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Theorem keridl 26596
Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
keridl.1  |-  G  =  ( 1st `  S
)
keridl.2  |-  Z  =  (GId `  G )
Assertion
Ref Expression
keridl  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
)  e.  ( Idl `  R ) )

Proof of Theorem keridl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2435 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
3 keridl.1 . . . 4  |-  G  =  ( 1st `  S
)
4 eqid 2435 . . . 4  |-  ran  G  =  ran  G
51, 2, 3, 4rngohomf 26536 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  G )
6 cnvimass 5216 . . . 4  |-  ( `' F " { Z } )  C_  dom  F
7 fdm 5587 . . . 4  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  dom  F  =  ran  ( 1st `  R ) )
86, 7syl5sseq 3388 . . 3  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  ( `' F " { Z } )  C_  ran  ( 1st `  R
) )
95, 8syl 16 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
)  C_  ran  ( 1st `  R ) )
10 eqid 2435 . . . . 5  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
111, 2, 10rngo0cl 21976 . . . 4  |-  ( R  e.  RingOps  ->  (GId `  ( 1st `  R ) )  e.  ran  ( 1st `  R ) )
12113ad2ant1 978 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  (GId `  ( 1st `  R ) )  e.  ran  ( 1st `  R ) )
13 keridl.2 . . . . 5  |-  Z  =  (GId `  G )
141, 10, 3, 13rngohom0 26542 . . . 4  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 1st `  R ) ) )  =  Z )
15 fvex 5734 . . . . 5  |-  ( F `
 (GId `  ( 1st `  R ) ) )  e.  _V
1615elsnc 3829 . . . 4  |-  ( ( F `  (GId `  ( 1st `  R ) ) )  e.  { Z }  <->  ( F `  (GId `  ( 1st `  R
) ) )  =  Z )
1714, 16sylibr 204 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 1st `  R ) ) )  e.  { Z }
)
18 ffn 5583 . . . 4  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  F  Fn  ran  ( 1st `  R ) )
19 elpreima 5842 . . . 4  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
(GId `  ( 1st `  R ) )  e.  ( `' F " { Z } )  <->  ( (GId `  ( 1st `  R
) )  e.  ran  ( 1st `  R )  /\  ( F `  (GId `  ( 1st `  R
) ) )  e. 
{ Z } ) ) )
205, 18, 193syl 19 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (GId `  ( 1st `  R
) )  e.  ( `' F " { Z } )  <->  ( (GId `  ( 1st `  R
) )  e.  ran  ( 1st `  R )  /\  ( F `  (GId `  ( 1st `  R
) ) )  e. 
{ Z } ) ) )
2112, 17, 20mpbir2and 889 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  (GId `  ( 1st `  R ) )  e.  ( `' F " { Z } ) )
22 an4 798 . . . . . . . 8  |-  ( ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  e.  { Z } )  /\  (
y  e.  ran  ( 1st `  R )  /\  ( F `  y )  e.  { Z }
) )  <->  ( (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  /\  ( ( F `  x )  e.  { Z }  /\  ( F `  y )  e.  { Z } ) ) )
231, 2, 3rngohomadd 26539 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( F `  (
x ( 1st `  R
) y ) )  =  ( ( F `
 x ) G ( F `  y
) ) )
2423adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  /\  (
( F `  x
)  =  Z  /\  ( F `  y )  =  Z ) )  ->  ( F `  ( x ( 1st `  R ) y ) )  =  ( ( F `  x ) G ( F `  y ) ) )
25 oveq12 6082 . . . . . . . . . . . . . 14  |-  ( ( ( F `  x
)  =  Z  /\  ( F `  y )  =  Z )  -> 
( ( F `  x ) G ( F `  y ) )  =  ( Z G Z ) )
2625adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  /\  (
( F `  x
)  =  Z  /\  ( F `  y )  =  Z ) )  ->  ( ( F `
 x ) G ( F `  y
) )  =  ( Z G Z ) )
273rngogrpo 21968 . . . . . . . . . . . . . . . 16  |-  ( S  e.  RingOps  ->  G  e.  GrpOp )
284, 13grpoidcl 21795 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  GrpOp  ->  Z  e.  ran  G )
294, 13grpolid 21797 . . . . . . . . . . . . . . . . 17  |-  ( ( G  e.  GrpOp  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
3028, 29mpdan 650 . . . . . . . . . . . . . . . 16  |-  ( G  e.  GrpOp  ->  ( Z G Z )  =  Z )
3127, 30syl 16 . . . . . . . . . . . . . . 15  |-  ( S  e.  RingOps  ->  ( Z G Z )  =  Z )
32313ad2ant2 979 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( Z G Z )  =  Z )
3332ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  /\  (
( F `  x
)  =  Z  /\  ( F `  y )  =  Z ) )  ->  ( Z G Z )  =  Z )
3424, 26, 333eqtrd 2471 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  /\  (
( F `  x
)  =  Z  /\  ( F `  y )  =  Z ) )  ->  ( F `  ( x ( 1st `  R ) y ) )  =  Z )
3534ex 424 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( ( F `
 x )  =  Z  /\  ( F `
 y )  =  Z )  ->  ( F `  ( x
( 1st `  R
) y ) )  =  Z ) )
36 fvex 5734 . . . . . . . . . . . . 13  |-  ( F `
 x )  e. 
