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Theorem prsubrtr 25502
Description: The product of a subset  B of  X by an element of  X is the image of  B by a right translation. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
trfun.2  |-  F  =  ( x  e.  X  |->  ( x G A ) )
trinv.1  |-  X  =  ran  G
prsubrtr.1  |-  H  =  ( cset `  G
)
Assertion
Ref Expression
prsubrtr  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( B H { A } )  =  ( F " B ) )
Distinct variable groups:    x, A    x, B    x, G    x, X
Allowed substitution hints:    F( x)    H( x)

Proof of Theorem prsubrtr
Dummy variables  a 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5882 . . . . . . 7  |-  ( y  =  A  ->  (
x G y )  =  ( x G A ) )
21eqeq2d 2307 . . . . . 6  |-  ( y  =  A  ->  (
a  =  ( x G y )  <->  a  =  ( x G A ) ) )
32rexsng 3686 . . . . 5  |-  ( A  e.  X  ->  ( E. y  e.  { A } a  =  ( x G y )  <-> 
a  =  ( x G A ) ) )
43rexbidv 2577 . . . 4  |-  ( A  e.  X  ->  ( E. x  e.  B  E. y  e.  { A } a  =  ( x G y )  <->  E. x  e.  B  a  =  ( x G A ) ) )
543ad2ant2 977 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( E. x  e.  B  E. y  e. 
{ A } a  =  ( x G y )  <->  E. x  e.  B  a  =  ( x G A ) ) )
6 simp1 955 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  G  e.  GrpOp )
7 trinv.1 . . . . . . . . 9  |-  X  =  ran  G
8 grporndm 20893 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
97, 8syl5eq 2340 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  =  dom  dom  G )
109pweqd 3643 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ~P X  =  ~P dom  dom  G
)
1110eleq2d 2363 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( B  e.  ~P X  <->  B  e.  ~P dom  dom  G )
)
1211biimpa 470 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  ~P X )  ->  B  e.  ~P dom  dom 
G )
13123adant2 974 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  B  e.  ~P dom  dom 
G )
149eleq2d 2363 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( A  e.  X  <->  A  e.  dom  dom 
G ) )
1514biimpa 470 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  A  e.  dom  dom  G )
16153adant3 975 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  A  e.  dom  dom  G
)
17 snelpwi 4236 . . . . 5  |-  ( A  e.  dom  dom  G  ->  { A }  e.  ~P dom  dom  G )
1816, 17syl 15 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  { A }  e.  ~P dom  dom  G )
19 eqid 2296 . . . . 5  |-  dom  dom  G  =  dom  dom  G
20 prsubrtr.1 . . . . 5  |-  H  =  ( cset `  G
)
2119, 20iscst3 25279 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  ~P dom  dom  G  /\  { A }  e.  ~P dom  dom  G )  ->  ( a  e.  ( B H { A } )  <->  E. x  e.  B  E. y  e.  { A } a  =  ( x G y ) ) )
226, 13, 18, 21syl3anc 1182 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( B H { A } )  <->  E. x  e.  B  E. y  e.  { A } a  =  ( x G y ) ) )
23 df-ima 4718 . . . . . 6  |-  ( F
" B )  =  ran  ( F  |`  B )
24 trfun.2 . . . . . . . . 9  |-  F  =  ( x  e.  X  |->  ( x G A ) )
2524reseq1i 4967 . . . . . . . 8  |-  ( F  |`  B )  =  ( ( x  e.  X  |->  ( x G A ) )  |`  B )
26 elpwi 3646 . . . . . . . . . 10  |-  ( B  e.  ~P X  ->  B  C_  X )
27263ad2ant3 978 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  B  C_  X )
28 resmpt 5016 . . . . . . . . 9  |-  ( B 
C_  X  ->  (
( x  e.  X  |->  ( x G A ) )  |`  B )  =  ( x  e.  B  |->  ( x G A ) ) )
2927, 28syl 15 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( ( x  e.  X  |->  ( x G A ) )  |`  B )  =  ( x  e.  B  |->  ( x G A ) ) )
3025, 29syl5eq 2340 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F  |`  B )  =  ( x  e.  B  |->  ( x G A ) ) )
3130rneqd 4922 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  ->  ran  ( F  |`  B )  =  ran  ( x  e.  B  |->  ( x G A ) ) )
3223, 31syl5eq 2340 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( F " B
)  =  ran  (
x  e.  B  |->  ( x G A ) ) )
3332eleq2d 2363 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( F " B )  <-> 
a  e.  ran  (
x  e.  B  |->  ( x G A ) ) ) )
34 eqid 2296 . . . . 5  |-  ( x  e.  B  |->  ( x G A ) )  =  ( x  e.  B  |->  ( x G A ) )
35 ovex 5899 . . . . 5  |-  ( x G A )  e. 
_V
3634, 35elrnmpti 4946 . . . 4  |-  ( a  e.  ran  ( x  e.  B  |->  ( x G A ) )  <->  E. x  e.  B  a  =  ( x G A ) )
3733, 36syl6bb 252 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( F " B )  <->  E. x  e.  B  a  =  ( x G A ) ) )
385, 22, 373bitr4d 276 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( a  e.  ( B H { A } )  <->  a  e.  ( F " B ) ) )
3938eqrdv 2294 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e. 
~P X )  -> 
( B H { A } )  =  ( F " B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   {csn 3653    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   csetccst 25275
This theorem is referenced by:  caytr  25503
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-grpo 20874  df-cst 25276
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