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Theorem cntzmhm 16503
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 2457 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2457 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
31, 2mhmf 16098 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
4 cntzmhm.z . . . . 5  |-  Z  =  (Cntz `  G )
51, 4cntzssv 16493 . . . 4  |-  ( Z `
 S )  C_  ( Base `  G )
65sseli 3495 . . 3  |-  ( A  e.  ( Z `  S )  ->  A  e.  ( Base `  G
) )
7 ffvelrn 6030 . . 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 2457 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
109, 4cntzi 16494 . . . . . . 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 5876 . . . . 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 16492 . . . . . . . . 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 3499 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  x  e.  ( Base `  G ) )
19 eqid 2457 . . . . . . 7  |-  ( +g  `  H )  =  ( +g  `  H )
201, 9, 19mhmlin 16100 . . . . . 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 1228 . . . . 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 16100 . . . . . 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 1228 . . . . 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 2506 . . . 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 2871 . . 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 5737 . . . . 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 6304 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( F `  A
) ( +g  `  H
) y )  =  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) ) )
30 oveq1 6303 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y ( +g  `  H
) ( F `  A ) )  =  ( ( F `  x ) ( +g  `  H ) ( F `
 A ) ) )
3129, 30eqeq12d 2479 . . . . 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 6153 . . . 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 5358 . . . 4  |-  ( F
" S )  C_  ran  F
36 frn 5743 . . . . 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 3510 . . 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 16487 . . 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 922 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 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   MndHom cmhm 16091  Cntzccntz 16480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-mhm 16093  df-cntz 16482
This theorem is referenced by:  cntzmhm2  16504
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