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Theorem gaorb 14761
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 5866 . . . . . 6  |-  ( x  =  A  ->  (
g  .(+)  x )  =  ( g  .(+)  A ) )
2 eqeq12 2295 . . . . . 6  |-  ( ( ( g  .(+)  x )  =  ( g  .(+)  A )  /\  y  =  B )  ->  (
( g  .(+)  x )  =  y  <->  ( g  .(+)  A )  =  B ) )
31, 2sylan 457 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( g  .(+)  x )  =  y  <->  ( g  .(+)  A )  =  B ) )
43rexbidv 2564 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. g  e.  X  ( g  .(+)  x )  =  y  <->  E. g  e.  X  ( g  .(+)  A )  =  B ) )
5 oveq1 5865 . . . . . 6  |-  ( g  =  h  ->  (
g  .(+)  A )  =  ( h  .(+)  A ) )
65eqeq1d 2291 . . . . 5  |-  ( g  =  h  ->  (
( g  .(+)  A )  =  B  <->  ( h  .(+) 
A )  =  B ) )
76cbvrexv 2765 . . . 4  |-  ( E. g  e.  X  ( g  .(+)  A )  =  B  <->  E. h  e.  X  ( h  .(+)  A )  =  B )
84, 7syl6bb 252 . . 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 2791 . . . . . . 7  |-  x  e. 
_V
11 vex 2791 . . . . . . 7  |-  y  e. 
_V
1210, 11prss 3769 . . . . . 6  |-  ( ( x  e.  Y  /\  y  e.  Y )  <->  { x ,  y } 
C_  Y )
1312anbi1i 676 . . . . 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 4083 . . . 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 2306 . . 3  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  Y  /\  y  e.  Y )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }
168, 15brab2ga 4763 . 2  |-  ( A  .~  B  <->  ( ( A  e.  Y  /\  B  e.  Y )  /\  E. h  e.  X  ( h  .(+)  A )  =  B ) )
17 df-3an 936 . 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 243 1  |-  ( A  .~  B  <->  ( A  e.  Y  /\  B  e.  Y  /\  E. h  e.  X  ( h  .(+) 
A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   {cpr 3641   class class class wbr 4023   {copab 4076  (class class class)co 5858
This theorem is referenced by:  gaorber  14762  orbsta  14767  sylow2alem1  14928  sylow2alem2  14929  sylow3lem3  14940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-iota 5219  df-fv 5263  df-ov 5861
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