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Theorem wemaplem1 7507
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 5719 . . . . . 6  |-  ( x  =  P  ->  (
x `  z )  =  ( P `  z ) )
2 fveq1 5719 . . . . . 6  |-  ( y  =  Q  ->  (
y `  z )  =  ( Q `  z ) )
31, 2breqan12d 4219 . . . . 5  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( x `  z ) S ( y `  z )  <-> 
( P `  z
) S ( Q `
 z ) ) )
4 fveq1 5719 . . . . . . . 8  |-  ( x  =  P  ->  (
x `  w )  =  ( P `  w ) )
5 fveq1 5719 . . . . . . . 8  |-  ( y  =  Q  ->  (
y `  w )  =  ( Q `  w ) )
64, 5eqeqan12d 2450 . . . . . . 7  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( x `  w )  =  ( y `  w )  <-> 
( P `  w
)  =  ( Q `
 w ) ) )
76imbi2d 308 . . . . . 6  |-  ( ( x  =  P  /\  y  =  Q )  ->  ( ( w R z  ->  ( x `  w )  =  ( y `  w ) )  <->  ( w R z  ->  ( P `  w )  =  ( Q `  w ) ) ) )
87ralbidv 2717 . . . . 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 692 . . . 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 2718 . . 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 5720 . . . . . 6  |-  ( z  =  a  ->  ( P `  z )  =  ( P `  a ) )
12 fveq2 5720 . . . . . 6  |-  ( z  =  a  ->  ( Q `  z )  =  ( Q `  a ) )
1311, 12breq12d 4217 . . . . 5  |-  ( z  =  a  ->  (
( P `  z
) S ( Q `
 z )  <->  ( P `  a ) S ( Q `  a ) ) )
14 breq2 4208 . . . . . . . 8  |-  ( z  =  a  ->  (
w R z  <->  w R
a ) )
1514imbi1d 309 . . . . . . 7  |-  ( z  =  a  ->  (
( w R z  ->  ( P `  w )  =  ( Q `  w ) )  <->  ( w R a  ->  ( P `  w )  =  ( Q `  w ) ) ) )
1615ralbidv 2717 . . . . . 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 4207 . . . . . . . 8  |-  ( w  =  b  ->  (
w R a  <->  b R
a ) )
18 fveq2 5720 . . . . . . . . 9  |-  ( w  =  b  ->  ( P `  w )  =  ( P `  b ) )
19 fveq2 5720 . . . . . . . . 9  |-  ( w  =  b  ->  ( Q `  w )  =  ( Q `  b ) )
2018, 19eqeq12d 2449 . . . . . . . 8  |-  ( w  =  b  ->  (
( P `  w
)  =  ( Q `
 w )  <->  ( P `  b )  =  ( Q `  b ) ) )
2117, 20imbi12d 312 . . . . . . 7  |-  ( w  =  b  ->  (
( w R a  ->  ( P `  w )  =  ( Q `  w ) )  <->  ( b R a  ->  ( P `  b )  =  ( Q `  b ) ) ) )
2221cbvralv 2924 . . . . . 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 253 . . . . 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 692 . . . 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 2925 . . 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 253 . 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 4461 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   {copab 4257   ` cfv 5446
This theorem is referenced by:  wemaplem2  7508  wemaplem3  7509  wemappo  7510  wemapso2lem  7511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-iota 5410  df-fv 5454
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