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Theorem cntzmhm 15856
Description: Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
cntzmhm.z  |-  Z  =  (Cntz `  G )
cntzmhm.y  |-  Y  =  (Cntz `  H )
Assertion
Ref Expression
cntzmhm  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F `  A )  e.  ( Y `  ( F " S ) ) )

Proof of Theorem cntzmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2443 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
31, 2mhmf 15469 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
4 cntzmhm.z . . . . 5  |-  Z  =  (Cntz `  G )
51, 4cntzssv 15846 . . . 4  |-  ( Z `
 S )  C_  ( Base `  G )
65sseli 3352 . . 3  |-  ( A  e.  ( Z `  S )  ->  A  e.  ( Base `  G
) )
7 ffvelrn 5841 . . 3  |-  ( ( F : ( Base `  G ) --> ( Base `  H )  /\  A  e.  ( Base `  G
) )  ->  ( F `  A )  e.  ( Base `  H
) )
83, 6, 7syl2an 477 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F `  A )  e.  ( Base `  H
) )
9 eqid 2443 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
109, 4cntzi 15847 . . . . . . 7  |-  ( ( A  e.  ( Z `
 S )  /\  x  e.  S )  ->  ( A ( +g  `  G ) x )  =  ( x ( +g  `  G ) A ) )
1110adantll 713 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( A ( +g  `  G ) x )  =  ( x ( +g  `  G ) A ) )
1211fveq2d 5695 . . . . 5  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( F `  ( A ( +g  `  G
) x ) )  =  ( F `  ( x ( +g  `  G ) A ) ) )
13 simpll 753 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  F  e.  ( G MndHom  H ) )
146ad2antlr 726 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  A  e.  ( Base `  G ) )
151, 4cntzrcl 15845 . . . . . . . . 9  |-  ( A  e.  ( Z `  S )  ->  ( G  e.  _V  /\  S  C_  ( Base `  G
) ) )
1615adantl 466 . . . . . . . 8  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( G  e.  _V  /\  S  C_  ( Base `  G
) ) )
1716simprd 463 . . . . . . 7  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  S  C_  ( Base `  G
) )
1817sselda 3356 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  x  e.  ( Base `  G ) )
19 eqid 2443 . . . . . . 7  |-  ( +g  `  H )  =  ( +g  `  H )
201, 9, 19mhmlin 15471 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Base `  G
)  /\  x  e.  ( Base `  G )
)  ->  ( F `  ( A ( +g  `  G ) x ) )  =  ( ( F `  A ) ( +g  `  H
) ( F `  x ) ) )
2113, 14, 18, 20syl3anc 1218 . . . . 5  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( F `  ( A ( +g  `  G
) x ) )  =  ( ( F `
 A ) ( +g  `  H ) ( F `  x
) ) )
221, 9, 19mhmlin 15471 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  x  e.  ( Base `  G
)  /\  A  e.  ( Base `  G )
)  ->  ( F `  ( x ( +g  `  G ) A ) )  =  ( ( F `  x ) ( +g  `  H
) ( F `  A ) ) )
2313, 18, 14, 22syl3anc 1218 . . . . 5  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( F `  (
x ( +g  `  G
) A ) )  =  ( ( F `
 x ) ( +g  `  H ) ( F `  A
) ) )
2412, 21, 233eqtr3d 2483 . . . 4  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) )  =  ( ( F `
 x ) ( +g  `  H ) ( F `  A
) ) )
2524ralrimiva 2799 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  A. x  e.  S  ( ( F `  A )
( +g  `  H ) ( F `  x
) )  =  ( ( F `  x
) ( +g  `  H
) ( F `  A ) ) )
263adantr 465 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
27 ffn 5559 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  F  Fn  ( Base `  G )
)
2826, 27syl 16 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  F  Fn  ( Base `  G
) )
29 oveq2 6099 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( F `  A
) ( +g  `  H
) y )  =  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) ) )
30 oveq1 6098 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y ( +g  `  H
) ( F `  A ) )  =  ( ( F `  x ) ( +g  `  H ) ( F `
 A ) ) )
3129, 30eqeq12d 2457 . . . . 5  |-  ( y  =  ( F `  x )  ->  (
( ( F `  A ) ( +g  `  H ) y )  =  ( y ( +g  `  H ) ( F `  A
) )  <->  ( ( F `  A )
( +g  `  H ) ( F `  x
) )  =  ( ( F `  x
) ( +g  `  H
) ( F `  A ) ) ) )
3231ralima 5957 . . . 4  |-  ( ( F  Fn  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( A. y  e.  ( F " S ) ( ( F `  A
) ( +g  `  H
) y )  =  ( y ( +g  `  H ) ( F `
 A ) )  <->  A. x  e.  S  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) )  =  ( ( F `
 x ) ( +g  `  H ) ( F `  A
) ) ) )
3328, 17, 32syl2anc 661 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( A. y  e.  ( F " S ) ( ( F `  A
) ( +g  `  H
) y )  =  ( y ( +g  `  H ) ( F `
 A ) )  <->  A. x  e.  S  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) )  =  ( ( F `
 x ) ( +g  `  H ) ( F `  A
) ) ) )
3425, 33mpbird 232 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  A. y  e.  ( F " S
) ( ( F `
 A ) ( +g  `  H ) y )  =  ( y ( +g  `  H
) ( F `  A ) ) )
35 imassrn 5180 . . . 4  |-  ( F
" S )  C_  ran  F
36 frn 5565 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  ran  F  C_  ( Base `  H )
)
3726, 36syl 16 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ran  F 
C_  ( Base `  H
) )
3835, 37syl5ss 3367 . . 3  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F " S )  C_  ( Base `  H )
)
39 cntzmhm.y . . . 4  |-  Y  =  (Cntz `  H )
402, 19, 39elcntz 15840 . . 3  |-  ( ( F " S ) 
C_  ( Base `  H
)  ->  ( ( F `  A )  e.  ( Y `  ( F " S ) )  <-> 
( ( F `  A )  e.  (
Base `  H )  /\  A. y  e.  ( F " S ) ( ( F `  A ) ( +g  `  H ) y )  =  ( y ( +g  `  H ) ( F `  A
) ) ) ) )
4138, 40syl 16 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  (
( F `  A
)  e.  ( Y `
 ( F " S ) )  <->  ( ( F `  A )  e.  ( Base `  H
)  /\  A. y  e.  ( F " S
) ( ( F `
 A ) ( +g  `  H ) y )  =  ( y ( +g  `  H
) ( F `  A ) ) ) ) )
428, 34, 41mpbir2and 913 1  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F `  A )  e.  ( Y `  ( F " S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    C_ wss 3328   ran crn 4841   "cima 4843    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   MndHom cmhm 15462  Cntzccntz 15833
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-mhm 15464  df-cntz 15835
This theorem is referenced by:  cntzmhm2  15857
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