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Theorem axcontlem1 24394
Description: Lemma for axcont 24406. 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 2529 . . . . 5  |-  ( x  =  y  ->  (
x  e.  D  <->  y  e.  D ) )
32adantr 465 . . . 4  |-  ( ( x  =  y  /\  t  =  s )  ->  ( x  e.  D  <->  y  e.  D ) )
4 eleq1 2529 . . . . . 6  |-  ( t  =  s  ->  (
t  e.  ( 0 [,) +oo )  <->  s  e.  ( 0 [,) +oo ) ) )
54adantl 466 . . . . 5  |-  ( ( x  =  y  /\  t  =  s )  ->  ( t  e.  ( 0 [,) +oo )  <->  s  e.  ( 0 [,) +oo ) ) )
6 fveq1 5871 . . . . . . . 8  |-  ( x  =  y  ->  (
x `  i )  =  ( y `  i ) )
7 oveq2 6304 . . . . . . . . . 10  |-  ( t  =  s  ->  (
1  -  t )  =  ( 1  -  s ) )
87oveq1d 6311 . . . . . . . . 9  |-  ( t  =  s  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  i
) ) )
9 oveq1 6303 . . . . . . . . 9  |-  ( t  =  s  ->  (
t  x.  ( U `
 i ) )  =  ( s  x.  ( U `  i
) ) )
108, 9oveq12d 6314 . . . . . . . 8  |-  ( t  =  s  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  i ) )  +  ( s  x.  ( U `  i )
) ) )
116, 10eqeqan12d 2480 . . . . . . 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 2896 . . . . . 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 5872 . . . . . . . 8  |-  ( i  =  j  ->  (
y `  i )  =  ( y `  j ) )
14 fveq2 5872 . . . . . . . . . 10  |-  ( i  =  j  ->  ( Z `  i )  =  ( Z `  j ) )
1514oveq2d 6312 . . . . . . . . 9  |-  ( i  =  j  ->  (
( 1  -  s
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  j
) ) )
16 fveq2 5872 . . . . . . . . . 10  |-  ( i  =  j  ->  ( U `  i )  =  ( U `  j ) )
1716oveq2d 6312 . . . . . . . . 9  |-  ( i  =  j  ->  (
s  x.  ( U `
 i ) )  =  ( s  x.  ( U `  j
) ) )
1815, 17oveq12d 6314 . . . . . . . 8  |-  ( i  =  j  ->  (
( ( 1  -  s )  x.  ( Z `  i )
)  +  ( s  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j )
) ) )
1913, 18eqeq12d 2479 . . . . . . 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 3084 . . . . . 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 261 . . . . 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 710 . . . 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 710 . . 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 4526 . 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 2486 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {copab 4514   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642    - cmin 9824   [,)cico 11556   ...cfz 11697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-iota 5557  df-fv 5602  df-ov 6299
This theorem is referenced by:  axcontlem6  24399  axcontlem11  24404
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