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Theorem axcontlem6 25910
Description: Lemma for axcont 25917. State the defining properties of the value of  F (Contributed by Scott Fenton, 19-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
axcontlem6  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  ( F `
 P ) )  x.  ( Z `  i ) )  +  ( ( F `  P )  x.  ( U `  i )
) ) ) )
Distinct variable groups:    t, D, x    i, p, t, x, N    P, i, t, x    U, i, p, t, x   
i, Z, p, t, x    i, F
Allowed substitution hints:    D( i, p)    P( p)    F( x, t, p)

Proof of Theorem axcontlem6
Dummy variables  s 
y  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( F `
 P )  =  ( F `  P
)
2 axcontlem5.1 . . . 4  |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }
3 axcontlem5.2 . . . . 5  |-  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 )
) ) ) ) }
43axcontlem1 25905 . . . 4  |-  F  =  { <. y ,  s
>.  |  ( y  e.  D  /\  (
s  e.  ( 0 [,)  +oo )  /\  A. j  e.  ( 1 ... N ) ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j )
) ) ) ) }
52, 4axcontlem5 25909 . . 3  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  =  ( F `
 P )  <->  ( ( F `  P )  e.  ( 0 [,)  +oo )  /\  A. j  e.  ( 1 ... N
) ( P `  j )  =  ( ( ( 1  -  ( F `  P
) )  x.  ( Z `  j )
)  +  ( ( F `  P )  x.  ( U `  j ) ) ) ) ) )
61, 5mpbii 204 . 2  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  e.  ( 0 [,)  +oo )  /\  A. j  e.  ( 1 ... N ) ( P `  j )  =  ( ( ( 1  -  ( F `
 P ) )  x.  ( Z `  j ) )  +  ( ( F `  P )  x.  ( U `  j )
) ) ) )
7 fveq2 5730 . . . . 5  |-  ( j  =  i  ->  ( P `  j )  =  ( P `  i ) )
8 fveq2 5730 . . . . . . 7  |-  ( j  =  i  ->  ( Z `  j )  =  ( Z `  i ) )
98oveq2d 6099 . . . . . 6  |-  ( j  =  i  ->  (
( 1  -  ( F `  P )
)  x.  ( Z `
 j ) )  =  ( ( 1  -  ( F `  P ) )  x.  ( Z `  i
) ) )
10 fveq2 5730 . . . . . . 7  |-  ( j  =  i  ->  ( U `  j )  =  ( U `  i ) )
1110oveq2d 6099 . . . . . 6  |-  ( j  =  i  ->  (
( F `  P
)  x.  ( U `
 j ) )  =  ( ( F `
 P )  x.  ( U `  i
) ) )
129, 11oveq12d 6101 . . . . 5  |-  ( j  =  i  ->  (
( ( 1  -  ( F `  P
) )  x.  ( Z `  j )
)  +  ( ( F `  P )  x.  ( U `  j ) ) )  =  ( ( ( 1  -  ( F `
 P ) )  x.  ( Z `  i ) )  +  ( ( F `  P )  x.  ( U `  i )
) ) )
137, 12eqeq12d 2452 . . . 4  |-  ( j  =  i  ->  (
( P `  j
)  =  ( ( ( 1  -  ( F `  P )
)  x.  ( Z `
 j ) )  +  ( ( F `
 P )  x.  ( U `  j
) ) )  <->  ( P `  i )  =  ( ( ( 1  -  ( F `  P
) )  x.  ( Z `  i )
)  +  ( ( F `  P )  x.  ( U `  i ) ) ) ) )
1413cbvralv 2934 . . 3  |-  ( A. j  e.  ( 1 ... N ) ( P `  j )  =  ( ( ( 1  -  ( F `
 P ) )  x.  ( Z `  j ) )  +  ( ( F `  P )  x.  ( U `  j )
) )  <->  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  ( F `  P
) )  x.  ( Z `  i )
)  +  ( ( F `  P )  x.  ( U `  i ) ) ) )
1514anbi2i 677 . 2  |-  ( ( ( F `  P
)  e.  ( 0 [,)  +oo )  /\  A. j  e.  ( 1 ... N ) ( P `  j )  =  ( ( ( 1  -  ( F `
 P ) )  x.  ( Z `  j ) )  +  ( ( F `  P )  x.  ( U `  j )
) ) )  <->  ( ( F `  P )  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( P `  i )  =  ( ( ( 1  -  ( F `  P
) )  x.  ( Z `  i )
)  +  ( ( F `  P )  x.  ( U `  i ) ) ) ) )
166, 15sylib 190 1  |-  ( ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N
)  /\  U  e.  ( EE `  N ) )  /\  Z  =/= 
U )  /\  P  e.  D )  ->  (
( F `  P
)  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  ( ( ( 1  -  ( F `
 P ) )  x.  ( Z `  i ) )  +  ( ( F `  P )  x.  ( U `  i )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   <.cop 3819   class class class wbr 4214   {copab 4267   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    +oocpnf 9119    - cmin 9293   NNcn 10002   [,)cico 10920   ...cfz 11045   EEcee 25829    Btwn cbtwn 25830
This theorem is referenced by:  axcontlem7  25911  axcontlem8  25912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-z 10285  df-uz 10491  df-ico 10924  df-icc 10925  df-fz 11046  df-ee 25832  df-btwn 25833
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