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Theorem rngohomco 26605
Description: The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngohomco  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )

Proof of Theorem rngohomco
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
2 eqid 2283 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3 eqid 2283 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
4 eqid 2283 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
51, 2, 3, 4rngohomf 26597 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  ->  G : ran  ( 1st `  S
) --> ran  ( 1st `  T ) )
653expa 1151 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  G : ran  ( 1st `  S ) --> ran  ( 1st `  T ) )
763adantl1 1111 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  G : ran  ( 1st `  S
) --> ran  ( 1st `  T ) )
87adantrl 696 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  G : ran  ( 1st `  S
) --> ran  ( 1st `  T ) )
9 eqid 2283 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
10 eqid 2283 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
119, 10, 1, 2rngohomf 26597 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  ( 1st `  S ) )
12113expa 1151 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R ) --> ran  ( 1st `  S ) )
13123adantl3 1113 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  ( 1st `  S ) )
1413adantrr 697 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  F : ran  ( 1st `  R
) --> ran  ( 1st `  S ) )
15 fco 5398 . . 3  |-  ( ( G : ran  ( 1st `  S ) --> ran  ( 1st `  T
)  /\  F : ran  ( 1st `  R
) --> ran  ( 1st `  S ) )  -> 
( G  o.  F
) : ran  ( 1st `  R ) --> ran  ( 1st `  T
) )
168, 14, 15syl2anc 642 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F ) : ran  ( 1st `  R
) --> ran  ( 1st `  T ) )
17 eqid 2283 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
18 eqid 2283 . . . . . . 7  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
1910, 17, 18rngo1cl 21096 . . . . . 6  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
20193ad2ant1 976 . . . . 5  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
2120adantr 451 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  (GId `  ( 2nd `  R
) )  e.  ran  ( 1st `  R ) )
22 fvco3 5596 . . . 4  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )  -> 
( ( G  o.  F ) `  (GId `  ( 2nd `  R
) ) )  =  ( G `  ( F `  (GId `  ( 2nd `  R ) ) ) ) )
2314, 21, 22syl2anc 642 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  (
( G  o.  F
) `  (GId `  ( 2nd `  R ) ) )  =  ( G `
 ( F `  (GId `  ( 2nd `  R
) ) ) ) )
24 eqid 2283 . . . . . . . . 9  |-  ( 2nd `  S )  =  ( 2nd `  S )
25 eqid 2283 . . . . . . . . 9  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
2617, 18, 24, 25rngohom1 26599 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
27263expa 1151 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
28273adantl3 1113 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
2928adantrr 697 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
3029fveq2d 5529 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G `  ( F `  (GId `  ( 2nd `  R ) ) ) )  =  ( G `
 (GId `  ( 2nd `  S ) ) ) )
31 eqid 2283 . . . . . . . 8  |-  ( 2nd `  T )  =  ( 2nd `  T )
32 eqid 2283 . . . . . . . 8  |-  (GId `  ( 2nd `  T ) )  =  (GId `  ( 2nd `  T ) )
3324, 25, 31, 32rngohom1 26599 . . . . . . 7  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  ->  ( G `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  T ) ) )
34333expa 1151 . . . . . 6  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  -> 
( G `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  T ) ) )
35343adantl1 1111 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  ( G `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  T ) ) )
3635adantrl 696 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  T ) ) )
3730, 36eqtrd 2315 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G `  ( F `  (GId `  ( 2nd `  R ) ) ) )  =  (GId `  ( 2nd `  T ) ) )
3823, 37eqtrd 2315 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  (
( G  o.  F
) `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  T ) ) )
399, 10, 1rngohomadd 26600 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ( 1st `  S ) ( F `  y
) ) )
4039ex 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 ( 1st `  R
) y ) )  =  ( ( F `
 x ) ( 1st `  S ) ( F `  y
) ) ) )
41403expa 1151 . . . . . . . . . 10  |-  ( ( ( 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 ) ( 1st `  S ) ( F `  y
) ) ) )
42413adantl3 1113 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  -> 
( F `  (
x ( 1st `  R
) y ) )  =  ( ( F `
 x ) ( 1st `  S ) ( F `  y
) ) ) )
4342imp 418 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( x
( 1st `  R
) y ) )  =  ( ( F `
 x ) ( 1st `  S ) ( F `  y
) ) )
4443adantlrr 701 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( F `  (
x ( 1st `  R
) y ) )  =  ( ( F `
 x ) ( 1st `  S ) ( F `  y
) ) )
4544fveq2d 5529 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( G `  ( F `  ( x
( 1st `  R
) y ) ) )  =  ( G `
 ( ( F `
 x ) ( 1st `  S ) ( F `  y
) ) ) )
469, 10, 1, 2rngohomcl 26598 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  x  e. 