_V
3736elsnc 3829 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  { Z }  <->  ( F `  x )  =  Z )
38 fvex 5734 . . . . . . . . . . . . 13  |-  ( F `
 y )  e. 
_V
3938elsnc 3829 . . . . . . . . . . . 12  |-  ( ( F `  y )  e.  { Z }  <->  ( F `  y )  =  Z )
4037, 39anbi12i 679 . . . . . . . . . . 11  |-  ( ( ( F `  x
)  e.  { Z }  /\  ( F `  y )  e.  { Z } )  <->  ( ( F `  x )  =  Z  /\  ( F `  y )  =  Z ) )
41 fvex 5734 . . . . . . . . . . . 12  |-  ( F `
 ( x ( 1st `  R ) y ) )  e. 
_V
4241elsnc 3829 . . . . . . . . . . 11  |-  ( ( F `  ( x ( 1st `  R
) y ) )  e.  { Z }  <->  ( F `  ( x ( 1st `  R
) y ) )  =  Z )
4335, 40, 423imtr4g 262 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( ( F `
 x )  e. 
{ Z }  /\  ( F `  y )  e.  { Z }
)  ->  ( F `  ( x ( 1st `  R ) y ) )  e.  { Z } ) )
4443imdistanda 675 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  /\  ( ( F `  x )  e.  { Z }  /\  ( F `  y
)  e.  { Z } ) )  -> 
( ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  /\  ( F `  ( x
( 1st `  R
) y ) )  e.  { Z }
) ) )
451, 2rngogcl 21969 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) )
46453expib 1156 . . . . . . . . . . 11  |-  ( R  e.  RingOps  ->  ( ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) ) )
47463ad2ant1 978 . . . . . . . . . 10  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  -> 
( x ( 1st `  R ) y )  e.  ran  ( 1st `  R ) ) )
4847anim1d 548 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  /\  ( F `
 ( x ( 1st `  R ) y ) )  e. 
{ Z } )  ->  ( ( x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
)  /\  ( F `  ( x ( 1st `  R ) y ) )  e.  { Z } ) ) )
4944, 48syld 42 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  /\  ( ( F `  x )  e.  { Z }  /\  ( F `  y
)  e.  { Z } ) )  -> 
( ( x ( 1st `  R ) y )  e.  ran  ( 1st `  R )  /\  ( F `  ( x ( 1st `  R ) y ) )  e.  { Z } ) ) )
5022, 49syl5bi 209 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  e.  { Z } )  /\  (
y  e.  ran  ( 1st `  R )  /\  ( F `  y )  e.  { Z }
) )  ->  (
( x ( 1st `  R ) y )  e.  ran  ( 1st `  R )  /\  ( F `  ( x
( 1st `  R
) y ) )  e.  { Z }
) ) )
51 elpreima 5842 . . . . . . . . 9  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
x  e.  ( `' F " { Z } )  <->  ( x  e.  ran  ( 1st `  R
)  /\  ( F `  x )  e.  { Z } ) ) )
525, 18, 513syl 19 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( x  e.  ( `' F " { Z } )  <->  ( x  e.  ran  ( 1st `  R
)  /\  ( F `  x )  e.  { Z } ) ) )
53 elpreima 5842 . . . . . . . . 9  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
y  e.  ( `' F " { Z } )  <->  ( y  e.  ran  ( 1st `  R
)  /\  ( F `  y )  e.  { Z } ) ) )
545, 18, 533syl 19 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( y  e.  ( `' F " { Z } )  <->  ( y  e.  ran  ( 1st `  R
)  /\  ( F `  y )  e.  { Z } ) ) )
5552, 54anbi12d 692 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ( `' F " { Z } )  /\  y  e.  ( `' F " { Z } ) )  <-> 
( ( x  e. 
