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Theorem gaorb 16323
Description: The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.)
Hypothesis
Ref Expression
gaorb.1  |-  .~  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  Y  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
Assertion
Ref Expression
gaorb  |-  ( A  .~  B  <->  ( A  e.  Y  /\  B  e.  Y  /\  E. h  e.  X  ( h  .(+) 
A )  =  B ) )
Distinct variable groups:    g, h, x, y, A    B, g, h, x, y    .~ , h    .(+) ,
g, h, x, y   
g, X, h, x, y    h, Y, x, y
Allowed substitution hints:    .~ ( x, y, g)    Y( g)

Proof of Theorem gaorb
StepHypRef Expression
1 oveq2 6289 . . . . . 6  |-  ( x  =  A  ->  (
g  .(+)  x )  =  ( g  .(+)  A ) )
2 eqeq12 2462 . . . . . 6  |-  ( ( ( g  .(+)  x )  =  ( g  .(+)  A )  /\  y  =  B )  ->  (
( g  .(+)  x )  =  y  <->  ( g  .(+)  A )  =  B ) )
31, 2sylan 471 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( g  .(+)  x )  =  y  <->  ( g  .(+)  A )  =  B ) )
43rexbidv 2954 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. g  e.  X  ( g  .(+)  x )  =  y  <->  E. g  e.  X  ( g  .(+)  A )  =  B ) )
5 oveq1 6288 . . . . . 6  |-  ( g  =  h  ->  (
g  .(+)  A )  =  ( h  .(+)  A ) )
65eqeq1d 2445 . . . . 5  |-  ( g  =  h  ->  (
( g  .(+)  A )  =  B  <->  ( h  .(+) 
A )  =  B ) )
76cbvrexv 3071 . . . 4  |-  ( E. g  e.  X  ( g  .(+)  A )  =  B  <->  E. h  e.  X  ( h  .(+)  A )  =  B )
84, 7syl6bb 261 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. g  e.  X  ( g  .(+)  x )  =  y  <->  E. h  e.  X  ( h  .(+) 
A )  =  B ) )
9 gaorb.1 . . . 4  |-  .~  =  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  Y  /\  E. g  e.  X  (
g  .(+)  x )  =  y ) }
10 vex 3098 . . . . . . 7  |-  x  e. 
_V
11 vex 3098 . . . . . . 7  |-  y  e. 
_V
1210, 11prss 4169 . . . . . 6  |-  ( ( x  e.  Y  /\  y  e.  Y )  <->  { x ,  y } 
C_  Y )
1312anbi1i 695 . . . . 5  |-  ( ( ( x  e.  Y  /\  y  e.  Y
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y )  <->  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) )
1413opabbii 4501 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  Y  /\  y  e.  Y
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }
159, 14eqtr4i 2475 . . 3  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  Y  /\  y  e.  Y )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }
168, 15brab2ga 5065 . 2  |-  ( A  .~  B  <->  ( ( A  e.  Y  /\  B  e.  Y )  /\  E. h  e.  X  ( h  .(+)  A )  =  B ) )
17 df-3an 976 . 2  |-  ( ( A  e.  Y  /\  B  e.  Y  /\  E. h  e.  X  ( h  .(+)  A )  =  B )  <->  ( ( A  e.  Y  /\  B  e.  Y )  /\  E. h  e.  X  ( h  .(+)  A )  =  B ) )
1816, 17bitr4i 252 1  |-  ( A  .~  B  <->  ( A  e.  Y  /\  B  e.  Y  /\  E. h  e.  X  ( h  .(+) 
A )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E.wrex 2794    C_ wss 3461   {cpr 4016   class class class wbr 4437   {copab 4494  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-iota 5541  df-fv 5586  df-ov 6284
This theorem is referenced by:  gaorber  16324  orbsta  16329  sylow2alem1  16615  sylow2alem2  16616  sylow3lem3  16627
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