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Theorem axcontlem5 24103
Description: Lemma for axcont 24111. 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 24100 . . . . 5  |-  ( ( ( N  e.  NN  /\  Z  e.  ( EE
`  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F : D -1-1-onto-> ( 0 [,) +oo ) )
4 f1of 5822 . . . . 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 6032 . . 3  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  ( F `  P )  e.  ( 0 [,) +oo ) )
7 eleq1 2539 . . 3  |-  ( ( F `  P )  =  T  ->  (
( F `  P
)  e.  ( 0 [,) +oo )  <->  T  e.  ( 0 [,) +oo ) ) )
86, 7syl5ibcom 220 . 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 457 . . 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 5823 . . . . . . 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 5914 . . . . . 6  |-  ( ( F  Fn  D  /\  P  e.  D )  ->  ( ( F `  P )  =  T  <-> 
P F T ) )
1412, 13sylan 471 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  =  T  <->  P F T ) )
15143adant3 1016 . . . 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 2539 . . . . . . . 8  |-  ( x  =  P  ->  (
x  e.  D  <->  P  e.  D ) )
17 fveq1 5871 . . . . . . . . . . 11  |-  ( x  =  P  ->  (
x `  i )  =  ( P `  i ) )
1817eqeq1d 2469 . . . . . . . . . 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 2906 . . . . . . . . 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 703 . . . . . . . 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 710 . . . . . . 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 2539 . . . . . . . . . 10  |-  ( t  =  T  ->  (
t  e.  ( 0 [,) +oo )  <->  T  e.  ( 0 [,) +oo ) ) )
23 oveq2 6303 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  (
1  -  t )  =  ( 1  -  T ) )
2423oveq1d 6310 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  T )  x.  ( Z `  i
) ) )
25 oveq1 6302 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
t  x.  ( U `
 i ) )  =  ( T  x.  ( U `  i ) ) )
2624, 25oveq12d 6313 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) )
2726eqeq2d 2481 . . . . . . . . . . 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 2906 . . . . . . . . . 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 710 . . . . . . . . 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 703 . . . . . . . 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 649 . . . . . . . . . . 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 644 . . . . . . . . . . . 12  |-  ( ( T  e.  ( 0 [,) +oo )  /\  T  e.  ( 0 [,) +oo ) )  <-> 
T  e.  ( 0 [,) +oo ) )
3332anbi2i 694 . . . . . . . . . . 11  |-  ( ( P  e.  D  /\  ( T  e.  (
0 [,) +oo )  /\  T  e.  (
0 [,) +oo )
) )  <->  ( P  e.  D  /\  T  e.  ( 0 [,) +oo ) ) )
3431, 33bitr2i 250 . . . . . . . . . 10  |-  ( ( P  e.  D  /\  T  e.  ( 0 [,) +oo ) )  <-> 
( ( P  e.  D  /\  T  e.  ( 0 [,) +oo ) )  /\  T  e.  ( 0 [,) +oo ) ) )
3534anbi1i 695 . . . . . . . . 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 649 . . . . . . . . 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 649 . . . . . . . . 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 275 . . . . . . . 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 261 . . . . . . 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 4772 . . . . . 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 878 . . . . 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 1014 . . . 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 253 . . 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 1198 . 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 354 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821   <.cop 4039   class class class wbr 4453   {copab 4510    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509   +oocpnf 9637    - cmin 9817   NNcn 10548   [,)cico 11543   ...cfz 11684   EEcee 24023    Btwn cbtwn 24024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-z 10877  df-uz 11095  df-ico 11547  df-icc 11548  df-fz 11685  df-ee 24026  df-btwn 24027
This theorem is referenced by:  axcontlem6  24104
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