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Theorem axcontlem5 25907
Description: Lemma for axcont 25915. Compute the value of  F. (Contributed by Scott Fenton, 18-Jun-2013.)
Hypotheses
Ref Expression
axcontlem5.1  |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }
axcontlem5.2  |-  F  =  { <. x ,  t
>.  |  ( x  e.  D  /\  (
t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i )
) ) ) ) }
Assertion
Ref Expression
axcontlem5  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) ) )
Distinct variable groups:    t, D, x    i, p, t, x, N    P, i, t, x   
x, T, i, t    U, i, p, t, x   
i, Z, p, t, x
Allowed substitution hints:    D( i, p)    P( p)    T( p)    F( x, t, i, p)

Proof of Theorem axcontlem5
StepHypRef Expression
1 axcontlem5.1 . . . . . 6  |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }
2 axcontlem5.2 . . . . . 6  |-  F  =  { <. x ,  t
>.  |  ( x  e.  D  /\  (
t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i )
) ) ) ) }
31, 2axcontlem2 25904 . . . . 5  |-  ( ( ( N  e.  NN  /\  Z  e.  ( EE
`  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F : D -1-1-onto-> ( 0 [,)  +oo ) )
4 f1of 5674 . . . . 5  |-  ( F : D -1-1-onto-> ( 0 [,)  +oo )  ->  F : D --> ( 0 [,)  +oo ) )
53, 4syl 16 . . . 4  |-  ( ( ( N  e.  NN  /\  Z  e.  ( EE
`  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F : D --> ( 0 [,)  +oo ) )
65ffvelrnda 5870 . . 3  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  ( F `  P )  e.  ( 0 [,)  +oo ) )
7 eleq1 2496 . . 3  |-  ( ( F `  P )  =  T  ->  (
( F `  P
)  e.  ( 0 [,)  +oo )  <->  T  e.  ( 0 [,)  +oo ) ) )
86, 7syl5ibcom 212 . 2  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  =  T  ->  T  e.  ( 0 [,)  +oo ) ) )
9 simpl 444 . . 3  |-  ( ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) )  ->  T  e.  ( 0 [,)  +oo ) )
109a1i 11 . 2  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  T
)  x.  ( Z `
 i ) )  +  ( T  x.  ( U `  i ) ) ) )  ->  T  e.  ( 0 [,)  +oo ) ) )
11 f1ofn 5675 . . . . . . 7  |-  ( F : D -1-1-onto-> ( 0 [,)  +oo )  ->  F  Fn  D
)
123, 11syl 16 . . . . . 6  |-  ( ( ( N  e.  NN  /\  Z  e.  ( EE
`  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F  Fn  D )
13 fnbrfvb 5767 . . . . . 6  |-  ( ( F  Fn  D  /\  P  e.  D )  ->  ( ( F `  P )  =  T  <-> 
P F T ) )
1412, 13sylan 458 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  =  T  <->  P F T ) )
15143adant3 977 . . . 4  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  ->  (
( F `  P
)  =  T  <->  P F T ) )
16 eleq1 2496 . . . . . . . 8  |-  ( x  =  P  ->  (
x  e.  D  <->  P  e.  D ) )
17 fveq1 5727 . . . . . . . . . . 11  |-  ( x  =  P  ->  (
x `  i )  =  ( P `  i ) )
1817eqeq1d 2444 . . . . . . . . . 10  |-  ( x  =  P  ->  (
( x `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) )  <->  ( P `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) ) ) )
1918ralbidv 2725 . . . . . . . . 9  |-  ( x  =  P  ->  ( A. i  e.  (
1 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i )
) )  <->  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) ) ) )
2019anbi2d 685 . . . . . . . 8  |-  ( x  =  P  ->  (
( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) ) )  <-> 
( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) ) ) ) )
2116, 20anbi12d 692 . . . . . . 7  |-  ( x  =  P  ->  (
( x  e.  D  /\  ( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) ) ) )  <->  ( P  e.  D  /\  ( t  e.  ( 0 [,) 
+oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i )
) ) ) ) ) )
22 eleq1 2496 . . . . . . . . . 10  |-  ( t  =  T  ->  (
t  e.  ( 0 [,)  +oo )  <->  T  e.  ( 0 [,)  +oo ) ) )
23 oveq2 6089 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  (
1  -  t )  =  ( 1  -  T ) )
2423oveq1d 6096 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  T )  x.  ( Z `  i
) ) )
25 oveq1 6088 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
t  x.  ( U `
 i ) )  =  ( T  x.  ( U `  i ) ) )
2624, 25oveq12d 6099 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) )
2726eqeq2d 2447 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
( P `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) )  <->  ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) )
2827ralbidv 2725 . . . . . . . . . 10  |-  ( t  =  T  ->  ( A. i  e.  (
