MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elghomlem2OLD Structured version   Unicode version

Theorem elghomlem2OLD 26088
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 26089. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
elghomlem1OLD.1  |-  S  =  { f  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) }
Assertion
Ref Expression
elghomlem2OLD  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
Distinct variable groups:    x, f,
y, F    f, G, x, y    f, H, x, y
Allowed substitution hints:    S( x, y, f)

Proof of Theorem elghomlem2OLD
StepHypRef Expression
1 elghomlem1OLD.1 . . . 4  |-  S  =  { f  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) }
21elghomlem1OLD 26087 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
32eleq2d 2492 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  F  e.  S
) )
4 elex 3089 . . . . 5  |-  ( F  e.  S  ->  F  e.  _V )
5 feq1 5728 . . . . . . . 8  |-  ( f  =  F  ->  (
f : ran  G --> ran  H  <->  F : ran  G --> ran  H ) )
6 fveq1 5880 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
7 fveq1 5880 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
86, 7oveq12d 6323 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
) H ( f `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
9 fveq1 5880 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
108, 9eqeq12d 2444 . . . . . . . . 9  |-  ( f  =  F  ->  (
( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
11102ralbidv 2866 . . . . . . . 8  |-  ( f  =  F  ->  ( A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
125, 11anbi12d 715 . . . . . . 7  |-  ( f  =  F  ->  (
( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x
) H ( f `
 y ) )  =  ( f `  ( x G y ) ) )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1312, 1elab2g 3219 . . . . . 6  |-  ( F  e.  _V  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1413biimpd 210 . . . . 5  |-  ( F  e.  _V  ->  ( F  e.  S  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) ) )
154, 14mpcom 37 . . . 4  |-  ( F  e.  S  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
16 rnexg 6739 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
17 fex 6153 . . . . . . . 8  |-  ( ( F : ran  G --> ran  H  /\  ran  G  e.  _V )  ->  F  e.  _V )
1817expcom 436 . . . . . . 7  |-  ( ran 
G  e.  _V  ->  ( F : ran  G --> ran  H  ->  F  e.  _V ) )
1916, 18syl 17 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( F : ran  G --> ran  H  ->  F  e.  _V )
)
2019adantrd 469 . . . . 5  |-  ( G  e.  GrpOp  ->  ( ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  _V ) )
2113biimprd 226 . . . . 5  |-  ( F  e.  _V  ->  (
( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )  ->  F  e.  S )
)
2220, 21syli 38 . . . 4  |-  ( G  e.  GrpOp  ->  ( ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  S
) )
2315, 22impbid2 207 . . 3  |-  ( G  e.  GrpOp  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
2423adantr 466 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
253, 24bitrd 256 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407   A.wral 2771   _Vcvv 3080   ran crn 4854   -->wf 5597   ` cfv 5601  (class class class)co 6305   GrpOpcgr 25912   GrpOpHom cghomOLD 26083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-ghomOLD 26084
This theorem is referenced by:  elghomOLD  26089
  Copyright terms: Public domain W3C validator