ran  ( 1st `  R
)  /\  ( F `  x )  e.  { Z } )  /\  (
y  e.  ran  ( 1st `  R )  /\  ( F `  y )  e.  { Z }
) ) ) )
56 elpreima 5842 . . . . . . . 8  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
( x ( 1st `  R ) y )  e.  ( `' F " { Z } )  <-> 
( ( x ( 1st `  R ) y )  e.  ran  ( 1st `  R )  /\  ( F `  ( x ( 1st `  R ) y ) )  e.  { Z } ) ) )
575, 18, 563syl 19 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x ( 1st `  R
) y )  e.  ( `' F " { Z } )  <->  ( (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
)  /\  ( F `  ( x ( 1st `  R ) y ) )  e.  { Z } ) ) )
5850, 55, 573imtr4d 260 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ( `' F " { Z } )  /\  y  e.  ( `' F " { Z } ) )  ->  ( x ( 1st `  R ) y )  e.  ( `' F " { Z } ) ) )
5958impl 604 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ( `' F " { Z }
) )  /\  y  e.  ( `' F " { Z } ) )  ->  ( x ( 1st `  R ) y )  e.  ( `' F " { Z } ) )
6059ralrimiva 2781 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ( `' F " { Z } ) )  ->  A. y  e.  ( `' F " { Z } ) ( x ( 1st `  R
) y )  e.  ( `' F " { Z } ) )
6137anbi2i 676 . . . . . . 7  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  e.  { Z }
)  <->  ( x  e. 
ran  ( 1st `  R
)  /\  ( F `  x )  =  Z ) )
62 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  R )  =  ( 2nd `  R )
631, 62, 2rngocl 21960 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R
)  /\  x  e.  ran  ( 1st `  R
) )  ->  (
z ( 2nd `  R
) x )  e. 
ran  ( 1st `  R
) )
64633expb 1154 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
z  e.  ran  ( 1st `  R )  /\  x  e.  ran  ( 1st `  R ) ) )  ->  ( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R ) )
65643ad2antl1 1119 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( z  e.  ran  ( 1st `  R )  /\  x  e.  ran  ( 1st `  R
) ) )  -> 
( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R ) )
6665anass1rs 783 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ran  ( 1st `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R ) )
6766adantlrr 702 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
z ( 2nd `  R
) x )  e. 
ran  ( 1st `  R
) )
68 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  S )  =  ( 2nd `  S )
691, 2, 62, 68rngohommul 26540 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( z  e.  ran  ( 1st `  R )  /\  x  e.  ran  ( 1st `  R
) ) )  -> 
( F `  (
z ( 2nd `  R
) x ) )  =  ( ( F `
 z ) ( 2nd `  S ) ( F `  x
) ) )
7069anass1rs 783 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ran  ( 1st `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( F `  (
z ( 2nd `  R
) x ) )  =  ( ( F `
 z ) ( 2nd `  S ) ( F `  x
) ) )
7170adantlrr 702 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( F `  ( z
( 2nd `  R
) x ) )  =  ( ( F `
 z ) ( 2nd `  S ) ( F `  x
) ) )
72 oveq2 6081 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  =  Z  ->  (
( F `  z
) ( 2nd `  S
) ( F `  x ) )  =  ( ( F `  z ) ( 2nd `  S ) Z ) )
7372adantl 453 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  -> 
( ( F `  z ) ( 2nd `  S ) ( F `
 x ) )  =  ( ( F `
 z ) ( 2nd `  S ) Z ) )
7473ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
( F `  z
) ( 2nd `  S
) ( F `  x ) )  =  ( ( F `  z ) ( 2nd `  S ) Z ) )
751, 2, 3, 4rngohomcl 26537 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  ( F `  z )  e.  ran  G )
7613, 4, 3, 68rngorz 21980 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  RingOps  /\  ( F `  z )  e.  ran  G )  -> 
( ( F `  z ) ( 2nd `  S ) Z )  =  Z )
77763ad2antl2 1120 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F `
 z )  e. 
ran  G )  -> 
( ( F `  z ) ( 2nd `  S ) Z )  =  Z )
7875, 77syldan 457 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  (
( F `  z
) ( 2nd `  S
) Z )  =  Z )
7978adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
( F `  z
) ( 2nd `  S
) Z )  =  Z )
8071, 74, 793eqtrd 2471 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( F `  ( z
( 2nd `  R
) x ) )  =  Z )
81 fvex 5734 . . . . . . . . . . . . 13  |-  ( F `
 ( z ( 2nd `  R ) x ) )  e. 