ran  ( 1st `  R
) )  ->  ( F `  x )  e.  ran  ( 1st `  S
) )
479, 10, 1, 2rngohomcl 26598 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  y  e. 
ran  ( 1st `  R
) )  ->  ( F `  y )  e.  ran  ( 1st `  S
) )
4846, 47anim12da 26332 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )
4948ex 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 )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) ) )
50493expa 1151 . . . . . . . . . 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.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) ) )
51503adantl3 1113 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  -> 
( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) ) )
5251imp 418 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  (
( F `  x
)  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )
5352adantlrr 701 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )
541, 2, 3rngohomadd 26600 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  /\  ( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S
) ) )  -> 
( G `  (
( F `  x
) ( 1st `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) )
5554ex 423 . . . . . . . . . . 11  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  ->  ( (
( F `  x
)  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) )  -> 
( G `  (
( F `  x
) ( 1st `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) ) )
56553expa 1151 . . . . . . . . . 10  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  -> 
( ( ( F `
 x )  e. 
ran  ( 1st `  S
)  /\  ( F `  y )  e.  ran  ( 1st `  S ) )  ->  ( G `  ( ( F `  x ) ( 1st `  S ) ( F `
 y ) ) )  =  ( ( G `  ( F `
 x ) ) ( 1st `  T
) ( G `  ( F `  y ) ) ) ) )
57563adantl1 1111 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  ( (
( F `  x
)  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) )  -> 
( G `  (
( F `  x
) ( 1st `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) ) )
5857imp 418 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  /\  ( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )  ->  ( G `  ( ( F `  x )
( 1st `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) )
5958adantlrl 700 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( ( F `
 x )  e. 
ran  ( 1st `  S
)  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )  ->  ( G `  ( ( F `  x )
( 1st `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) )
6053, 59syldan 456 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( G `  (
( F `  x
) ( 1st `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) )
6145, 60eqtrd 2315 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( G `  ( F `  ( x
( 1st `  R
) y ) ) )  =  ( ( G `  ( F `
 x ) ) ( 1st `  T
) ( G `  ( F `  y ) ) ) )
629, 10rngogcl 21058 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) )
63623expb 1152 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 1st `  R ) y )  e.  ran  ( 1st `  R ) )
64633ad2antl1 1117 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 1st `  R ) y )  e.  ran  ( 1st `  R ) )
6564adantlr 695 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( x ( 1st `  R ) y )  e.  ran  ( 1st `  R ) )
66 fvco3 5596 . . . . . . 7  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  ( x
( 1st `  R
) y )  e. 
ran  ( 1st `  R
) )  ->  (
( G  o.  F
) `  ( x
( 1st `  R
) y ) )  =  ( G `  ( F `  ( x ( 1st `  R
) y ) ) ) )
6714, 66sylan 457 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x ( 1st `  R ) y )  e.  ran  ( 1st `  R ) )  ->  ( ( G  o.  F ) `  ( x ( 1st `  R ) y ) )  =  ( G `
 ( F `  ( x ( 1st `  R ) y ) ) ) )
6865, 67syldan 456 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( G  o.  F ) `  (
x ( 1st `  R
) y ) )  =  ( G `  ( F `  ( x ( 1st `  R
) y ) ) ) )
69 fvco3 5596 . . . . . . . 8  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  x  e.  ran  ( 1st `  R
) )  ->  (
( G  o.  F
) `  x )  =  ( G `  ( F `  x ) ) )
7014, 69sylan 457 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  x  e.  ran  ( 1st `  R ) )  ->  ( ( G  o.  F ) `  x )  =  ( G `  ( F `
 x ) ) )
71 fvco3 5596 . . . . . . . 8  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
( G  o.  F
) `  y )  =  ( G `  ( F `  y ) ) )
