L )  <_  1  <->  S  <_  ( L  x.  1 ) ) )
127123, 126mpbird 232 . . . . . 6  |-  ( ph  ->  ( S  /  L
)  <_  1 )
128 0re 9585 . . . . . . 7  |-  0  e.  RR
129 1re 9584 . . . . . . 7  |-  1  e.  RR
130128, 129elicc2i 11593 . . . . . 6  |-  ( ( S  /  L )  e.  ( 0 [,] 1 )  <->  ( ( S  /  L )  e.  RR  /\  0  <_ 
( S  /  L
)  /\  ( S  /  L )  <_  1
) )
131118, 121, 127, 130syl3anbrc 1178 . . . . 5  |-  ( ph  ->  ( S  /  L
)  e.  ( 0 [,] 1 ) )
132117, 39, 1313jca 1174 . . . 4  |-  ( ph  ->  ( ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  e.  D  /\  ( X `  z
)  e.  D  /\  ( S  /  L
)  e.  ( 0 [,] 1 ) ) )
13321, 44cvxcl 23512 . . . 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 666 . . 3  |-  ( ph  ->  ( ( ( S  /  L )  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) )  e.  D
)
135116, 134eqeltrd 2542 . 2  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  ( B  u.  {
z } ) ) )  /  L )  e.  D )
13643, 134ffvelrnd 6008 . . . 4  |-  ( ph  ->  ( F `  (
( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  +  ( ( 1  -  ( S  /  L
) )  x.  ( X `  z )
) ) )  e.  RR )
13743, 117ffvelrnd 6008 . . . . . 6  |-  ( ph  ->  ( F `  (
(fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) )  e.  RR )
138118, 137remulcld 9613 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  ( F `  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )  e.  RR )
13943, 39ffvelrnd 6008 . . . . . . 7  |-  ( ph  ->  ( F `  ( X `  z )
)  e.  RR )
14055, 139remulcld 9613 . . . . . 6  |-  ( ph  ->  ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  e.  RR )
141140, 57, 67redivcld 10368 . . . . 5  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  e.  RR )
142138, 141readdcld 9612 . . . 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 5723 . . . . . . . . . . 11  |-  ( ( F : D --> RR  /\  X : A --> D )  ->  ( F  o.  X ) : A --> RR )
14443, 20, 143syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  X
) : A --> RR )
14515, 19, 144, 5, 5, 23off 6527 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  ( F  o.  X ) ) : A --> RR )
146145, 7fssresd 5734 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) : B --> RR )
147146, 9, 27fdmfifsupp 7831 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) finSupp  0 )
1481, 4, 9, 13, 146, 147gsumsubgcl 17131 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  e.  RR )
149148, 52, 102redivcld 10368 . . . . . 6  |-  ( ph  ->  ( (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S )  e.  RR )
150118, 149remulcld 9613 . . . . 5  |-  ( ph  ->  ( ( S  /  L )  x.  (
(fld  gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  /  S ) )  e.  RR )
151 resubcl 9874 . . . . . . 7  |-  ( ( 1  e.  RR  /\  ( S  /  L
)  e.  RR )  ->  ( 1  -  ( S  /  L
) )  e.  RR )
152129, 118, 151sylancr 661 . . . . . 6  |-  ( ph  ->  ( 1  -  ( S  /  L ) )  e.  RR )
153152, 139remulcld 9613 . . . . 5  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  e.  RR )
154150, 153readdcld 9612 . . . 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 6278 . . . . . . . . . . . 12  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  (
t  x.  x )  =  ( t  x.  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
156155oveq1d 6285 . . . . . . . . . . 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 5852 . . . . . . . . . 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 5848 . . . . . . . . . . . 12  |-  ( x  =  ( (fld  gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S )  ->  ( F `  x )  =  ( F `  ( (fld 
gsumg  ( ( T  oF  x.  X )  |`  B ) )  /  S ) ) )
159158oveq2d 6286 . . . . . . . . . . 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 6285 . . . . . . . . . 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 4452 . . . . . . . . 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 314 . . . . . . . 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 6278 . . . . . . . . . . . 12  |-  ( y  =  ( X `  z )  ->  (
( 1  -  t
)  x.  y )  =  ( ( 1  -  t )  x.  ( X `  z
) ) )
164163oveq2d 6286 . . . . . . . . . . 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 5852 . . . . . . . . . 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 5848 . . . . . . . . . . . 12  |-  ( y  =  ( X `  z )  ->  ( F `  y )  =  ( F `  ( X `  z ) ) )
167166oveq2d 6286 . . . . . . . . . . 11  |-  ( y  =  ( X `  z )  ->  (
( 1  -  t
)  x.  ( F `
 y ) )  =  ( ( 1  -  t )  x.  ( F `  ( X `  z )
) ) )
168167oveq2d 6286 . . . . . . . . . 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 4452 . . . . . . . . 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 314 . . . . . . . 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 6277 . . . . . . . . . . . 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 6278 . . . . . . . . . . . . 13  |-  ( t  =  ( S  /  L )  ->  (
1  -  t )  =  ( 1  -  ( S  /  L
) ) )
173172oveq1d 6285 . . . . . . . . . . . 12  |-  ( t  =  ( S  /  L )  ->  (
( 1  -  t
)  x.  ( X `
 z ) )  =  ( ( 1  -  ( S  /  L ) )  x.  ( X `  z
) ) )
174171, 173oveq12d 6288 . . . . . . . . . . 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 5852 . . . . . . . . . 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 6277 . . . . . . . . . . 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 6285 . . . . . . . . . . 11  |-  ( t  =  ( S  /  L )  ->  (
( 1  -  t
)  x.  ( F `
 ( X `  z ) ) )  =  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z )
) ) )
178176, 177oveq12d 6288 . . . . . . . . . 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 4452 . . . . . . . . 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 314 . . . . . . . 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 433 . . . . . . . 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 3174 . . . . . . 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 1226 . . . . . 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 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 )
