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Theorem keridl 26657
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 2283 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2283 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
3 keridl.1 . . . 4  |-  G  =  ( 1st `  S
)
4 eqid 2283 . . . 4  |-  ran  G  =  ran  G
51, 2, 3, 4rngohomf 26597 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  G )
6 cnvimass 5033 . . . 4  |-  ( `' F " { Z } )  C_  dom  F
7 fdm 5393 . . . 4  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  dom  F  =  ran  ( 1st `  R ) )
86, 7syl5sseq 3226 . . 3  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  ( `' F " { Z } )  C_  ran  ( 1st `  R
) )
95, 8syl 15 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( `' F " { Z }
)  C_  ran  ( 1st `  R ) )
10 eqid 2283 . . . . 5  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
111, 2, 10rngo0cl 21065 . . . 4  |-  ( R  e.  RingOps  ->  (GId `  ( 1st `  R ) )  e.  ran  ( 1st `  R ) )
12113ad2ant1 976 . . 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 26603 . . . 4  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 1st `  R ) ) )  =  Z )
15 fvex 5539 . . . . 5  |-  ( F `
 (GId `  ( 1st `  R ) ) )  e.  _V
1615elsnc 3663 . . . 4  |-  ( ( F `  (GId `  ( 1st `  R ) ) )  e.  { Z }  <->  ( F `  (GId `  ( 1st `  R
) ) )  =  Z )
1714, 16sylibr 203 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 1st `  R ) ) )  e.  { Z }
)
18 ffn 5389 . . . 4  |-  ( F : ran  ( 1st `  R ) --> ran  G  ->  F  Fn  ran  ( 1st `  R ) )
19 elpreima 5645 . . . 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 18 . . 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 888 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  (GId `  ( 1st `  R ) )  e.  ( `' F " { Z } ) )
22 an4 797 . . . . . . . 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 26600 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 5867 . . . . . . . . . . . . . 14  |-  ( ( ( F `  x
)  =  Z  /\  ( F `  y )  =  Z )  -> 
( ( F `  x ) G ( F `  y ) )  =  ( Z G Z ) )
2625adantl 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 ) G ( F `  y
) )  =  ( Z G Z ) )
273rngogrpo 21057 . . . . . . . . . . . . . . . 16  |-  ( S  e.  RingOps  ->  G  e.  GrpOp )
284, 13grpoidcl 20884 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  GrpOp  ->  Z  e.  ran  G )
294, 13grpolid 20886 . . . . . . . . . . . . . . . . 17  |-  ( ( G  e.  GrpOp  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
3028, 29mpdan 649 . . . . . . . . . . . . . . . 16  |-  ( G  e.  GrpOp  ->  ( Z G Z )  =  Z )
3127, 30syl 15 . . . . . . . . . . . . . . 15  |-  ( S  e.  RingOps  ->  ( Z G Z )  =  Z )
32313ad2ant2 977 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( Z G Z )  =  Z )
3332ad2antrr 706 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 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 423 . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  ( F `
 x )  e. 
_V
3736elsnc 3663 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  { Z }  <->  ( F `  x )  =  Z )
38 fvex 5539 . . . . . . . . . . . . 13  |-  ( F `
 y )  e. 
_V
3938elsnc 3663 . . . . . . . . . . . 12  |-  ( ( F `  y )  e.  { Z }  <->  ( F `  y )  =  Z )
4037, 39anbi12i 678 . . . . . . . . . . 11  |-  ( ( ( F `  x
)  e.  { Z }  /\  ( F `  y )  e.  { Z } )  <->  ( ( F `  x )  =  Z  /\  ( F `  y )  =  Z ) )
41 fvex 5539 . . . . . . . . . . . 12  |-  ( F `
 ( x ( 1st `  R ) y ) )  e. 
_V
4241elsnc 3663 . . . . . . . . . . 11  |-  ( ( F `  ( x ( 1st `  R
) y ) )  e.  { Z }  <->  ( F `  ( x ( 1st `  R
) y ) )  =  Z )
4335, 40, 423imtr4g 261 . . . . . . . . . 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 674 . . . . . . . . 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 21058 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) )
46453expib 1154 . . . . . . . . . . 11  |-  ( R  e.  RingOps  ->  ( ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) ) )
47463ad2ant1 976 . . . . . . . . . 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 547 . . . . . . . . 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 40 . . . . . . . 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 208 . . . . . . 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 5645 . . . . . . . . 9  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
x  e.  ( `' F " { Z } )  <->  ( x  e.  ran  ( 1st `  R
)  /\  ( F `  x )  e.  { Z } ) ) )
525, 18, 513syl 18 . . . . . . . 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 5645 . . . . . . . . 9  |-  ( F  Fn  ran  ( 1st `  R )  ->  (
y  e.  ( `' F " { Z } )  <->  ( y  e.  ran  ( 1st `  R
)  /\  ( F `  y )  e.  { Z } ) ) )
545, 18, 533syl 18 . . . . . . . 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 691 . . . . . . 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 5645 . . . . . . . 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 18 . . . . . . 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 259 . . . . . 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 603 . . . . 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 2626 . . . 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 675 . . . . . . 7  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  e.  { Z }
)  <->  ( x  e. 
