Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrngohom Unicode version

Theorem isrngohom 26699
Description: The predicate "is a ring homomorphism from  R to  S." (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomval.1  |-  G  =  ( 1st `  R
)
rnghomval.2  |-  H  =  ( 2nd `  R
)
rnghomval.3  |-  X  =  ran  G
rnghomval.4  |-  U  =  (GId `  H )
rnghomval.5  |-  J  =  ( 1st `  S
)
rnghomval.6  |-  K  =  ( 2nd `  S
)
rnghomval.7  |-  Y  =  ran  J
rnghomval.8  |-  V  =  (GId `  K )
Assertion
Ref Expression
isrngohom  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
Distinct variable groups:    x, y, F    y, Y    x, R, y    x, S, y    x, X, y
Allowed substitution hints:    U( x, y)    G( x, y)    H( x, y)    J( x, y)    K( x, y)    V( x, y)    Y( x)

Proof of Theorem isrngohom
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 rnghomval.1 . . . 4  |-  G  =  ( 1st `  R
)
2 rnghomval.2 . . . 4  |-  H  =  ( 2nd `  R
)
3 rnghomval.3 . . . 4  |-  X  =  ran  G
4 rnghomval.4 . . . 4  |-  U  =  (GId `  H )
5 rnghomval.5 . . . 4  |-  J  =  ( 1st `  S
)
6 rnghomval.6 . . . 4  |-  K  =  ( 2nd `  S
)
7 rnghomval.7 . . . 4  |-  Y  =  ran  J
8 rnghomval.8 . . . 4  |-  V  =  (GId `  K )
91, 2, 3, 4, 5, 6, 7, 8rngohomval 26698 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngHom  S )  =  { f  e.  ( Y  ^m  X )  |  ( ( f `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( (
f `  ( x G y ) )  =  ( ( f `
 x ) J ( f `  y
) )  /\  (
f `  ( x H y ) )  =  ( ( f `
 x ) K ( f `  y
) ) ) ) } )
109eleq2d 2363 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  F  e.  { f  e.  ( Y  ^m  X )  |  ( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( f `
 ( x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `
 ( x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) } ) )
11 fvex 5555 . . . . . . . 8  |-  ( 1st `  S )  e.  _V
125, 11eqeltri 2366 . . . . . . 7  |-  J  e. 
_V
1312rnex 4958 . . . . . 6  |-  ran  J  e.  _V
147, 13eqeltri 2366 . . . . 5  |-  Y  e. 
_V
15 fvex 5555 . . . . . . . 8  |-  ( 1st `  R )  e.  _V
161, 15eqeltri 2366 . . . . . . 7  |-  G  e. 
_V
1716rnex 4958 . . . . . 6  |-  ran  G  e.  _V
183, 17eqeltri 2366 . . . . 5  |-  X  e. 
_V
1914, 18elmap 6812 . . . 4  |-  ( F  e.  ( Y  ^m  X )  <->  F : X
--> Y )
2019anbi1i 676 . . 3  |-  ( ( F  e.  ( Y  ^m  X )  /\  ( ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `
 ( x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `
 ( x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) ) ) )  <-> 
( F : X --> Y  /\  ( ( F `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
21 fveq1 5540 . . . . . 6  |-  ( f  =  F  ->  (
f `  U )  =  ( F `  U ) )
2221eqeq1d 2304 . . . . 5  |-  ( f  =  F  ->  (
( f `  U
)  =  V  <->  ( F `  U )  =  V ) )
23 fveq1 5540 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
24 fveq1 5540 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
25 fveq1 5540 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2624, 25oveq12d 5892 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) J ( f `
 y ) )  =  ( ( F `
 x ) J ( F `  y
) ) )
2723, 26eqeq12d 2310 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  <->  ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) ) ) )
28 fveq1 5540 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x H y ) )  =  ( F `  ( x H y ) ) )
2924, 25oveq12d 5892 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) K ( f `
 y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )
3028, 29eqeq12d 2310 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) )  <->  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) )
3127, 30anbi12d 691 . . . . . 6  |-  ( f  =  F  ->  (
( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) )  <->  ( ( F `
 ( x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `
 ( x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) ) ) )
32312ralbidv 2598 . . . . 5  |-  ( f  =  F  ->  ( A. x  e.  X  A. y  e.  X  ( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) )  <->  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) ) )
3322, 32anbi12d 691 . . . 4  |-  ( f  =  F  ->  (
( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( f `
 ( x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `
 ( x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) )  <->  ( ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
3433elrab 2936 . . 3  |-  ( F  e.  { f  e.  ( Y  ^m  X
)  |  ( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }  <->  ( F  e.  ( Y  ^m  X
)  /\  ( ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
35 3anass 938 . . 3  |-  ( ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) )  <-> 
( F : X --> Y  /\  ( ( F `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
3620, 34, 353bitr4i 268 . 2  |-  ( F  e.  { f  e.  ( Y  ^m  X
)  |  ( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }  <->  ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) )
3710, 36syl6bb 252 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788  GIdcgi 20870   RingOpscrngo 21058    RngHom crnghom 26694
This theorem is referenced by:  rngohomf  26700  rngohom1  26702  rngohomadd  26703  rngohommul  26704  rngohomco  26708  rngoisocnv  26715
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-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-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-rngohom 26697
  Copyright terms: Public domain W3C validator