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Theorem axcontlem5 23214
Description: Lemma for axcont 23222. 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 23211 . . . . 5  |-  ( ( ( N  e.  NN  /\  Z  e.  ( EE
`  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F : D -1-1-onto-> ( 0 [,) +oo ) )
4 f1of 5641 . . . . 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 5843 . . 3  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  ( F `  P )  e.  ( 0 [,) +oo ) )
7 eleq1 2503 . . 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 5642 . . . . . . 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 5732 . . . . . 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 1008 . . . 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 2503 . . . . . . . 8  |-  ( x  =  P  ->  (
x  e.  D  <->  P  e.  D ) )
17 fveq1 5690 . . . . . . . . . . 11  |-  ( x  =  P  ->  (
x `  i )  =  ( P `  i ) )
1817eqeq1d 2451 . . . . . . . . . 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 2735 . . . . . . . . 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 2503 . . . . . . . . . 10  |-  ( t  =  T  ->  (
t  e.  ( 0 [,) +oo )  <->  T  e.  ( 0 [,) +oo ) ) )
23 oveq2 6099 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  (
1  -  t )  =  ( 1  -  T ) )
2423oveq1d 6106 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  T )  x.  ( Z `  i
) ) )
25 oveq1 6098 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
t  x.  ( U `
 i ) )  =  ( T  x.  ( U `  i ) ) )
2624, 25oveq12d 6109 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i )
) ) )
2726eqeq2d 2454 . . . . . . . . . . 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 2735 . . . . . . . . . 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 4608 . . . . . 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 875 . . . . 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 1006 . . . 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 1189 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719   <.cop 3883   class class class wbr 4292   {copab 4349    Fn wfn 5413   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287   +oocpnf 9415    - cmin 9595   NNcn 10322   [,)cico 11302   ...cfz 11437   EEcee 23134    Btwn cbtwn 23135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-z 10647  df-uz 10862  df-ico 11306  df-icc 11307  df-fz 11438  df-ee 23137  df-btwn 23138
This theorem is referenced by:  axcontlem6  23215
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