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Theorem keridl 28678
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 28618 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  G )
6 cnvimass 5179 . . . 4  |-  ( `' F " { Z } )  C_  dom  F
7 fdm 5553 . . . 4  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  dom  F  =  ran  ( 1st `  R ) )
86, 7syl5sseq 3394 . . 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 23710 . . . 4  |-  ( R  e.  RingOps  ->  (GId `  ( 1st `  R ) )  e.  ran  ( 1st `  R ) )
12113ad2ant1 1004 . . 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 28624 . . . 4  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 1st `  R ) ) )  =  Z )
15 fvex 5691 . . . . 5  |-  ( F `
 (GId `  ( 1st `  R ) ) )  e.  _V
1615elsnc 3891 . . . 4  |-  ( ( F `  (GId `  ( 1st `  R ) ) )  e.  { Z }  <->  ( F `  (GId `  ( 1st `  R
) ) )  =  Z )
1714, 16sylibr 212 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 1st `  R ) ) )  e.  { Z }
)
18 ffn 5549 . . . 4  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  F  Fn  ran  ( 1st `  R ) )
19 elpreima 5813 . . . 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 20 . . 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 908 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  (GId `  ( 1st `  R ) )  e.  ( `' F " { Z } ) )
22 an4 815 . . . . . . . 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 28621 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 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 6091 . . . . . . . . . . . . . 14  |-  ( ( ( F `  x
)  =  Z  /\  ( F `  y )  =  Z )  -> 
( ( F `  x ) G ( F `  y ) )  =  ( Z G Z ) )
2625adantl 463 . . . . . . . . . . . . 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 23702 . . . . . . . . . . . . . . . 16  |-  ( S  e.  RingOps  ->  G  e.  GrpOp )
284, 13grpoidcl 23529 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  GrpOp  ->  Z  e.  ran  G )
294, 13grpolid 23531 . . . . . . . . . . . . . . . . 17  |-  ( ( G  e.  GrpOp  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
3028, 29mpdan 663 . . . . . . . . . . . . . . . 16  |-  ( G  e.  GrpOp  ->  ( Z G Z )  =  Z )
3127, 30syl 16 . . . . . . . . . . . . . . 15  |-  ( S  e.  RingOps  ->  ( Z G Z )  =  Z )
32313ad2ant2 1005 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( Z G Z )  =  Z )
3332ad2antrr 720 . . . . . . . . . . . . 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 434 . . . . . . . . . . 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 5691 . . . . . . . . . . . . 13  |-  ( F `
 x )  e. 
_V
3736elsnc 3891 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  { Z }  <->  ( F `  x )  =  Z )
38 fvex 5691 . . . . . . . . . . . . 13  |-  ( F `
 y )  e. 
_V
3938elsnc 3891 . . . . . . . . . . . 12  |-  ( ( F `  y )  e.  { Z }  <->  ( F `  y )  =  Z )
4037, 39anbi12i 692 . . . . . . . . . . 11  |-  ( ( ( F `  x
)  e.  { Z }  /\  ( F `  y )  e.  { Z } )  <->  ( ( F `  x )  =  Z  /\  ( F `  y )  =  Z ) )
41 fvex 5691 . . . . . . . . . . . 12  |-  ( F `
 ( x ( 1st `  R ) y ) )  e. 
_V
4241elsnc 3891 . . . . . . . . . . 11  |-  ( ( F `  ( x ( 1st `  R
) y ) )  e.  { Z }  <->  ( F `  ( x ( 1st `  R
) y ) )  =  Z )
4335, 40, 423imtr4g 270 . . . . . . . . . 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 688 . . . . . . . . 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 23703 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) )
46453expib 1185 . . . . . . . . . . 11  |-  ( R  e.  RingOps  ->  ( ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) ) )
47463ad2ant1 1004 . . . . . . . . . 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 561 . . . . . . . . 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 44 . . . . . . . 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 217 . . . . . . 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 5813 . . . . . . . . 9  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
x  e.  ( `' F " { Z } )  <->  ( x  e.  ran  ( 1st `  R
)  /\  ( F `  x )  e.  { Z } ) ) )
525, 18, 513syl 20 . . . . . . . 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 5813 . . . . . . . . 9  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
y  e.  ( `' F " { Z } )  <->  ( y  e.  ran  ( 1st `  R
)  /\  ( F `  y )  e.  { Z } ) ) )
545, 18, 533syl 20 . . . . . . . 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 705 . . . . . . 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 5813 . . . . . . . 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 20 . . . . . . 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 268 . . . . . 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 617 . . . . 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 2791 . . . 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 689 . . . . . . 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 23694 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R
)  /\  x  e.  ran  ( 1st `  R
) )  ->  (
z ( 2nd `  R
) x )  e. 