7214, 71sylan 457 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  y  e.  ran  ( 1st `  R ) )  ->  ( ( G  o.  F ) `  y )  =  ( G `  ( F `
 y ) ) )
7370, 72anim12da 26332 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( ( G  o.  F ) `  x )  =  ( G `  ( F `
 x ) )  /\  ( ( G  o.  F ) `  y )  =  ( G `  ( F `
 y ) ) ) )
74 oveq12 5867 . . . . . 6  |-  ( ( ( ( G  o.  F ) `  x
)  =  ( G `
 ( F `  x ) )  /\  ( ( G  o.  F ) `  y
)  =  ( G `
 ( F `  y ) ) )  ->  ( ( ( G  o.  F ) `
 x ) ( 1st `  T ) ( ( G  o.  F ) `  y
) )  =  ( ( G `  ( F `  x )
) ( 1st `  T
) ( G `  ( F `  y ) ) ) )
7573, 74syl 15 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( ( G  o.  F ) `  x ) ( 1st `  T ) ( ( G  o.  F ) `
 y ) )  =  ( ( G `
 ( F `  x ) ) ( 1st `  T ) ( G `  ( F `  y )
) ) )
7661, 68, 753eqtr4d 2325 . . . 4  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( G  o.  F ) `  (
x ( 1st `  R
) y ) )  =  ( ( ( G  o.  F ) `
 x ) ( 1st `  T ) ( ( G  o.  F ) `  y
) ) )
779, 10, 17, 24rngohommul 26601 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( F `  (
x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) )
7877ex 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 ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) )
79783expa 1151 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) )
80793adantl3 1113 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) )  -> 
( F `  (
x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) )
8180imp 418 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) )
8281adantlrr 701 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( F `  (
x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) )
8382fveq2d 5529 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( G `  ( F `  ( x
( 2nd `  R
) y ) ) )  =  ( G `
 ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) )
841, 2, 24, 31rngohommul 26601 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  /\  ( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S
) ) )  -> 
( G `  (
( F `  x
) ( 2nd `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) )
8584ex 423 . . . . . . . . . . 11  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  ->  ( (
( F `  x
)  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) )  -> 
( G `  (
( F `  x
) ( 2nd `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) ) )
86853expa 1151 . . . . . . . . . 10  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  -> 
( ( ( F `
 x )  e. 
ran  ( 1st `  S
)  /\  ( F `  y )  e.  ran  ( 1st `  S ) )  ->  ( G `  ( ( F `  x ) ( 2nd `  S ) ( F `
 y ) ) )  =  ( ( G `  ( F `
 x ) ) ( 2nd `  T
) ( G `  ( F `  y ) ) ) ) )
87863adantl1 1111 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  ( (
( F `  x
)  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) )  -> 
( G `  (
( F `  x
) ( 2nd `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) ) )
8887imp 418 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  /\  ( ( F `  x )  e.  ran  ( 1st `  S )  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )  ->  ( G `  ( ( F `  x )
( 2nd `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) )
8988adantlrl 700 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( ( F `
 x )  e. 
ran  ( 1st `  S
)  /\  ( F `  y )  e.  ran  ( 1st `  S ) ) )  ->  ( G `  ( ( F `  x )
( 2nd `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) )
9053, 89syldan 456 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( G `  (
( F `  x
) ( 2nd `  S
) ( F `  y ) ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) )
9183, 90eqtrd 2315 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( G `  ( F `  ( x
( 2nd `  R
) y ) ) )  =  ( ( G `  ( F `
 x ) ) ( 2nd `  T
) ( G `  ( F `  y ) ) ) )
929, 17, 10rngocl 21049 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) y )  e. 
ran  ( 1st `  R
) )
93923expb 1152 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 2nd `  R ) y )  e.  ran  ( 1st `  R ) )
94933ad2antl1 1117 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 2nd `  R ) y )  e.  ran  ( 1st `  R ) )
9594adantlr 695 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( x ( 2nd `  R ) y )  e.  ran  ( 1st `  R ) )
96 fvco3 5596 . . . . . . 7  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  ( x
( 2nd `  R
) y )  e. 
