Theorem cntzmhm 17771

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 2622	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		eqid 2622	. . . 4 |- ( Base ` H ) = ( Base ` H )
3	1, 2	mhmf 17340	. . 3 |- ( F e. ( G MndHom H ) -> F : ( Base ` G ) --> ( Base ` H ) )
4		cntzmhm.z	. . . . 5 |- Z = (Cntz ` G )
5	1, 4	cntzssv 17761	. . . 4 |- ( Z ` S ) C_ ( Base ` G )
6	5	sseli 3599	. . 3 |- ( A e. ( Z ` S ) -> A e. ( Base ` G ) )
7		ffvelrn 6357	. . 3 |- ( ( F : ( Base ` G ) --> ( Base ` H ) /\ A e. ( Base ` G ) ) -> ( F ` A ) e. ( Base ` H ) )
8	3, 6, 7	syl2an 494	. 2 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( F ` A ) e. ( Base ` H ) )
9		eqid 2622	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	9, 4	cntzi 17762	. . . . . . 7 |- ( ( A e. ( Z ` S ) /\ x e. S ) -> ( A ( +g ` G ) x ) = ( x ( +g ` G ) A ) )
11	10	adantll 750	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> ( A ( +g ` G ) x ) = ( x ( +g ` G ) A ) )
12	11	fveq2d 6195	. . . . 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 790	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> F e. ( G MndHom H ) )
14	6	ad2antlr 763	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> A e. ( Base ` G ) )
15	1, 4	cntzrcl 17760	. . . . . . . . 9 |- ( A e. ( Z ` S ) -> ( G e. _V /\ S C_ ( Base ` G ) ) )
16	15	adantl 482	. . . . . . . 8 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( G e. _V /\ S C_ ( Base ` G ) ) )
17	16	simprd 479	. . . . . . 7 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> S C_ ( Base ` G ) )
18	17	sselda 3603	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) /\ x e. S ) -> x e. ( Base ` G ) )
19		eqid 2622	. . . . . . 7 |- ( +g ` H ) = ( +g ` H )
20	1, 9, 19	mhmlin 17342	. . . . . 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 1326	. . . . 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 17342	. . . . . 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 1326	. . . . 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 2664	. . . 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 2966	. . 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 481	. . . . 5 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> F : ( Base ` G ) --> ( Base ` H ) )
27		ffn 6045	. . . . 5 |- ( F : ( Base ` G ) --> ( Base ` H ) -> F Fn ( Base ` G ) )
28	26, 27	syl 17	. . . 4 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> F Fn ( Base ` G ) )
29		oveq2 6658	. . . . . 6 |- ( y = ( F ` x ) -> ( ( F ` A ) ( +g ` H ) y ) = ( ( F ` A ) ( +g ` H ) ( F ` x ) ) )
30		oveq1 6657	. . . . . 6 |- ( y = ( F ` x ) -> ( y ( +g ` H ) ( F ` A ) ) = ( ( F ` x ) ( +g ` H ) ( F ` A ) ) )
31	29, 30	eqeq12d 2637	. . . . 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 ) ) ) )
32	31	ralima 6498	. . . 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 ) ) ) )
33	28, 17, 32	syl2anc 693	. . 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 ) ) ) )
34	25, 33	mpbird 247	. 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 5477	. . . 4 |- ( F " S ) C_ ran F
36		frn 6053	. . . . 5 |- ( F : ( Base ` G ) --> ( Base ` H ) -> ran F C_ ( Base ` H ) )
37	26, 36	syl 17	. . . 4 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ran F C_ ( Base ` H ) )
38	35, 37	syl5ss 3614	. . 3 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( F " S ) C_ ( Base ` H ) )
39		cntzmhm.y	. . . 4 |- Y = (Cntz ` H )
40	2, 19, 39	elcntz 17755	. . 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 ) ) ) ) )
41	38, 40	syl 17	. 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 ) ) ) ) )
42	8, 34, 41	mpbir2and 957	1 |- ( ( F e. ( G MndHom H ) /\ A e. ( Z ` S ) ) -> ( F ` A ) e. ( Y ` ( F " S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   MndHom cmhm 17333  Cntzccntz 17748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-mhm 17335  df-cntz 17750

This theorem is referenced by:  cntzmhm2  17772