1 ... N ) ( P `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i )
) )  <->  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) )
2922, 28anbi12d 692 . . . . . . . . 9  |-  ( t  =  T  ->  (
( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) ) )  <-> 
( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  T
)  x.  ( Z `
 i ) )  +  ( T  x.  ( U `  i ) ) ) ) ) )
3029anbi2d 685 . . . . . . . 8  |-  ( t  =  T  ->  (
( P  e.  D  /\  ( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) ) ) )  <->  ( P  e.  D  /\  ( T  e.  ( 0 [,) 
+oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) ) ) )
31 anass 631 . . . . . . . . . . 11  |-  ( ( ( P  e.  D  /\  T  e.  (
0 [,)  +oo ) )  /\  T  e.  ( 0 [,)  +oo )
)  <->  ( P  e.  D  /\  ( T  e.  ( 0 [,) 
+oo )  /\  T  e.  ( 0 [,)  +oo ) ) ) )
32 anidm 626 . . . . . . . . . . . 12  |-  ( ( T  e.  ( 0 [,)  +oo )  /\  T  e.  ( 0 [,)  +oo ) )  <->  T  e.  ( 0 [,)  +oo ) )
3332anbi2i 676 . . . . . . . . . . 11  |-  ( ( P  e.  D  /\  ( T  e.  (
0 [,)  +oo )  /\  T  e.  ( 0 [,)  +oo ) ) )  <-> 
( P  e.  D  /\  T  e.  (
0 [,)  +oo ) ) )
3431, 33bitr2i 242 . . . . . . . . . 10  |-  ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  <->  ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  T  e.  ( 0 [,)  +oo ) ) )
3534anbi1i 677 . . . . . . . . 9  |-  ( ( ( P  e.  D  /\  T  e.  (
0 [,)  +oo ) )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) )  <->  ( ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  T  e.  ( 0 [,)  +oo ) )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) )
36 anass 631 . . . . . . . . 9  |-  ( ( ( P  e.  D  /\  T  e.  (
0 [,)  +oo ) )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) )  <->  ( P  e.  D  /\  ( T  e.  ( 0 [,) 
+oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) ) )
37 anass 631 . . . . . . . . 9  |-  ( ( ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  T  e.  ( 0 [,)  +oo ) )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) )  <->  ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  ( T  e.  (
0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) ) )
3835, 36, 373bitr3i 267 . . . . . . . 8  |-  ( ( P  e.  D  /\  ( T  e.  (
0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) )  <-> 
( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) ) )
3930, 38syl6bb 253 . . . . . . 7  |-  ( t  =  T  ->  (
( P  e.  D  /\  ( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  t
)  x.  ( Z `
 i ) )  +  ( t  x.  ( U `  i
) ) ) ) )  <->  ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) ) ) )
4021, 39, 2brabg 4474 . . . . . 6  |-  ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  -> 
( P F T  <-> 
( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  /\  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) ) ) ) )
4140bianabs 851 . . . . 5  |-  ( ( P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  -> 
( P F T  <-> 
( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i
)  =  ( ( ( 1  -  T
)  x.  ( Z `
 i ) )  +  ( T  x.  ( U `  i ) ) ) ) ) )
42413adant1 975 . . . 4  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  ->  ( P F T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) ) )
4315, 42bitrd 245 . . 3  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D  /\  T  e.  ( 0 [,)  +oo ) )  ->  (
( F `  P
)  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) ) )
44433expia 1155 . 2  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  ( T  e.  ( 0 [,)  +oo )  ->  (
( F `  P
)  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) ) ) )
458, 10, 44pm5.21ndd 344 1  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  T )  x.  ( Z `  i )
)  +  ( T  x.  ( U `  i ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709   <.cop 3817   class class class wbr 4212   {copab 4265    Fn wfn 5449   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    +oocpnf 9117    - cmin 9291   NNcn 10000   [,)cico 10918   ...cfz 11043   EEcee 25827    Btwn cbtwn 25828
This theorem is referenced by:  axcontlem6  25908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-z 10283  df-uz 10489  df-ico 10922  df-icc 10923  df-fz 11044  df-ee 25830  df-btwn 25831
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