Theorem resmgmhm 41798

Description: Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)

Hypothesis

Ref	Expression
resmgmhm.u	|- U = ( S ↾s X )

Assertion

Ref	Expression
resmgmhm	|- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( F |` X ) e. ( U MgmHom T ) )

Proof of Theorem resmgmhm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmhmrcl 41781	. . . 4 |- ( F e. ( S MgmHom T ) -> ( S e. Mgm /\ T e. Mgm ) )
2	1	simprd 479	. . 3 |- ( F e. ( S MgmHom T ) -> T e. Mgm )
3		resmgmhm.u	. . . 4 |- U = ( S ↾s X )
4	3	submgmmgm 41795	. . 3 |- ( X e. (SubMgm ` S ) -> U e. Mgm )
5	2, 4	anim12ci 591	. 2 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( U e. Mgm /\ T e. Mgm ) )
6		eqid 2622	. . . . . 6 |- ( Base ` S ) = ( Base ` S )
7		eqid 2622	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
8	6, 7	mgmhmf 41784	. . . . 5 |- ( F e. ( S MgmHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
9	6	submgmss 41792	. . . . 5 |- ( X e. (SubMgm ` S ) -> X C_ ( Base ` S ) )
10		fssres 6070	. . . . 5 |- ( ( F : ( Base ` S ) --> ( Base ` T ) /\ X C_ ( Base ` S ) ) -> ( F |` X ) : X --> ( Base ` T ) )
11	8, 9, 10	syl2an 494	. . . 4 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( F |` X ) : X --> ( Base ` T ) )
12	9	adantl 482	. . . . . 6 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> X C_ ( Base ` S ) )
13	3, 6	ressbas2 15931	. . . . . 6 |- ( X C_ ( Base ` S ) -> X = ( Base ` U ) )
14	12, 13	syl 17	. . . . 5 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> X = ( Base ` U ) )
15	14	feq2d 6031	. . . 4 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( ( F |` X ) : X --> ( Base ` T ) <-> ( F |` X ) : ( Base ` U ) --> ( Base ` T ) ) )
16	11, 15	mpbid 222	. . 3 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( F |` X ) : ( Base ` U ) --> ( Base ` T ) )
17		simpll 790	. . . . . . 7 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> F e. ( S MgmHom T ) )
18	9	ad2antlr 763	. . . . . . . 8 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> X C_ ( Base ` S ) )
19		simprl 794	. . . . . . . 8 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
20	18, 19	sseldd 3604	. . . . . . 7 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> x e. ( Base ` S ) )
21		simprr 796	. . . . . . . 8 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
22	18, 21	sseldd 3604	. . . . . . 7 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> y e. ( Base ` S ) )
23		eqid 2622	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
24		eqid 2622	. . . . . . . 8 |- ( +g ` T ) = ( +g ` T )
25	6, 23, 24	mgmhmlin 41786	. . . . . . 7 |- ( ( F e. ( S MgmHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
26	17, 20, 22, 25	syl3anc 1326	. . . . . 6 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
27	23	submgmcl 41794	. . . . . . . . 9 |- ( ( X e. (SubMgm ` S ) /\ x e. X /\ y e. X ) -> ( x ( +g ` S ) y ) e. X )
28	27	3expb 1266	. . . . . . . 8 |- ( ( X e. (SubMgm ` S ) /\ ( x e. X /\ y e. X ) ) -> ( x ( +g ` S ) y ) e. X )
29	28	adantll 750	. . . . . . 7 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( x ( +g ` S ) y ) e. X )
30		fvres 6207	. . . . . . 7 |- ( ( x ( +g ` S ) y ) e. X -> ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
31	29, 30	syl 17	. . . . . 6 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( F ` ( x ( +g ` S ) y ) ) )
32		fvres 6207	. . . . . . . 8 |- ( x e. X -> ( ( F |` X ) ` x ) = ( F ` x ) )
33		fvres 6207	. . . . . . . 8 |- ( y e. X -> ( ( F |` X ) ` y ) = ( F ` y ) )
34	32, 33	oveqan12d 6669	. . . . . . 7 |- ( ( x e. X /\ y e. X ) -> ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
35	34	adantl 482	. . . . . 6 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
36	26, 31, 35	3eqtr4d 2666	. . . . 5 |- ( ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) /\ ( x e. X /\ y e. X ) ) -> ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) )
37	36	ralrimivva 2971	. . . 4 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> A. x e. X A. y e. X ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) )
38	3, 23	ressplusg 15993	. . . . . . . . . 10 |- ( X e. (SubMgm ` S ) -> ( +g ` S ) = ( +g ` U ) )
39	38	adantl 482	. . . . . . . . 9 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( +g ` S ) = ( +g ` U ) )
40	39	oveqd 6667	. . . . . . . 8 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( x ( +g ` S ) y ) = ( x ( +g ` U ) y ) )
41	40	fveq2d 6195	. . . . . . 7 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( F |` X ) ` ( x ( +g ` U ) y ) ) )
42	41	eqeq1d 2624	. . . . . 6 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) <-> ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
43	14, 42	raleqbidv 3152	. . . . 5 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( A. y e. X ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) <-> A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
44	14, 43	raleqbidv 3152	. . . 4 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( A. x e. X A. y e. X ( ( F |` X ) ` ( x ( +g ` S ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) <-> A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
45	37, 44	mpbid 222	. . 3 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) )
46	16, 45	jca 554	. 2 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( ( F |` X ) : ( Base ` U ) --> ( Base ` T ) /\ A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) )
47		eqid 2622	. . 3 |- ( Base ` U ) = ( Base ` U )
48		eqid 2622	. . 3 |- ( +g ` U ) = ( +g ` U )
49	47, 7, 48, 24	ismgmhm 41783	. 2 |- ( ( F |` X ) e. ( U MgmHom T ) <-> ( ( U e. Mgm /\ T e. Mgm ) /\ ( ( F |` X ) : ( Base ` U ) --> ( Base ` T ) /\ A. x e. ( Base ` U ) A. y e. ( Base ` U ) ( ( F |` X ) ` ( x ( +g ` U ) y ) ) = ( ( ( F |` X ) ` x ) ( +g ` T ) ( ( F |` X ) ` y ) ) ) ) )
50	5, 46, 49	sylanbrc 698	1 |- ( ( F e. ( S MgmHom T ) /\ X e. (SubMgm ` S ) ) -> ( F |` X ) e. ( U MgmHom T ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941  Mgmcmgm 17240   MgmHom cmgmhm 41777  SubMgmcsubmgm 41778

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mgm 17242  df-mgmhm 41779  df-submgm 41780

This theorem is referenced by: (None)