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Theorem rngohomco 32175
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 2423 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
2 eqid 2423 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3 eqid 2423 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
4 eqid 2423 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
51, 2, 3, 4rngohomf 32167 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  ->  G : ran  ( 1st `  S
) --> ran  ( 1st `  T ) )
653expa 1206 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  G : ran  ( 1st `  S ) --> ran  ( 1st `  T ) )
763adantl1 1162 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  G : ran  ( 1st `  S
) --> ran  ( 1st `  T ) )
87adantrl 721 . . 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 2423 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
10 eqid 2423 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
119, 10, 1, 2rngohomf 32167 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  ( 1st `  S ) )
12113expa 1206 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R ) --> ran  ( 1st `  S ) )
13123adantl3 1164 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  F : ran  ( 1st `  R
) --> ran  ( 1st `  S ) )
1413adantrr 722 . . 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 5755 . . 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 666 . 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 2423 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
18 eqid 2423 . . . . . . 7  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
1910, 17, 18rngo1cl 26153 . . . . . 6  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
20193ad2ant1 1027 . . . . 5  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
2120adantr 467 . . . 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 5957 . . . 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 666 . . 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 2423 . . . . . . . . 9  |-  ( 2nd `  S )  =  ( 2nd `  S )
25 eqid 2423 . . . . . . . . 9  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
2617, 18, 24, 25rngohom1 32169 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
27263expa 1206 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
28273adantl3 1164 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
2928adantrr 722 . . . . 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 5884 . . . 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 2423 . . . . . . . 8  |-  ( 2nd `  T )  =  ( 2nd `  T )
32 eqid 2423 . . . . . . . 8  |-  (GId `  ( 2nd `  T ) )  =  (GId `  ( 2nd `  T ) )
3324, 25, 31, 32rngohom1 32169 . . . . . . 7  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngHom  T ) )  ->  ( G `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  T ) ) )
34333expa 1206 . . . . . 6  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  -> 
( G `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  T ) ) )
35343adantl1 1162 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngHom  T ) )  ->  ( G `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  T ) ) )
3635adantrl 721 . . . 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 2464 . . 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 2464 . 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 32170 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 1206 . . . . . . . . . 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 1164 . . . . . . . . 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 431 . . . . . . . 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 726 . . . . . . 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 5884 . . . . . 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 32168 . . . . . . . . . . . . 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 32168 . . . . . . . . . . . . 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 32000 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 1206 . . . . . . . . . 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 1164 . . . . . . . . 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 431 . . . . . . . 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 726 . . . . . . 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 32170 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 1206 . . . . . . . . . 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 1162 . . . . . . . . 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 431 . . . . . . . 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 725 . . . . . . 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 473 . . . . . 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 2464 . . . . 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 26115 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 1st `  R
) y )  e. 
ran  ( 1st `  R
) )
63623expb 1207 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 1st `  R ) y )  e.  ran  ( 1st `  R ) )
64633ad2antl1 1168 . . . . . . 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 720 . . . . . 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 5957 . . . . . . 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 474 . . . . . 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 473 . . . . 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 5957 . . . . . . . 8  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  x  e.  ran  ( 1st `  R
) )  ->  (
( G  o.  F
) `  x )  =  ( G `  ( F `  x ) ) )
7014, 69sylan 474 . . . . . . 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 5957 . . . . . . . 8  |-  ( ( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
( G  o.  F
) `  y )  =  ( G `  ( F `  y ) ) )
7214, 71sylan 474 . . . . . . 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 32000 . . . . . 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 6313 . . . . . 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 17 . . . . 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 2474 . . . 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 32171 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 1206 . . . . . . . . . 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 1164 . . . . . . . . 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 431 . . . . . . . 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 726 . . . . . . 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 5884 . . . . . 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 32171 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 1206 . . . . . . . . . 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 1162 . . . . . . . . 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 431 . . . . . . . 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 725 . . . . . . 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 473 . . . . . 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 2464 . . . . 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 26106 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  x  e.  ran  ( 1st `  R
)  /\  y  e.  ran  ( 1st `  R
) )  ->  (
x ( 2nd `  R
) y )  e. 
ran  ( 1st `  R
) )
93923expb 1207 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  (
x  e.  ran  ( 1st `  R )  /\  y  e.  ran  ( 1st `  R ) ) )  ->  ( x ( 2nd `  R ) y )  e.  ran  ( 1st `  R ) )
94933ad2antl1 1168 . . . . . . 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 720 . . . . . 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 5957 . . . . . . 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 474 . . . . . 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 473 . . . . 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 6313 . . . . . 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 17 . . . . 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 2474 . . . 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 535 . . 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 2847 . 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 32166 . . . 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 1025 . . 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 467 . 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 1189 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   ran crn 4853    o. ccom 4856   -->wf 5596   ` cfv 5600  (class class class)co 6304   1stc1st 6804   2ndc2nd 6805  GIdcgi 25911   RingOpscrngo 26099    RngHom crnghom 32161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-riota 6266  df-ov 6307  df-oprab 6308  df-mpt2 6309  df-1st 6806  df-2nd 6807  df-map 7484  df-grpo 25915  df-gid 25916  df-ablo 26006  df-ass 26037  df-exid 26039  df-mgmOLD 26043  df-sgrOLD 26055  df-mndo 26062  df-rngo 26100  df-rngohom 32164
This theorem is referenced by:  rngoisoco  32183
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