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Theorem rngohomval 28768
Description: The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
rnghomval.1  |-  G  =  ( 1st `  R
)
rnghomval.2  |-  H  =  ( 2nd `  R
)
rnghomval.3  |-  X  =  ran  G
rnghomval.4  |-  U  =  (GId `  H )
rnghomval.5  |-  J  =  ( 1st `  S
)
rnghomval.6  |-  K  =  ( 2nd `  S
)
rnghomval.7  |-  Y  =  ran  J
rnghomval.8  |-  V  =  (GId `  K )
Assertion
Ref Expression
rngohomval  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngHom  S )  =  { f  e.  ( Y  ^m  X )  |  ( ( f `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( (
f `  ( x G y ) )  =  ( ( f `
 x ) J ( f `  y
) )  /\  (
f `  ( x H y ) )  =  ( ( f `
 x ) K ( f `  y
) ) ) ) } )
Distinct variable groups:    x, f,
y    f, G    f, H    f, J    f, Y, y   
f, K    R, f, x, y    S, f, x, y    f, X, x, y    U, f    f, V
Allowed substitution hints:    U( x, y)    G( x, y)    H( x, y)    J( x, y)    K( x, y)    V( x, y)    Y( x)

Proof of Theorem rngohomval
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  s  =  S )
21fveq2d 5694 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 1st `  s
)  =  ( 1st `  S ) )
3 rnghomval.5 . . . . . . 7  |-  J  =  ( 1st `  S
)
42, 3syl6eqr 2492 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 1st `  s
)  =  J )
54rneqd 5066 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( 1st `  s
)  =  ran  J
)
6 rnghomval.7 . . . . 5  |-  Y  =  ran  J
75, 6syl6eqr 2492 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( 1st `  s
)  =  Y )
8 simpl 457 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  r  =  R )
98fveq2d 5694 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 1st `  r
)  =  ( 1st `  R ) )
10 rnghomval.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
119, 10syl6eqr 2492 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 1st `  r
)  =  G )
1211rneqd 5066 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( 1st `  r
)  =  ran  G
)
13 rnghomval.3 . . . . 5  |-  X  =  ran  G
1412, 13syl6eqr 2492 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ran  ( 1st `  r
)  =  X )
157, 14oveq12d 6108 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ran  ( 1st `  s )  ^m  ran  ( 1st `  r ) )  =  ( Y  ^m  X ) )
168fveq2d 5694 . . . . . . . . 9  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 2nd `  r
)  =  ( 2nd `  R ) )
17 rnghomval.2 . . . . . . . . 9  |-  H  =  ( 2nd `  R
)
1816, 17syl6eqr 2492 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 2nd `  r
)  =  H )
1918fveq2d 5694 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  (GId `  ( 2nd `  r ) )  =  (GId `  H )
)
20 rnghomval.4 . . . . . . 7  |-  U  =  (GId `  H )
2119, 20syl6eqr 2492 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  (GId `  ( 2nd `  r ) )  =  U )
2221fveq2d 5694 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f `  (GId `  ( 2nd `  r
) ) )  =  ( f `  U
) )
231fveq2d 5694 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 2nd `  s
)  =  ( 2nd `  S ) )
24 rnghomval.6 . . . . . . . 8  |-  K  =  ( 2nd `  S
)
2523, 24syl6eqr 2492 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( 2nd `  s
)  =  K )
2625fveq2d 5694 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  (GId `  ( 2nd `  s ) )  =  (GId `  K )
)
27 rnghomval.8 . . . . . 6  |-  V  =  (GId `  K )
2826, 27syl6eqr 2492 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  (GId `  ( 2nd `  s ) )  =  V )
2922, 28eqeq12d 2456 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( f `  (GId `  ( 2nd `  r
) ) )  =  (GId `  ( 2nd `  s ) )  <->  ( f `  U )  =  V ) )
3011oveqd 6107 . . . . . . . . 9  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x ( 1st `  r ) y )  =  ( x G y ) )
3130fveq2d 5694 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f `  (
x ( 1st `  r
) y ) )  =  ( f `  ( x G y ) ) )
324oveqd 6107 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( f `  x ) ( 1st `  s ) ( f `
 y ) )  =  ( ( f `
 x ) J ( f `  y
) ) )
3331, 32eqeq12d 2456 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( f `  ( x ( 1st `  r ) y ) )  =  ( ( f `  x ) ( 1st `  s
) ( f `  y ) )  <->  ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) ) ) )
3418oveqd 6107 . . . . . . . . 9  |-  ( ( r  =  R  /\  s  =  S )  ->  ( x ( 2nd `  r ) y )  =  ( x H y ) )
3534fveq2d 5694 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f `  (
x ( 2nd `  r
) y ) )  =  ( f `  ( x H y ) ) )
3625oveqd 6107 . . . . . . . 8  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( f `  x ) ( 2nd `  s ) ( f `
 y ) )  =  ( ( f `
 x ) K ( f `  y
) ) )
3735, 36eqeq12d 2456 . . . . . . 7  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( f `  ( x ( 2nd `  r ) y ) )  =  ( ( f `  x ) ( 2nd `  s
) ( f `  y ) )  <->  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) ) )
3833, 37anbi12d 710 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( ( f `
 ( x ( 1st `  r ) y ) )  =  ( ( f `  x ) ( 1st `  s ) ( f `
 y ) )  /\  ( f `  ( x ( 2nd `  r ) y ) )  =  ( ( f `  x ) ( 2nd `  s
) ( f `  y ) ) )  <-> 
( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) ) ) )
3914, 38raleqbidv 2930 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ( A. y  e. 
