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

Theorem isrngohom 28783
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 28782 . . 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 2510 . 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 5713 . . . . . . . 8  |-  ( 1st `  S )  e.  _V
125, 11eqeltri 2513 . . . . . . 7  |-  J  e. 
_V
1312rnex 6524 . . . . . 6  |-  ran  J  e.  _V
147, 13eqeltri 2513 . . . . 5  |-  Y  e. 
_V
15 fvex 5713 . . . . . . . 8  |-  ( 1st `  R )  e.  _V
161, 15eqeltri 2513 . . . . . . 7  |-  G  e. 
_V
1716rnex 6524 . . . . . 6  |-  ran  G  e.  _V
183, 17eqeltri 2513 . . . . 5  |-  X  e. 
_V
1914, 18elmap 7253 . . . 4  |-  ( F  e.  ( Y  ^m  X )  <->  F : X
--> Y )
2019anbi1i 695 . . 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 5702 . . . . . 6  |-  ( f  =  F  ->  (
f `  U )  =  ( F `  U ) )
2221eqeq1d 2451 . . . . 5  |-  ( f  =  F  ->  (
( f `  U
)  =  V  <->  ( F `  U )  =  V ) )
23 fveq1 5702 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
24 fveq1 5702 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
25 fveq1 5702 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2624, 25oveq12d 6121 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) J ( f `
 y ) )  =  ( ( F `
 x ) J ( F `  y
) ) )
2723, 26eqeq12d 2457 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  <->  ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) ) ) )
28 fveq1 5702 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x H y ) )  =  ( F `  ( x H y ) ) )
2924, 25oveq12d 6121 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) K ( f `
 y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )
3028, 29eqeq12d 2457 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) )  <->  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) )
3127, 30anbi12d 710 . . . . . 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 2769 . . . . 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 710 . . . 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 3129 . . 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 969 . . 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 277 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   {crab 2731   _Vcvv 2984   ran crn 4853   -->wf 5426   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588    ^m cmap 7226  GIdcgi 23686   RingOpscrngo 23874    RngHom crnghom 28778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-map 7228  df-rngohom 28781
This theorem is referenced by:  rngohomf  28784  rngohom1  28786  rngohomadd  28787  rngohommul  28788  rngohomco  28792  rngoisocnv  28799
  Copyright terms: Public domain W3C validator