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Theorem cntzmhm 15137
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 2436 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2436 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
31, 2mhmf 14743 . . 3  |-  ( F  e.  ( G MndHom  H
)  ->  F :
( Base `  G ) --> ( Base `  H )
)
4 cntzmhm.z . . . . 5  |-  Z  =  (Cntz `  G )
51, 4cntzssv 15127 . . . 4  |-  ( Z `
 S )  C_  ( Base `  G )
65sseli 3344 . . 3  |-  ( A  e.  ( Z `  S )  ->  A  e.  ( Base `  G
) )
7 ffvelrn 5868 . . 3  |-  ( ( F : ( Base `  G ) --> ( Base `  H )  /\  A  e.  ( Base `  G
) )  ->  ( F `  A )  e.  ( Base `  H
) )
83, 6, 7syl2an 464 . 2  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( F `  A )  e.  ( Base `  H
) )
9 eqid 2436 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
109, 4cntzi 15128 . . . . . . 7  |-  ( ( A  e.  ( Z `
 S )  /\  x  e.  S )  ->  ( A ( +g  `  G ) x )  =  ( x ( +g  `  G ) A ) )
1110adantll 695 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  ( A ( +g  `  G ) x )  =  ( x ( +g  `  G ) A ) )
1211fveq2d 5732 . . . . 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 731 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  F  e.  ( G MndHom  H ) )
146ad2antlr 708 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  A  e.  ( Base `  G ) )
151, 4cntzrcl 15126 . . . . . . . . 9  |-  ( A  e.  ( Z `  S )  ->  ( G  e.  _V  /\  S  C_  ( Base `  G
) ) )
1615adantl 453 . . . . . . . 8  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  ( G  e.  _V  /\  S  C_  ( Base `  G
) ) )
1716simprd 450 . . . . . . 7  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  S  C_  ( Base `  G
) )
1817sselda 3348 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  /\  x  e.  S )  ->  x  e.  ( Base `  G ) )
19 eqid 2436 . . . . . . 7  |-  ( +g  `  H )  =  ( +g  `  H )
201, 9, 19mhmlin 14745 . . . . . 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 1184 . . . . 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 14745 . . . . . 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 1184 . . . . 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 2476 . . . 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 2789 . . 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 452 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S
) )  ->  F : ( Base `  G
) --> ( Base `  H
) )
27 ffn 5591 . . . . 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 6089 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( F `  A
) ( +g  `  H
) y )  =  ( ( F `  A ) ( +g  `  H ) ( F `
 x ) ) )
30 oveq1 6088 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y ( +g  `  H
) ( F `  A ) )  =  ( ( F `  x ) ( +g  `  H ) ( F `
 A ) ) )
3129, 30eqeq12d 2450 . . . . 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 5978 . . . 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 643 . . 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 224 . 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 5216 . . . 4  |-  ( F
" S )  C_  ran  F
36 frn 5597 . . . . 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 3359 . . 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 15121 . . 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 889 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    C_ wss 3320   ran crn 4879   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   MndHom cmhm 14736  Cntzccntz 15114
This theorem is referenced by:  cntzmhm2  15138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-mhm 14738  df-cntz 15116
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