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Theorem rngohomco 25753
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 2316 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
2 eqid 2316 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3 eqid 2316 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
4 eqid 2316 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
51, 2, 3, 4rngohomf 25745 . . . . . 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 2316 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
10 eqid 2316 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
119, 10, 1, 2rngohomf 25745 . . . . . 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 5436 . . 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 2316 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
18 eqid 2316 . . . . . . 7  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
1910, 17, 18rngo1cl 21149 . . . . . 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 5634 . . . 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 2316 . . . . . . . . 9  |-  ( 2nd `  S )  =  ( 2nd `  S )
25 eqid 2316 . . . . . . . . 9  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
2617, 18, 24, 25rngohom1 25747 . . . . . . . 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 5567 . . . 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 2316 . . . . . . . 8  |-  ( 2nd `  T )  =  ( 2nd `  T )
32 eqid 2316 . . . . . . . 8  |-  (GId `  ( 2nd `  T ) )  =  (GId `  ( 2nd `  T ) )
3324, 25, 31, 32rngohom1 25747 . . . . . . 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 2348 . . 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 2348 . 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 25748 . . . . . . . . . . . 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 5567 . . . . . 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 25746 . . . . . . . . . . . . 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 25746 . . . . . . . . . . . . 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 25481 . . . . . . . . . . . 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 25748 . . . . . . . . . . . 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 2348 . . . . 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 21111 . . . . . . . . 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 5634 . . . . . . 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 5634 . . . . . . . 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 5634 . . . . . . . 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 25481 . . . . . 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 5909 . . . . . 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 2358 . . . 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 25749 . . . . . . . . . . . 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 5567 . . . . . 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 25749 . . . . . . . . . . . 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 2348 . . . . 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 21102 . . . . . . . . 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 5634 . . . . . . 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 5909 . . . . . 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 2358 . . . 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 2669 . 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 25744 . . . 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 1633    e. wcel 1701   A.wral 2577   ran crn 4727    o. ccom 4730   -->wf 5288   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163  GIdcgi 20907   RingOpscrngo 21095    RngHom crnghom 25739
This theorem is referenced by:  rngoisoco  25761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-map 6817  df-grpo 20911  df-gid 20912  df-ablo 21002  df-ass 21033  df-exid 21035  df-mgm 21039  df-sgr 21051  df-mndo 21058  df-rngo 21096  df-rngohom 25742
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