_V
8281elsnc 3829 . . . . . . . . . . . 12  |-  ( ( F `  ( z ( 2nd `  R
) x ) )  e.  { Z }  <->  ( F `  ( z ( 2nd `  R
) x ) )  =  Z )
8380, 82sylibr 204 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( F `  ( z
( 2nd `  R
) x ) )  e.  { Z }
)
84 elpreima 5842 . . . . . . . . . . . . 13  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  <-> 
( ( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R )  /\  ( F `  ( z ( 2nd `  R ) x ) )  e.  { Z } ) ) )
855, 18, 843syl 19 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
z ( 2nd `  R
) x )  e.  ( `' F " { Z } )  <->  ( (
z ( 2nd `  R
) x )  e. 
ran  ( 1st `  R
)  /\  ( F `  ( z ( 2nd `  R ) x ) )  e.  { Z } ) ) )
8685ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  <-> 
( ( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R )  /\  ( F `  ( z ( 2nd `  R ) x ) )  e.  { Z } ) ) )
8767, 83, 86mpbir2and 889 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
z ( 2nd `  R
) x )  e.  ( `' F " { Z } ) )
881, 62, 2rngocl 21960 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  z  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) z )  e. 
ran  ( 1st `  R
) )
89883expb 1154 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  z  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R ) )
90893ad2antl1 1119 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  z  e.  ran  ( 1st `  R
) ) )  -> 
( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R ) )
9190anassrs 630 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ran  ( 1st `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R ) )
9291adantlrr 702 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) z )  e. 
ran  ( 1st `  R
) )
931, 2, 62, 68rngohommul 26540 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  z  e.  ran  ( 1st `  R
) ) )  -> 
( F `  (
x ( 2nd `  R
) z ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  z
) ) )
9493anassrs 630 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ran  ( 1st `  R ) )  /\  z  e.  ran  ( 1st `  R ) )  -> 
( F `  (
x ( 2nd `  R
) z ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  z
) ) )
9594adantlrr 702 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( F `  ( x
( 2nd `  R
) z ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  z
) ) )
96 oveq1 6080 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  =  Z  ->  (
( F `  x
) ( 2nd `  S
) ( F `  z ) )  =  ( Z ( 2nd `  S ) ( F `
 z ) ) )
9796adantl 453 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  -> 
( ( F `  x ) ( 2nd `  S ) ( F `
 z ) )  =  ( Z ( 2nd `  S ) ( F `  z
) ) )
9897ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
( F `  x
) ( 2nd `  S
) ( F `  z ) )  =  ( Z ( 2nd `  S ) ( F `
 z ) ) )
9913, 4, 3, 68rngolz 21979 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  RingOps  /\  ( F `  z )  e.  ran  G )  -> 
( Z ( 2nd `  S ) ( F `
 z ) )  =  Z )
100993ad2antl2 1120 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F `
 z )  e. 
ran  G )  -> 
( Z ( 2nd `  S ) ( F `
 z ) )  =  Z )
10175, 100syldan 457 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  ( Z ( 2nd `  S
) ( F `  z ) )  =  Z )
102101adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( Z ( 2nd `  S
) ( F `  z ) )  =  Z )
10395, 98, 1023eqtrd 2471 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( F `  ( x
( 2nd `  R
) z ) )  =  Z )
104 fvex 5734 . . . . . . . . . . . . 13  |-  ( F `
 ( x ( 2nd `  R ) z ) )  e. 
