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Theorem cntzmhm 14830
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 2296 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2296 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
31, 2mhmf 14436 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
4 cntzmhm.z . . . . 5  |-  Z  =  (Cntz `  G )
51, 4cntzssv 14820 . . . 4  |-  ( Z `
 S )  C_  ( Base `  G )
65sseli 3189 . . 3  |-  ( A  e.  ( Z `  S )  ->  A  e.  ( Base `  G
) )
7 ffvelrn 5679 . . 3  |-  ( ( F : ( Base `  G ) --> ( Base `  H )  /\  A  e.  ( Base `  G
) )  ->  ( F `  A )  e.  ( Base `  H
) )
83, 6, 7syl2an 463 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F `  A )  e.  ( Base `  H
) )
9 eqid 2296 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
109, 4cntzi 14821 . . . . . . 7  |-  ( ( A  e.  ( Z `
 S )  /\  x  e.  S )  ->  ( A ( +g  `  G ) x )  =  ( x ( +g  `  G ) A ) )
1110adantll 694 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( A ( +g  `  G ) x )  =  ( x ( +g  `  G ) A ) )
1211fveq2d 5545 . . . . 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 730 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  F  e.  ( G MndHom  H ) )
146ad2antlr 707 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  A  e.  ( Base `  G ) )
151, 4cntzrcl 14819 . . . . . . . . 9  |-  ( A  e.  ( Z `  S )  ->  ( G  e.  _V  /\  S  C_  ( Base `  G
) ) )
1615adantl 452 . . . . . . . 8  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( G  e.  _V  /\  S  C_  ( Base `  G
) ) )
1716simprd 449 . . . . . . 7  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  S  C_  ( Base `  G
) )
1817sselda 3193 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  x  e.  ( Base `  G ) )
19 eqid 2296 . . . . . . 7  |-  ( +g  `  H )  =  ( +g  `  H )
201, 9, 19mhmlin 14438 . . . . . 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 1182 . . . . 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 14438 . . . . . 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 1182 . . . . 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 2336 . . . 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 2639 . . 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 451 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
27 ffn 5405 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  F  Fn  ( Base `  G )
)
2826, 27syl 15 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  F  Fn  ( Base `  G
) )
29 oveq2 5882 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( F `  A
) ( +g  `  H
) y )  =  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) ) )
30 oveq1 5881 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y ( +g  `  H
) ( F `  A ) )  =  ( ( F `  x ) ( +g  `  H ) ( F `
 A ) ) )
3129, 30eqeq12d 2310 . . . . 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 5774 . . . 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 642 . . 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 223 . 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 5041 . . . 4  |-  ( F
" S )  C_  ran  F
36 frn 5411 . . . . 5  |-  ( F : ( Base `  G
) --> ( Base `  H
)  ->  ran  F  C_  ( Base `  H )
)
3726, 36syl 15 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ran  F 
C_  ( Base `  H
) )
3835, 37syl5ss 3203 . . 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 14814 . . 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 15 . 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 888 1  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F `  A )  e.  ( Y `  ( F " S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   MndHom cmhm 14429  Cntzccntz 14807
This theorem is referenced by:  cntzmhm2  14831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-mhm 14431  df-cntz 14809
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