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Theorem wemaplem1 7263
Description: Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
wemapso.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
Assertion
Ref Expression
wemaplem1  |-  ( ( P  e.  V  /\  Q  e.  W )  ->  ( P T Q  <->  E. a  e.  A  ( ( P `  a ) S ( Q `  a )  /\  A. b  e.  A  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) ) )
Distinct variable groups:    a, b, x    T, a, b    w, a, y, z, b, x, A    P, a, b, w, x, y, z    Q, a, b, w, x, y, z    R, a, b, w, x, y, z    S, a, b, w, x, y, z
Allowed substitution hints:    T( x, y, z, w)    V( x, y, z, w, a, b)    W( x, y, z, w, a, b)

Proof of Theorem wemaplem1
StepHypRef Expression
1 fveq1 5526 . . . . . 6  |-  ( x  =  P  ->  (
x `  z )  =  ( P `  z ) )
2 fveq1 5526 . . . . . 6  |-  ( y  =  Q  ->  (
y `  z )  =  ( Q `  z ) )
31, 2breqan12d 4040 . . . . 5  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( x `  z ) S ( y `  z )  <-> 
( P `  z
) S ( Q `
 z ) ) )
4 fveq1 5526 . . . . . . . 8  |-  ( x  =  P  ->  (
x `  w )  =  ( P `  w ) )
5 fveq1 5526 . . . . . . . 8  |-  ( y  =  Q  ->  (
y `  w )  =  ( Q `  w ) )
64, 5eqeqan12d 2300 . . . . . . 7  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( x `  w )  =  ( y `  w )  <-> 
( P `  w
)  =  ( Q `
 w ) ) )
76imbi2d 307 . . . . . 6  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( w R z  ->  ( x `  w )  =  ( y `  w ) )  <->  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) ) )
87ralbidv 2565 . . . . 5  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) )  <->  A. w  e.  A  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) ) )
93, 8anbi12d 691 . . . 4  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( ( x `
 z ) S ( y `  z
)  /\  A. w  e.  A  ( w R z  ->  (
x `  w )  =  ( y `  w ) ) )  <-> 
( ( P `  z ) S ( Q `  z )  /\  A. w  e.  A  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) ) ) )
109rexbidv 2566 . . 3  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( E. z  e.  A  ( ( x `
 z ) S ( y `  z
)  /\  A. w  e.  A  ( w R z  ->  (
x `  w )  =  ( y `  w ) ) )  <->  E. z  e.  A  ( ( P `  z ) S ( Q `  z )  /\  A. w  e.  A  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) ) ) )
11 fveq2 5527 . . . . . 6  |-  ( z  =  a  ->  ( P `  z )  =  ( P `  a ) )
12 fveq2 5527 . . . . . 6  |-  ( z  =  a  ->  ( Q `  z )  =  ( Q `  a ) )
1311, 12breq12d 4038 . . . . 5  |-  ( z  =  a  ->  (
( P `  z
) S ( Q `
 z )  <->  ( P `  a ) S ( Q `  a ) ) )
14 breq2 4029 . . . . . . . 8  |-  ( z  =  a  ->  (
w R z  <->  w R
a ) )
1514imbi1d 308 . . . . . . 7  |-  ( z  =  a  ->  (
( w R z  ->  ( P `  w )  =  ( Q `  w ) )  <->  ( w R a  ->  ( P `  w )  =  ( Q `  w ) ) ) )
1615ralbidv 2565 . . . . . 6  |-  ( z  =  a  ->  ( A. w  e.  A  ( w R z  ->  ( P `  w )  =  ( Q `  w ) )  <->  A. w  e.  A  ( w R a  ->  ( P `  w )  =  ( Q `  w ) ) ) )
17 breq1 4028 . . . . . . . 8  |-  ( w  =  b  ->  (
w R a  <->  b R
a ) )
18 fveq2 5527 . . . . . . . . 9  |-  ( w  =  b  ->  ( P `  w )  =  ( P `  b ) )
19 fveq2 5527 . . . . . . . . 9  |-  ( w  =  b  ->  ( Q `  w )  =  ( Q `  b ) )
2018, 19eqeq12d 2299 . . . . . . . 8  |-  ( w  =  b  ->  (
( P `  w
)  =  ( Q `
 w )  <->  ( P `  b )  =  ( Q `  b ) ) )
2117, 20imbi12d 311 . . . . . . 7  |-  ( w  =  b  ->  (
( w R a  ->  ( P `  w )  =  ( Q `  w ) )  <->  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) )
2221cbvralv 2766 . . . . . 6  |-  ( A. w  e.  A  (
w R a  -> 
( P `  w
)  =  ( Q `
 w ) )  <->  A. b  e.  A  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) )
2316, 22syl6bb 252 . . . . 5  |-  ( z  =  a  ->  ( A. w  e.  A  ( w R z  ->  ( P `  w )  =  ( Q `  w ) )  <->  A. b  e.  A  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) )
2413, 23anbi12d 691 . . . 4  |-  ( z  =  a  ->  (
( ( P `  z ) S ( Q `  z )  /\  A. w  e.  A  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) )  <->  ( ( P `  a ) S ( Q `  a )  /\  A. b  e.  A  (
b R a  -> 
( P `  b
)  =  ( Q `
 b ) ) ) ) )
2524cbvrexv 2767 . . 3  |-  ( E. z  e.  A  ( ( P `  z
) S ( Q `
 z )  /\  A. w  e.  A  ( w R z  -> 
( P `  w
)  =  ( Q `
 w ) ) )  <->  E. a  e.  A  ( ( P `  a ) S ( Q `  a )  /\  A. b  e.  A  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) )
2610, 25syl6bb 252 . 2  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( E. z  e.  A  ( ( x `
 z ) S ( y `  z
)  /\  A. w  e.  A  ( w R z  ->  (
x `  w )  =  ( y `  w ) ) )  <->  E. a  e.  A  ( ( P `  a ) S ( Q `  a )  /\  A. b  e.  A  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) ) )
27 wemapso.t . 2  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
2826, 27brabga 4281 1  |-  ( ( P  e.  V  /\  Q  e.  W )  ->  ( P T Q  <->  E. a  e.  A  ( ( P `  a ) S ( Q `  a )  /\  A. b  e.  A  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   class class class wbr 4025   {copab 4078   ` cfv 5257
This theorem is referenced by:  wemaplem2  7264  wemaplem3  7265  wemappo  7266  wemapso2lem  7267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-iota 5221  df-fv 5265
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