ran  ( 1st `  R
)  /\  ( F `  x )  =  Z ) )
62 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  R )  =  ( 2nd `  R )
631, 62, 2rngocl 21049 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  z  e.  ran  ( 1st `  R
)  /\  x  e.  ran  ( 1st `  R
) )  ->  (
z ( 2nd `  R
) x )  e. 
ran  ( 1st `  R
) )
64633expb 1152 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
z  e.  ran  ( 1st `  R )  /\  x  e.  ran  ( 1st `  R ) ) )  ->  ( z ( 2nd `  R ) x )  e.  ran  ( 1st `  R ) )
65643ad2antl1 1117 . . . . . . . . . . . . 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 782 . . . . . . . . . . . 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 701 . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  S )  =  ( 2nd `  S )
691, 2, 62, 68rngohommul 26601 . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . 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 701 . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  =  Z  ->  (
( F `  z
) ( 2nd `  S
) ( F `  x ) )  =  ( ( F `  z ) ( 2nd `  S ) Z ) )
7372adantl 452 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  -> 
( ( F `  z ) ( 2nd `  S ) ( F `
 x ) )  =  ( ( F `
 z ) ( 2nd `  S ) Z ) )
7473ad2antlr 707 . . . . . . . . . . . . 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 26598 . . . . . . . . . . . . . . 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 21069 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  RingOps  /\  ( F `  z )  e.  ran  G )  -> 
( ( F `  z ) ( 2nd `  S ) Z )  =  Z )
77763ad2antl2 1118 . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  (
( F `  z
) ( 2nd `  S
) Z )  =  Z )
7978adantlr 695 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  ( F `
 ( z ( 2nd `  R ) x ) )  e. 
_V
8281elsnc 3663 . . . . . . . . . . . 12  |-  ( ( F `  ( z ( 2nd `  R
) x ) )  e.  { Z }  <->  ( F `  ( z ( 2nd `  R
) x ) )  =  Z )
8380, 82sylibr 203 . . . . . . . . . . 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 5645 . . . . . . . . . . . . 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 18 . . . . . . . . . . . 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 706 . . . . . . . . . . 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 888 . . . . . . . . . 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 21049 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  z  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) z )  e. 
ran  ( 1st `  R
) )
89883expb 1152 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  z  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 2nd `  R ) z )  e.  ran  ( 1st `  R ) )
90893ad2antl1 1117 . . . . . . . . . . . . 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 629 . . . . . . . . . . . 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 701 . . . . . . . . . . 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 26601 . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . 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 701 . . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  =  Z  ->  (
( F `  x
) ( 2nd `  S
) ( F `  z ) )  =  ( Z ( 2nd `  S ) ( F `
 z ) ) )
9796adantl 452 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ran  ( 1st `  R )  /\  ( F `  x )  =  Z )  -> 
( ( F `  x ) ( 2nd `  S ) ( F `
 z ) )  =  ( Z ( 2nd `  S ) ( F `  z
) ) )
9897ad2antlr 707 . . . . . . . . . . . . 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 21068 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  RingOps  /\  ( F `  z )  e.  ran  G )  -> 
( Z ( 2nd `  S ) ( F `
 z ) )  =  Z )
100993ad2antl2 1118 . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  z  e. 
ran  ( 1st `  R
) )  ->  ( Z ( 2nd `  S
) ( F `  z ) )  =  Z )
102101adantlr 695 . . . . . . . . . . . . 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 2319 . . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  ( F `
 ( x ( 2nd `  R ) z ) )  e. 
_V
105104elsnc 3663 . . . . . . . . . . . 12  |-  ( ( F `  ( x ( 2nd `  R
) z ) )  e.  { Z }  <->  ( F `  ( x ( 2nd `  R
) z ) )  =  Z )
106103, 105sylibr 203 . . . . . . . . . . 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 5645 . . . . . . . . . . . . 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 18 . . . . . . . . . . . 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 706 . . . . . . . . . . 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 888 . . . . . . . . . 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 518 . . . . . . . . 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 2626 . . . . . . . 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 423 . . . . . . 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 208 . . . . . 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 206 . . . . 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 418 . . . 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 518 . . 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 2626 . 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 26639 . . 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 976 . 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 1135 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {csn 3640   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   RingOpscrngo 21042    RngHom crnghom 26591   Idlcidl 26632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-ghom 21025  df-rngo 21043  df-rngohom 26594  df-idl 26635
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