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Theorem axcontlem1 23033
Description: Lemma for axcont 23045. 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 2493 . . . . 5  |-  ( x  =  y  ->  (
x  e.  D  <->  y  e.  D ) )
32adantr 462 . . . 4  |-  ( ( x  =  y  /\  t  =  s )  ->  ( x  e.  D  <->  y  e.  D ) )
4 eleq1 2493 . . . . . 6  |-  ( t  =  s  ->  (
t  e.  ( 0 [,) +oo )  <->  s  e.  ( 0 [,) +oo ) ) )
54adantl 463 . . . . 5  |-  ( ( x  =  y  /\  t  =  s )  ->  ( t  e.  ( 0 [,) +oo )  <->  s  e.  ( 0 [,) +oo ) ) )
6 fveq1 5678 . . . . . . . 8  |-  ( x  =  y  ->  (
x `  i )  =  ( y `  i ) )
7 oveq2 6088 . . . . . . . . . 10  |-  ( t  =  s  ->  (
1  -  t )  =  ( 1  -  s ) )
87oveq1d 6095 . . . . . . . . 9  |-  ( t  =  s  ->  (
( 1  -  t
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  i
) ) )
9 oveq1 6087 . . . . . . . . 9  |-  ( t  =  s  ->  (
t  x.  ( U `
 i ) )  =  ( s  x.  ( U `  i
) ) )
108, 9oveq12d 6098 . . . . . . . 8  |-  ( t  =  s  ->  (
( ( 1  -  t )  x.  ( Z `  i )
)  +  ( t  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  i ) )  +  ( s  x.  ( U `  i )
) ) )
116, 10eqeqan12d 2448 . . . . . . 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 2725 . . . . . 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 5679 . . . . . . . 8  |-  ( i  =  j  ->  (
y `  i )  =  ( y `  j ) )
14 fveq2 5679 . . . . . . . . . 10  |-  ( i  =  j  ->  ( Z `  i )  =  ( Z `  j ) )
1514oveq2d 6096 . . . . . . . . 9  |-  ( i  =  j  ->  (
( 1  -  s
)  x.  ( Z `
 i ) )  =  ( ( 1  -  s )  x.  ( Z `  j
) ) )
16 fveq2 5679 . . . . . . . . . 10  |-  ( i  =  j  ->  ( U `  i )  =  ( U `  j ) )
1716oveq2d 6096 . . . . . . . . 9  |-  ( i  =  j  ->  (
s  x.  ( U `
 i ) )  =  ( s  x.  ( U `  j
) ) )
1815, 17oveq12d 6098 . . . . . . . 8  |-  ( i  =  j  ->  (
( ( 1  -  s )  x.  ( Z `  i )
)  +  ( s  x.  ( U `  i ) ) )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j )
) ) )
1913, 18eqeq12d 2447 . . . . . . 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 2937 . . . . . 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 703 . . . 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 703 . . 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 4349 . 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 2453 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 1362    e. wcel 1755   A.wral 2705   {copab 4337   ` cfv 5406  (class class class)co 6080   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275   +oocpnf 9403    - cmin 9583   [,)cico 11290   ...cfz 11424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-iota 5369  df-fv 5414  df-ov 6083
This theorem is referenced by:  axcontlem6  23038  axcontlem11  23043
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