ran  ( 1st `  R
) )  ->  (
( G  o.  F
) `  ( x
( 2nd `  R
) y ) )  =  ( G `  ( F `  ( x ( 2nd `  R
) y ) ) ) )
9714, 96sylan 457 . . . . . 6  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x ( 2nd `  R ) y )  e.  ran  ( 1st `  R ) )  ->  ( ( G  o.  F ) `  ( x ( 2nd `  R ) y ) )  =  ( G `
 ( F `  ( x ( 2nd `  R ) y ) ) ) )
9895, 97syldan 456 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( G  o.  F ) `  (
x ( 2nd `  R
) y ) )  =  ( G `  ( F `  ( x ( 2nd `  R
) y ) ) ) )
99 oveq12 5867 . . . . . 6  |-  ( ( ( ( G  o.  F ) `  x
)  =  ( G `
 ( F `  x ) )  /\  ( ( G  o.  F ) `  y
)  =  ( G `
 ( F `  y ) ) )  ->  ( ( ( G  o.  F ) `
 x ) ( 2nd `  T ) ( ( G  o.  F ) `  y
) )  =  ( ( G `  ( F `  x )
) ( 2nd `  T
) ( G `  ( F `  y ) ) ) )
10073, 99syl 15 . . . . 5  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( ( G  o.  F ) `  x ) ( 2nd `  T ) ( ( G  o.  F ) `
 y ) )  =  ( ( G `
 ( F `  x ) ) ( 2nd `  T ) ( G `  ( F `  y )
) ) )
10191, 98, 1003eqtr4d 2325 . . . 4  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( G  o.  F ) `  (
x ( 2nd `  R
) y ) )  =  ( ( ( G  o.  F ) `
 x ) ( 2nd `  T ) ( ( G  o.  F ) `  y
) ) )
10276, 101jca 518 . . 3  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  /\  ( x  e. 
ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) ) )  -> 
( ( ( G  o.  F ) `  ( x ( 1st `  R ) y ) )  =  ( ( ( G  o.  F
) `  x )
( 1st `  T
) ( ( G  o.  F ) `  y ) )  /\  ( ( G  o.  F ) `  (
x ( 2nd `  R
) y ) )  =  ( ( ( G  o.  F ) `
 x ) ( 2nd `  T ) ( ( G  o.  F ) `  y
) ) ) )
103102ralrimivva 2635 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  A. x  e.  ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( ( G  o.  F ) `
 ( x ( 1st `  R ) y ) )  =  ( ( ( G  o.  F ) `  x ) ( 1st `  T ) ( ( G  o.  F ) `
 y ) )  /\  ( ( G  o.  F ) `  ( x ( 2nd `  R ) y ) )  =  ( ( ( G  o.  F
) `  x )
( 2nd `  T
) ( ( G  o.  F ) `  y ) ) ) )
1049, 17, 10, 18, 3, 31, 4, 32isrngohom 26596 . . . 4  |-  ( ( R  e.  RingOps  /\  T  e.  RingOps )  ->  (
( G  o.  F
)  e.  ( R 
RngHom  T )  <->  ( ( G  o.  F ) : ran  ( 1st `  R
) --> ran  ( 1st `  T )  /\  (
( G  o.  F
) `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  T ) )  /\  A. x  e.  ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( ( G  o.  F ) `
 ( x ( 1st `  R ) y ) )  =  ( ( ( G  o.  F ) `  x ) ( 1st `  T ) ( ( G  o.  F ) `
 y ) )  /\  ( ( G  o.  F ) `  ( x ( 2nd `  R ) y ) )  =  ( ( ( G  o.  F
) `  x )
( 2nd `  T
) ( ( G  o.  F ) `  y ) ) ) ) ) )
1051043adant2 974 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  ->  ( ( G  o.  F )  e.  ( R  RngHom  T )  <-> 
( ( G  o.  F ) : ran  ( 1st `  R ) --> ran  ( 1st `  T
)  /\  ( ( G  o.  F ) `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  T ) )  /\  A. x  e. 
ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( ( G  o.  F ) `
 ( x ( 1st `  R ) y ) )  =  ( ( ( G  o.  F ) `  x ) ( 1st `  T ) ( ( G  o.  F ) `
 y ) )  /\  ( ( G  o.  F ) `  ( x ( 2nd `  R ) y ) )  =  ( ( ( G  o.  F
) `  x )
( 2nd `  T
) ( ( G  o.  F ) `  y ) ) ) ) ) )
106105adantr 451 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  (
( G  o.  F
)  e.  ( R 
RngHom  T )  <->  ( ( G  o.  F ) : ran  ( 1st `  R
) --> ran  ( 1st `  T )  /\  (
( G  o.  F
) `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  T ) )  /\  A. x  e.  ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( ( G  o.  F ) `
 ( x ( 1st `  R ) y ) )  =  ( ( ( G  o.  F ) `  x ) ( 1st `  T ) ( ( G  o.  F ) `
 y ) )  /\  ( ( G  o.  F ) `  ( x ( 2nd `  R ) y ) )  =  ( ( ( G  o.  F
) `  x )
( 2nd `  T
) ( ( G  o.  F ) `  y ) ) ) ) ) )
10716, 38, 103, 106mpbir3and 1135 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ran crn 4690    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042    RngHom crnghom 26591
This theorem is referenced by:  rngoisoco  26613
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-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-rmo 2551  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-fo 5261  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-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-rngohom 26594
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