ran  ( 1st `  r
) ( ( f `
 ( x ( 1st `  r ) y ) )  =  ( ( f `  x ) ( 1st `  s ) ( f `
 y ) )  /\  ( f `  ( x ( 2nd `  r ) y ) )  =  ( ( f `  x ) ( 2nd `  s
) ( f `  y ) ) )  <->  A. y  e.  X  ( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) ) ) )
4014, 39raleqbidv 2930 . . . 4  |-  ( ( r  =  R  /\  s  =  S )  ->  ( A. x  e. 
ran  ( 1st `  r
) A. y  e. 
ran  ( 1st `  r
) ( ( f `
 ( x ( 1st `  r ) y ) )  =  ( ( f `  x ) ( 1st `  s ) ( f `
 y ) )  /\  ( f `  ( x ( 2nd `  r ) y ) )  =  ( ( f `  x ) ( 2nd `  s
) ( f `  y ) ) )  <->  A. x  e.  X  A. y  e.  X  ( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) ) ) )
4129, 40anbi12d 710 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( ( f `
 (GId `  ( 2nd `  r ) ) )  =  (GId `  ( 2nd `  s ) )  /\  A. x  e.  ran  ( 1st `  r
) A. y  e. 
ran  ( 1st `  r
) ( ( f `
 ( x ( 1st `  r ) y ) )  =  ( ( f `  x ) ( 1st `  s ) ( f `
 y ) )  /\  ( f `  ( x ( 2nd `  r ) y ) )  =  ( ( f `  x ) ( 2nd `  s
) ( f `  y ) ) ) )  <->  ( ( f `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( (
f `  ( x G y ) )  =  ( ( f `
 x ) J ( f `  y
) )  /\  (
f `  ( x H y ) )  =  ( ( f `
 x ) K ( f `  y
) ) ) ) ) )
4215, 41rabeqbidv 2966 . 2  |-  ( ( r  =  R  /\  s  =  S )  ->  { f  e.  ( ran  ( 1st `  s
)  ^m  ran  ( 1st `  r ) )  |  ( ( f `  (GId `  ( 2nd `  r
) ) )  =  (GId `  ( 2nd `  s ) )  /\  A. x  e.  ran  ( 1st `  r ) A. y  e.  ran  ( 1st `  r ) ( ( f `  ( x ( 1st `  r
) y ) )  =  ( ( f `
 x ) ( 1st `  s ) ( f `  y
) )  /\  (
f `  ( x
( 2nd `  r
) y ) )  =  ( ( f `
 x ) ( 2nd `  s ) ( f `  y
) ) ) ) }  =  { f  e.  ( Y  ^m  X )  |  ( ( f `  U
)  =  V  /\  A. x  e.  X  A. y  e.  X  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) } )
43 df-rngohom 28767 . 2  |-  RngHom  =  ( r  e.  RingOps ,  s  e.  RingOps  |->  { f  e.  ( ran  ( 1st `  s )  ^m  ran  ( 1st `  r ) )  |  ( ( f `  (GId `  ( 2nd `  r ) ) )  =  (GId
`  ( 2nd `  s
) )  /\  A. x  e.  ran  ( 1st `  r ) A. y  e.  ran  ( 1st `  r
) ( ( f `
 ( x ( 1st `  r ) y ) )  =  ( ( f `  x ) ( 1st `  s ) ( f `
 y ) )  /\  ( f `  ( x ( 2nd `  r ) y ) )  =  ( ( f `  x ) ( 2nd `  s
) ( f `  y ) ) ) ) } )
44 ovex 6115 . . 3  |-  ( Y  ^m  X )  e. 
_V
4544rabex 4442 . 2  |-  { f  e.  ( Y  ^m  X )  |  ( ( f `  U
)  =  V  /\  A. x  e.  X  A. y  e.  X  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }  e.  _V
4642, 43, 45ovmpt2a 6220 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngHom  S )  =  { f  e.  ( Y  ^m  X )  |  ( ( f `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( (
f `  ( x G y ) )  =  ( ( f `
 x ) J ( f `  y
) )  /\  (
f `  ( x H y ) )  =  ( ( f `
 x ) K ( f `  y
) ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718   ran crn 4840   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575    ^m cmap 7213  GIdcgi 23673   RingOpscrngo 23861    RngHom crnghom 28764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-rngohom 28767
This theorem is referenced by:  isrngohom  28769
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