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Theorem gaorb 15086
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 6091 . . . . . 6  |-  ( x  =  A  ->  (
g  .(+)  x )  =  ( g  .(+)  A ) )
2 eqeq12 2450 . . . . . 6  |-  ( ( ( g  .(+)  x )  =  ( g  .(+)  A )  /\  y  =  B )  ->  (
( g  .(+)  x )  =  y  <->  ( g  .(+)  A )  =  B ) )
31, 2sylan 459 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( g  .(+)  x )  =  y  <->  ( g  .(+)  A )  =  B ) )
43rexbidv 2728 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. g  e.  X  ( g  .(+)  x )  =  y  <->  E. g  e.  X  ( g  .(+)  A )  =  B ) )
5 oveq1 6090 . . . . . 6  |-  ( g  =  h  ->  (
g  .(+)  A )  =  ( h  .(+)  A ) )
65eqeq1d 2446 . . . . 5  |-  ( g  =  h  ->  (
( g  .(+)  A )  =  B  <->  ( h  .(+) 
A )  =  B ) )
76cbvrexv 2935 . . . 4  |-  ( E. g  e.  X  ( g  .(+)  A )  =  B  <->  E. h  e.  X  ( h  .(+)  A )  =  B )
84, 7syl6bb 254 . . 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 2961 . . . . . . 7  |-  x  e. 
_V
11 vex 2961 . . . . . . 7  |-  y  e. 
_V
1210, 11prss 3954 . . . . . 6  |-  ( ( x  e.  Y  /\  y  e.  Y )  <->  { x ,  y } 
C_  Y )
1312anbi1i 678 . . . . 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 4274 . . . 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 2461 . . 3  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  Y  /\  y  e.  Y )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }
168, 15brab2ga 4953 . 2  |-  ( A  .~  B  <->  ( ( A  e.  Y  /\  B  e.  Y )  /\  E. h  e.  X  ( h  .(+)  A )  =  B ) )
17 df-3an 939 . 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 245 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   {cpr 3817   class class class wbr 4214   {copab 4267  (class class class)co 6083
This theorem is referenced by:  gaorber  15087  orbsta  15092  sylow2alem1  15253  sylow2alem2  15254  sylow3lem3  15265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-iota 5420  df-fv 5464  df-ov 6086
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