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Theorem axcontlem1 24592
Description: Lemma for axcont 24604. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)
Hypothesis
Ref Expression
axcontlem1.1  |-  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
axcontlem1  |-  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 )
) ) ) ) }
Distinct variable groups:    D, s,
t, x, y    i,
j, s, t, x, y, N    U, i,
j, s, t, x, y    i, Z, j, s, t, x, y
Allowed substitution hints:    D( i, j)    F( x, y, t, i, j, s)

Proof of Theorem axcontlem1
StepHypRef Expression
1 axcontlem1.1 . 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 )
) ) ) ) }
2 eleq1 2343 . . . . 5  |-  ( x  =  y  ->  (
x  e.  D  <->  y  e.  D ) )
32adantr 451 . . . 4  |-  ( ( x  =  y  /\  t  =  s )  ->  ( x  e.  D  <->  y  e.  D ) )
4 eleq1 2343 . . . . . 6  |-  ( t  =  s  ->  (
t  e.  ( 0 [,)  +oo )  <->  s  e.  ( 0 [,)  +oo ) ) )
54adantl 452 . . . . 5  |-  ( ( x  =  y  /\  t  =  s )  ->  ( t  e.  ( 0 [,)  +oo )  <->  s  e.  ( 0 [,) 
+oo ) ) )
6 fveq1 5524 . . . . . . . 8  |-  ( x  =  y  ->  (
x `  i )  =  ( y `  i ) )
7 oveq2 5866 . . . . . . . . . 10  |-  ( t  =  s  ->  (
1  -  t )  =  ( 1  -  s ) )
87oveq1d 5873 . . . . . . . . 9  |-  ( t  =  s  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  i
) ) )
9 oveq1 5865 . . . . . . . . 9  |-  ( t  =  s  ->  (
t  x.  ( U `
 i ) )  =  ( s  x.  ( U `  i
) ) )
108, 9oveq12d 5876 . . . . . . . 8  |-  ( t  =  s  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  i ) )  +  ( s  x.  ( U `  i )
) ) )
116, 10eqeqan12d 2298 . . . . . . 7  |-  ( ( x  =  y  /\  t  =  s )  ->  ( ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  <-> 
( y `  i
)  =  ( ( ( 1  -  s
)  x.  ( Z `
 i ) )  +  ( s  x.  ( U `  i
) ) ) ) )
1211ralbidv 2563 . . . . . 6  |-  ( ( x  =  y  /\  t  =  s )  ->  ( A. i  e.  ( 1 ... N
) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  <->  A. i  e.  (
1 ... N ) ( y `  i )  =  ( ( ( 1  -  s )  x.  ( Z `  i ) )  +  ( s  x.  ( U `  i )
) ) ) )
13 fveq2 5525 . . . . . . . 8  |-  ( i  =  j  ->  (
y `  i )  =  ( y `  j ) )
14 fveq2 5525 . . . . . . . . . 10  |-  ( i  =  j  ->  ( Z `  i )  =  ( Z `  j ) )
1514oveq2d 5874 . . . . . . . . 9  |-  ( i  =  j  ->  (
( 1  -  s
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  j
) ) )
16 fveq2 5525 . . . . . . . . . 10  |-  ( i  =  j  ->  ( U `  i )  =  ( U `  j ) )
1716oveq2d 5874 . . . . . . . . 9  |-  ( i  =  j  ->  (
s  x.  ( U `
 i ) )  =  ( s  x.  ( U `  j
) ) )
1815, 17oveq12d 5876 . . . . . . . 8  |-  ( i  =  j  ->  (
( ( 1  -  s )  x.  ( Z `  i )
)  +  ( s  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j )
) ) )
1913, 18eqeq12d 2297 . . . . . . 7  |-  ( i  =  j  ->  (
( y `  i
)  =  ( ( ( 1  -  s
)  x.  ( Z `
 i ) )  +  ( s  x.  ( U `  i
) ) )  <->  ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j )
)  +  ( s  x.  ( U `  j ) ) ) ) )
2019cbvralv 2764 . . . . . 6  |-  ( A. i  e.  ( 1 ... N ) ( y `  i )  =  ( ( ( 1  -  s )  x.  ( Z `  i ) )  +  ( s  x.  ( U `  i )
) )  <->  A. j  e.  ( 1 ... N
) ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j )
)  +  ( s  x.  ( U `  j ) ) ) )
2112, 20syl6bb 252 . . . . 5  |-  ( ( x  =  y  /\  t  =  s )  ->  ( A. i  e.  ( 1 ... N
) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  <->  A. j  e.  (
1 ... N ) ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j )
) ) ) )
225, 21anbi12d 691 . . . 4  |-  ( ( x  =  y  /\  t  =  s )  ->  ( ( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N
) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) ) )  <->  ( s  e.  ( 0 [,)  +oo )  /\  A. j  e.  ( 1 ... N
) ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j )
)  +  ( s  x.  ( U `  j ) ) ) ) ) )
233, 22anbi12d 691 . . 3  |-  ( ( x  =  y  /\  t  =  s )  ->  ( ( x  e.  D  /\  ( t  e.  ( 0 [,) 
+oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i )
) ) ) )  <-> 
( y  e.  D  /\  ( s  e.  ( 0 [,)  +oo )  /\  A. j  e.  ( 1 ... N ) ( y `  j
)  =  ( ( ( 1  -  s
)  x.  ( Z `
 j ) )  +  ( s  x.  ( U `  j
) ) ) ) ) ) )
2423cbvopabv 4088 . 2  |-  { <. 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
) ) ) ) ) }  =  { <. 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 )
) ) ) ) }
251, 24eqtri 2303 1  |-  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 )
) ) ) ) }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {copab 4076   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    - cmin 9037   [,)cico 10658   ...cfz 10782
This theorem is referenced by:  axcontlem6  24597  axcontlem11  24602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861
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