) ) ) )
185111oveq1d 6285 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  =  ( ( ( T `  z )  /  L )  x.  ( F `  ( X `  z )
) ) )
186139recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( F `  ( X `  z )
)  e.  CC )
18738, 186, 58, 67div23d 10353 . . . . . . 7  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  =  ( ( ( T `  z
)  /  L )  x.  ( F `  ( X `  z ) ) ) )
188185, 187eqtr4d 2498 . . . . . 6  |-  ( ph  ->  ( ( 1  -  ( S  /  L
) )  x.  ( F `  ( X `  z ) ) )  =  ( ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  /  L
) )
189188oveq2d 6286 . . . . 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 4463 . . . 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 2498 . . . . . 6  |-  ( ph  ->  ( ( ( T `
 z )  x.  ( F `  ( X `  z )
) )  /  L
)  =  ( ( 1  -  ( S  /  L ) )  x.  ( F `  ( X `  z ) ) ) )
192191oveq2d 6286 . . . . 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 10476 . . . . . . . 8  |-  ( ph  ->  0  <  ( S  /  L ) )
195 lemul2 10391 . . . . . . . 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 1230 . . . . . . 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 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 ) ) )
198138, 150, 153, 197leadd1dd 10162 . . . . 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 4459 . . . 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 9728 . . 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 5852 . . 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 9611 . . . . 5  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  e.  CC )
203140recnd 9611 . . . . 5  |-  ( ph  ->  ( ( T `  z )  x.  ( F `  ( X `  z ) ) )  e.  CC )
204202, 203, 58, 67divdird 10354 . . . 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 3487 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( T `  x )  e.  RR )
20643ffvelrnda 6007 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X `  x )  e.  D
)  ->  ( F `  ( X `  x
) )  e.  RR )
20778, 206syldan 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  ( X `  x ) )  e.  RR )
208205, 207remulcld 9613 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  RR )
209208recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  CC )
21073, 209syldan 468 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  e.  CC )
21184fveq2d 5852 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  ( X `  x ) )  =  ( F `  ( X `  z )
) )
21283, 211oveq12d 6288 . . . . . . 7  |-  ( x  =  z  ->  (
( T `  x
)  x.  ( F `
 ( X `  x ) ) )  =  ( ( T `
 z )  x.  ( F `  ( X `  z )
) ) )
21369, 70, 72, 9, 210, 36, 47, 203, 212gsumunsn 17182 . . . . . 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 5901 . . . . . . . . . . 11  |-  ( ph  ->  F  =  ( y  e.  D  |->  ( F `
 y ) ) )
215 fveq2 5848 . . . . . . . . . . 11  |-  ( y  =  ( X `  x )  ->  ( F `  y )  =  ( F `  ( X `  x ) ) )
21678, 88, 214, 215fmptco 6040 . . . . . . . . . 10  |-  ( ph  ->  ( F  o.  X
)  =  ( x  e.  A  |->  ( F `
 ( X `  x ) ) ) )
2175, 74, 207, 87, 216offval2 6529 . . . . . . . . 9  |-  ( ph  ->  ( T  oF  x.  ( F  o.  X ) )  =  ( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
218217reseq1d 5261 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) )  =  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  ( B  u.  { z } ) ) )
2196resmptd 5313 . . . . . . . 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 2495 . . . . . . 7  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  ( B  u.  { z } ) )  =  ( x  e.  ( B  u.  { z } )  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) )
221220oveq2d 6286 . . . . . 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 5261 . . . . . . . . 9  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B )  =  ( ( x  e.  A  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) )  |`  B )
)
2237resmptd 5313 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  A  |->  ( ( T `
 x )  x.  ( F `  ( X `  x )
) ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( F `
 ( X `  x ) ) ) ) )
224222, 223eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( T  oF  x.  ( F  o.  X ) )  |`  B )  =  ( x  e.  B  |->  ( ( T `  x
)  x.  ( F `
 ( X `  x ) ) ) ) )
225224oveq2d 6286 . . . . . . 7  |-  ( ph  ->  (fld 
gsumg  ( ( T  oF  x.  ( F  o.  X ) )  |`  B ) )  =  (fld 
gsumg  ( x  e.  B  |->  ( ( T `  x )  x.  ( F `  ( X `  x ) ) ) ) ) )
226225oveq1d 6285 . . . . . 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 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 ) ) ) ) )
228227oveq1d 6285 . . . 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 10345 . . . . 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 6288 . . . 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 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 ) ) ) ) )
232200, 201, 2313brtr4d 4469 . 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 530 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459    C_ wss 3461   {csn 4016   class class class wbr 4439    |-> cmpt 4497    |` cres 4990    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   +oocpnf 9614    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   RR+crp 11221   [,)cico 11534   [,]cicc 11535    gsumg cgsu 14930  SubGrpcsubg 16394  CMndccmn 16997   Abelcabl 16998   Ringcrg 17393   DivRingcdr 17591  SubRingcsubrg 17620  ℂfldccnfld 18615  RRfldcrefld 18813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-rp 11222  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-mulg 16259  df-subg 16397  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593  df-subrg 17622  df-cnfld 18616  df-refld 18814
This theorem is referenced by:  jensen  23516
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