_V
105104elsnc 3829 . . . . . . . . . . . 12  |-  ( ( F `  ( x ( 2nd `  R
) z ) )  e.  { Z }  <->  ( F `  ( x ( 2nd `  R
) z ) )  =  Z )
106103, 105sylibr 204 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  ( F `  ( x
( 2nd `  R
) z ) )  e.  { Z }
)
107 elpreima 5842 . . . . . . . . . . . . 13  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
( x ( 2nd `  R ) z )  e.  ( `' F " { Z } )  <-> 
( ( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R )  /\  ( F `  ( x ( 2nd `  R ) z ) )  e.  { Z } ) ) )
1085, 18, 1073syl 19 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x ( 2nd `  R
) z )  e.  ( `' F " { Z } )  <->  ( (
x ( 2nd `  R
) z )  e. 
ran  ( 1st `  R
)  /\  ( F `  ( x ( 2nd `  R ) z ) )  e.  { Z } ) ) )
109108ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
( x ( 2nd `  R ) z )  e.  ( `' F " { Z } )  <-> 
( ( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R )  /\  ( F `  ( x ( 2nd `  R ) z ) )  e.  { Z } ) ) )
11092, 106, 109mpbir2and 889 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) z )  e.  ( `' F " { Z } ) )
11187, 110jca 519 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  /\  z  e.  ran  ( 1st `  R
) )  ->  (
( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  ( x ( 2nd `  R ) z )  e.  ( `' F " { Z } ) ) )
112111ralrimiva 2781 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z ) )  ->  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  (
x ( 2nd `  R
) z )  e.  ( `' F " { Z } ) ) )
113112ex 424 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  ->  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  (
x ( 2nd `  R
) z )  e.  ( `' F " { Z } ) ) ) )
11461, 113syl5bi 209 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ran  ( 1st `  R )  /\  ( F `  x )  e.  { Z }
)  ->  A. z  e.  ran  ( 1st `  R
) ( ( z ( 2nd `  R
) x )  e.  ( `' F " { Z } )  /\  ( x ( 2nd `  R ) z )  e.  ( `' F " { Z } ) ) ) )
11552, 114sylbid 207 . . . . 5  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( x  e.  ( `' F " { Z } )  ->  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  (
x ( 2nd `  R
) z )  e.  ( `' F " { Z } ) ) ) )
116115imp 419 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ( `' F " { Z } ) )  ->  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  (
x ( 2nd `  R
) z )  e.  ( `' F " { Z } ) ) )
11760, 116jca 519 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e.  ( `' F " { Z } ) )  ->  ( A. y  e.  ( `' F " { Z } ) ( x ( 1st `  R
) y )  e.  ( `' F " { Z } )  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  ( x ( 2nd `  R ) z )  e.  ( `' F " { Z } ) ) ) )
118117ralrimiva 2781 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  A. x  e.  ( `' F " { Z } ) ( A. y  e.  ( `' F " { Z } ) ( x ( 1st `  R
) y )  e.  ( `' F " { Z } )  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  ( x ( 2nd `  R ) z )  e.  ( `' F " { Z } ) ) ) )
1191, 62, 2, 10isidl 26578 . . 3  |-  ( R  e.  RingOps  ->  ( ( `' F " { Z } )  e.  ( Idl `  R )  <-> 
( ( `' F " { Z } ) 
C_  ran  ( 1st `  R )  /\  (GId `  ( 1st `  R
) )  e.  ( `' F " { Z } )  /\  A. x  e.  ( `' F " { Z }
) ( A. y  e.  ( `' F " { Z } ) ( x ( 1st `  R
) y )  e.  ( `' F " { Z } )  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  ( x ( 2nd `  R ) z )  e.  ( `' F " { Z } ) ) ) ) ) )
1201193ad2ant1 978 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( ( `' F " { Z } )  e.  ( Idl `  R )  <-> 
( ( `' F " { Z } ) 
C_  ran  ( 1st `  R )  /\  (GId `  ( 1st `  R
) )  e.  ( `' F " { Z } )  /\  A. x  e.  ( `' F " { Z }
) ( A. y  e.  ( `' F " { Z } ) ( x ( 1st `  R
) y )  e.  ( `' F " { Z } )  /\  A. z  e.  ran  ( 1st `  R ) ( ( z ( 2nd `  R ) x )  e.  ( `' F " { Z } )  /\  ( x ( 2nd `  R ) z )  e.  ( `' F " { Z } ) ) ) ) ) )
1219, 21, 118, 120mpbir3and 1137 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
)  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   {csn 3806   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   GrpOpcgr 21764  GIdcgi 21765   RingOpscrngo 21953    RngHom crnghom 26530   Idlcidl 26571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-map 7012  df-grpo 21769  df-gid 21770  df-ginv 21771  df-ablo 21860  df-ghom 21936  df-rngo 21954  df-rngohom 26533  df-idl 26574
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