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Theorem axcontlem1 25619
Description: Lemma for axcont 25631. 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 2449 . . . . 5  |-  ( x  =  y  ->  (
x  e.  D  <->  y  e.  D ) )
32adantr 452 . . . 4  |-  ( ( x  =  y  /\  t  =  s )  ->  ( x  e.  D  <->  y  e.  D ) )
4 eleq1 2449 . . . . . 6  |-  ( t  =  s  ->  (
t  e.  ( 0 [,)  +oo )  <->  s  e.  ( 0 [,)  +oo ) ) )
54adantl 453 . . . . 5  |-  ( ( x  =  y  /\  t  =  s )  ->  ( t  e.  ( 0 [,)  +oo )  <->  s  e.  ( 0 [,) 
+oo ) ) )
6 fveq1 5669 . . . . . . . 8  |-  ( x  =  y  ->  (
x `  i )  =  ( y `  i ) )
7 oveq2 6030 . . . . . . . . . 10  |-  ( t  =  s  ->  (
1  -  t )  =  ( 1  -  s ) )
87oveq1d 6037 . . . . . . . . 9  |-  ( t  =  s  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  i
) ) )
9 oveq1 6029 . . . . . . . . 9  |-  ( t  =  s  ->  (
t  x.  ( U `
 i ) )  =  ( s  x.  ( U `  i
) ) )
108, 9oveq12d 6040 . . . . . . . 8  |-  ( t  =  s  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  i ) )  +  ( s  x.  ( U `  i )
) ) )
116, 10eqeqan12d 2404 . . . . . . 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 2671 . . . . . 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 5670 . . . . . . . 8  |-  ( i  =  j  ->  (
y `  i )  =  ( y `  j ) )
14 fveq2 5670 . . . . . . . . . 10  |-  ( i  =  j  ->  ( Z `  i )  =  ( Z `  j ) )
1514oveq2d 6038 . . . . . . . . 9  |-  ( i  =  j  ->  (
( 1  -  s
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  j
) ) )
16 fveq2 5670 . . . . . . . . . 10  |-  ( i  =  j  ->  ( U `  i )  =  ( U `  j ) )
1716oveq2d 6038 . . . . . . . . 9  |-  ( i  =  j  ->  (
s  x.  ( U `
 i ) )  =  ( s  x.  ( U `  j
) ) )
1815, 17oveq12d 6040 . . . . . . . 8  |-  ( i  =  j  ->  (
( ( 1  -  s )  x.  ( Z `  i )
)  +  ( s  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j )
) ) )
1913, 18eqeq12d 2403 . . . . . . 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 2877 . . . . . 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 253 . . . . 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 692 . . . 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 692 . . 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 4220 . 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 2409 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {copab 4208   ` cfv 5396  (class class class)co 6022   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    +oocpnf 9052    - cmin 9225   [,)cico 10852   ...cfz 10977
This theorem is referenced by:  axcontlem6  25624  axcontlem11  25629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-iota 5360  df-fv 5404  df-ov 6025
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