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Theorem gaorb 14777
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 5882 . . . . . 6  |-  ( x  =  A  ->  (
g  .(+)  x )  =  ( g  .(+)  A ) )
2 eqeq12 2308 . . . . . 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 2577 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. g  e.  X  ( g  .(+)  x )  =  y  <->  E. g  e.  X  ( g  .(+)  A )  =  B ) )
5 oveq1 5881 . . . . . 6  |-  ( g  =  h  ->  (
g  .(+)  A )  =  ( h  .(+)  A ) )
65eqeq1d 2304 . . . . 5  |-  ( g  =  h  ->  (
( g  .(+)  A )  =  B  <->  ( h  .(+) 
A )  =  B ) )
76cbvrexv 2778 . . . 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 2804 . . . . . . 7  |-  x  e. 
_V
11 vex 2804 . . . . . . 7  |-  y  e. 
_V
1210, 11prss 3785 . . . . . 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 4099 . . . 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 2319 . . 3  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  Y  /\  y  e.  Y )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }
168, 15brab2ga 4779 . 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   {cpr 3654   class class class wbr 4039   {copab 4092  (class class class)co 5874
This theorem is referenced by:  gaorber  14778  orbsta  14783  sylow2alem1  14944  sylow2alem2  14945  sylow3lem3  14956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-iota 5235  df-fv 5279  df-ov 5877
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