ran  ( 1st `  R
) )
64633expb 1183 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
z  e.  ran  ( 1st `  R )  /\  x  e.  ran  ( 1st `  R ) ) )  ->  ( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R ) )
65643ad2antl1 1145 . . . . . . . . . . . . 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 800 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 28622 . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  =  Z  ->  (
( F `  z
) ( 2nd `  S
) ( F `  x ) )  =  ( ( F `  z ) ( 2nd `  S ) Z ) )
7372adantl 463 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  -> 
( ( F `  z ) ( 2nd `  S ) ( F `
 x ) )  =  ( ( F `
 z ) ( 2nd `  S ) Z ) )
7473ad2antlr 721 . . . . . . . . . . . . 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 28619 . . . . . . . . . . . . . . 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 23714 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  RingOps  /\  ( F `  z )  e.  ran  G )  -> 
( ( F `  z ) ( 2nd `  S ) Z )  =  Z )
77763ad2antl2 1146 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  (
( F `  z
) ( 2nd `  S
) Z )  =  Z )
7978adantlr 709 . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . 13  |-  ( F `
 ( z ( 2nd `  R ) x ) )  e. 
_V
8281elsnc 3891 . . . . . . . . . . . 12  |-  ( ( F `  ( z ( 2nd `  R
) x ) )  e.  { Z }  <->  ( F `  ( z ( 2nd `  R
) x ) )  =  Z )
8380, 82sylibr 212 . . . . . . . . . . 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 5813 . . . . . . . . . . . . 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 20 . . . . . . . . . . . 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 720 . . . . . . . . . . 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 908 . . . . . . . . . 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 23694 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  z  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) z )  e. 
ran  ( 1st `  R
) )
89883expb 1183 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  z  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R ) )
90893ad2antl1 1145 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 28622 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 6089 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  =  Z  ->  (
( F `  x
) ( 2nd `  S
) ( F `  z ) )  =  ( Z ( 2nd `  S ) ( F `
 z ) ) )
9796adantl 463 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  -> 
( ( F `  x ) ( 2nd `  S ) ( F `
 z ) )  =  ( Z ( 2nd `  S ) ( F `  z
) ) )
9897ad2antlr 721 . . . . . . . . . . . . 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 23713 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  RingOps  /\  ( F `  z )  e.  ran  G )  -> 
( Z ( 2nd `  S ) ( F `
 z ) )  =  Z )
100993ad2antl2 1146 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  ( Z ( 2nd `  S
) ( F `  z ) )  =  Z )
102101adantlr 709 . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . 13  |-  ( F `
 ( x ( 2nd `  R ) z ) )  e. 
_V
105104elsnc 3891 . . . . . . . . . . . 12  |-  ( ( F `  ( x ( 2nd `  R
) z ) )  e.  { Z }  <->  ( F `  ( x ( 2nd `  R
) z ) )  =  Z )
106103, 105sylibr 212 . . . . . . . . . . 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 5813 . . . . . . . . . . . . 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 20 . . . . . . . . . . . 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 720 . . . . . . . . . . 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 908 . . . . . . . . . 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 529 . . . . . . . . 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 2791 . . . . . . . 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 434 . . . . . . 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 217 . . . . . 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 215 . . . . 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 429 . . . 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 529 . . 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 2791 . 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 28660 . . 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 1004 . 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 1166 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
)  e.  ( Idl `  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1757   A.wral 2707    C_ wss 3318   {csn 3867   `'ccnv 4828   dom cdm 4829   ran crn 4830   "cima 4832    Fn wfn 5403   -->wf 5404   ` cfv 5408  (class class class)co 6082   1stc1st 6566   2ndc2nd 6567   GrpOpcgr 23498  GIdcgi 23499   RingOpscrngo 23687    RngHom crnghom 28612   Idlcidl 28653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6568  df-2nd 6569  df-map 7206  df-grpo 23503  df-gid 23504  df-ginv 23505  df-ablo 23594  df-ghom 23670  df-rngo 23688  df-rngohom 28615  df-idl 28656
This theorem is referenced by: (None)
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