Theorem cntzmhm 14173

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.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		eqid 2238	. . . 4 |- ( Base ` H ) = ( Base ` H )
3	1, 2	mhmf 13831	. . 3 |- ( F e. ( G MndHom H ) -> F : ( Base ` G ) --> ( Base ` H ) )
4		cntzmhm.z	. . . . 5 |- Z = (Cntz ` G )
5	1, 4	cntzssv 14160	. . . 4 |- ( Z ` S ) C_ ( Base ` G )
6	5	sseli 3244	. . 3 |- ( A e. ( Z ` S ) -> A e. ( Base ` G ) )
7		ffvelcdm 5844	. . 3 |- ( ( F : ( Base ` G ) --> ( Base ` H ) /\ A e. ( Base ` G ) ) -> ( F ` A ) e. ( Base ` H ) )
8	3, 6, 7	syl2an 289	. 2 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( F ` A ) e. ( Base ` H ) )
9		eqid 2238	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	9, 4	cntzi 14162	. . . . . . 7 |- ( ( A e. ( Z ` S ) /\ x e. S ) -> ( A ( +g ` G ) x ) = ( x ( +g ` G ) A ) )
11	10	adantll 480	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> ( A ( +g ` G ) x ) = ( x ( +g ` G ) A ) )
12	11	fveq2d 5702	. . . . 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 531	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> F e. ( G MndHom H ) )
14	6	ad2antlr 493	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> A e. ( Base ` G ) )
15	1, 4	cntzrcl 14159	. . . . . . . . 9 |- ( A e. ( Z ` S ) -> ( G e. _V /\ S C_ ( Base ` G ) ) )
16	15	adantl 277	. . . . . . . 8 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( G e. _V /\ S C_ ( Base ` G ) ) )
17	16	simprd 114	. . . . . . 7 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> S C_ ( Base ` G ) )
18	17	sselda 3248	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> x e. ( Base ` G ) )
19		eqid 2238	. . . . . . 7 |- ( +g ` H ) = ( +g ` H )
20	1, 9, 19	mhmlin 13833	. . . . . 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 ) ) )
21	13, 14, 18, 20	syl3anc 1278	. . . . 5 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> ( F ` ( A ( +g ` G ) x ) ) = ( ( F ` A ) ( +g ` H ) ( F ` x ) ) )
22	1, 9, 19	mhmlin 13833	. . . . . 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 ) ) )
23	13, 18, 14, 22	syl3anc 1278	. . . . 5 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> ( F ` ( x ( +g ` G ) A ) ) = ( ( F ` x ) ( +g ` H ) ( F ` A ) ) )
24	12, 21, 23	3eqtr3d 2279	. . . 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 ) ) )
25	24	ralrimiva 2623	. . 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 ) ) )
26	3	adantr 276	. . . . 5 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> F : ( Base ` G ) --> ( Base ` H ) )
27	26	ffnd 5537	. . . 4 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> F Fn ( Base ` G ) )
28		oveq2 6096	. . . . . 6 |- ( y = ( F ` x ) -> ( ( F ` A ) ( +g ` H ) y ) = ( ( F ` A ) ( +g ` H ) ( F ` x ) ) )
29		oveq1 6095	. . . . . 6 |- ( y = ( F ` x ) -> ( y ( +g ` H ) ( F ` A ) ) = ( ( F ` x ) ( +g ` H ) ( F ` A ) ) )
30	28, 29	eqeq12d 2253	. . . . 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 ) ) ) )
31	30	ralima 5964	. . . 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 ) ) ) )
32	27, 17, 31	syl2anc 415	. . 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 ) ) ) )
33	25, 32	mpbird 167	. 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 ) ) )
34		imassrn 5140	. . . 4 |- ( F " S ) C_ ran F
35	26	frnd 5546	. . . 4 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ran F C_ ( Base ` H ) )
36	34, 35	sstrid 3259	. . 3 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( F " S ) C_ ( Base ` H ) )
37		cntzmhm.y	. . . 4 |- Y = (Cntz ` H )
38	2, 19, 37	elcntz 14154	. . 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 ) ) ) ) )
39	36, 38	syl 14	. 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 ) ) ) ) )
40	8, 33, 39	mpbir2and 957	1 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( F ` A ) e. ( Y ` ( F " S ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   ran crn 4777   "cima 4779   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   MndHom cmhm 13823  Cntzccntz 14146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-map 6928  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416  df-mhm 13825  df-cntz 14148

This theorem is used by:  cntzmhm2  14174