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Theorem wemaplem1 7758
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 5688 . . . . . 6  |-  ( x  =  P  ->  (
x `  z )  =  ( P `  z ) )
2 fveq1 5688 . . . . . 6  |-  ( y  =  Q  ->  (
y `  z )  =  ( Q `  z ) )
31, 2breqan12d 4305 . . . . 5  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( x `  z ) S ( y `  z )  <-> 
( P `  z
) S ( Q `
 z ) ) )
4 fveq1 5688 . . . . . . . 8  |-  ( x  =  P  ->  (
x `  w )  =  ( P `  w ) )
5 fveq1 5688 . . . . . . . 8  |-  ( y  =  Q  ->  (
y `  w )  =  ( Q `  w ) )
64, 5eqeqan12d 2456 . . . . . . 7  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( x `  w )  =  ( y `  w )  <-> 
( P `  w
)  =  ( Q `
 w ) ) )
76imbi2d 316 . . . . . 6  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( w R z  ->  ( x `  w )  =  ( y `  w ) )  <->  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) ) )
87ralbidv 2733 . . . . 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 710 . . . 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 2734 . . 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 5689 . . . . . 6  |-  ( z  =  a  ->  ( P `  z )  =  ( P `  a ) )
12 fveq2 5689 . . . . . 6  |-  ( z  =  a  ->  ( Q `  z )  =  ( Q `  a ) )
1311, 12breq12d 4303 . . . . 5  |-  ( z  =  a  ->  (
( P `  z
) S ( Q `
 z )  <->  ( P `  a ) S ( Q `  a ) ) )
14 breq2 4294 . . . . . . . 8  |-  ( z  =  a  ->  (
w R z  <->  w R
a ) )
1514imbi1d 317 . . . . . . 7  |-  ( z  =  a  ->  (
( w R z  ->  ( P `  w )  =  ( Q `  w ) )  <->  ( w R a  ->  ( P `  w )  =  ( Q `  w ) ) ) )
1615ralbidv 2733 . . . . . 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 4293 . . . . . . . 8  |-  ( w  =  b  ->  (
w R a  <->  b R
a ) )
18 fveq2 5689 . . . . . . . . 9  |-  ( w  =  b  ->  ( P `  w )  =  ( P `  b ) )
19 fveq2 5689 . . . . . . . . 9  |-  ( w  =  b  ->  ( Q `  w )  =  ( Q `  b ) )
2018, 19eqeq12d 2455 . . . . . . . 8  |-  ( w  =  b  ->  (
( P `  w
)  =  ( Q `
 w )  <->  ( P `  b )  =  ( Q `  b ) ) )
2117, 20imbi12d 320 . . . . . . 7  |-  ( w  =  b  ->  (
( w R a  ->  ( P `  w )  =  ( Q `  w ) )  <->  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) )
2221cbvralv 2945 . . . . . 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 261 . . . . 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 710 . . . 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 2946 . . 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 261 . 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 4601 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   class class class wbr 4290   {copab 4347   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-iota 5379  df-fv 5424
This theorem is referenced by:  wemaplem2  7759  wemaplem3  7760  wemappo  7761  wemapsolem  7762
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