Theorem reslmhm 19052

Description: Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
reslmhm.u	|- U = ( LSubSp ` S )
reslmhm.r	|- R = ( S ↾s X )

Assertion

Ref	Expression
reslmhm	|- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( F |` X ) e. ( R LMHom T ) )

Proof of Theorem reslmhm

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlmod1 19033	. . . 4 |- ( F e. ( S LMHom T ) -> S e. LMod )
2		reslmhm.r	. . . . 5 |- R = ( S ↾s X )
3		reslmhm.u	. . . . 5 |- U = ( LSubSp ` S )
4	2, 3	lsslmod 18960	. . . 4 |- ( ( S e. LMod /\ X e. U ) -> R e. LMod )
5	1, 4	sylan 488	. . 3 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> R e. LMod )
6		lmhmlmod2 19032	. . . 4 |- ( F e. ( S LMHom T ) -> T e. LMod )
7	6	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> T e. LMod )
8	5, 7	jca 554	. 2 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( R e. LMod /\ T e. LMod ) )
9		lmghm 19031	. . . . 5 |- ( F e. ( S LMHom T ) -> F e. ( S GrpHom T ) )
10	9	adantr 481	. . . 4 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> F e. ( S GrpHom T ) )
11	3	lsssubg 18957	. . . . 5 |- ( ( S e. LMod /\ X e. U ) -> X e. (SubGrp ` S ) )
12	1, 11	sylan 488	. . . 4 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> X e. (SubGrp ` S ) )
13	2	resghm 17676	. . . 4 |- ( ( F e. ( S GrpHom T ) /\ X e. (SubGrp ` S ) ) -> ( F |` X ) e. ( R GrpHom T ) )
14	10, 12, 13	syl2anc 693	. . 3 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( F |` X ) e. ( R GrpHom T ) )
15		eqid 2622	. . . . 5 |- (Scalar ` S ) = (Scalar ` S )
16		eqid 2622	. . . . 5 |- (Scalar ` T ) = (Scalar ` T )
17	15, 16	lmhmsca 19030	. . . 4 |- ( F e. ( S LMHom T ) -> (Scalar ` T ) = (Scalar ` S ) )
18	2, 15	resssca 16031	. . . 4 |- ( X e. U -> (Scalar ` S ) = (Scalar ` R ) )
19	17, 18	sylan9eq 2676	. . 3 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> (Scalar ` T ) = (Scalar ` R ) )
20		simpll 790	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> F e. ( S LMHom T ) )
21		simprl 794	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> a e. ( Base ` (Scalar ` S ) ) )
22		eqid 2622	. . . . . . . . . . 11 |- ( Base ` S ) = ( Base ` S )
23	22, 3	lssss 18937	. . . . . . . . . 10 |- ( X e. U -> X C_ ( Base ` S ) )
24	23	adantl 482	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> X C_ ( Base ` S ) )
25	24	adantr 481	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> X C_ ( Base ` S ) )
26	2, 22	ressbas2 15931	. . . . . . . . . . . 12 |- ( X C_ ( Base ` S ) -> X = ( Base ` R ) )
27	24, 26	syl 17	. . . . . . . . . . 11 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> X = ( Base ` R ) )
28	27	eleq2d 2687	. . . . . . . . . 10 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( b e. X <-> b e. ( Base ` R ) ) )
29	28	biimpar 502	. . . . . . . . 9 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ b e. ( Base ` R ) ) -> b e. X )
30	29	adantrl 752	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> b e. X )
31	25, 30	sseldd 3604	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> b e. ( Base ` S ) )
32		eqid 2622	. . . . . . . 8 |- ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` S ) )
33		eqid 2622	. . . . . . . 8 |- ( .s ` S ) = ( .s ` S )
34		eqid 2622	. . . . . . . 8 |- ( .s ` T ) = ( .s ` T )
35	15, 32, 22, 33, 34	lmhmlin 19035	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` S ) ) -> ( F ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( F ` b ) ) )
36	20, 21, 31, 35	syl3anc 1326	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> ( F ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( F ` b ) ) )
37	1	adantr 481	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> S e. LMod )
38	37	adantr 481	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> S e. LMod )
39		simplr 792	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> X e. U )
40	15, 33, 32, 3	lssvscl 18955	. . . . . . . 8 |- ( ( ( S e. LMod /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. X ) ) -> ( a ( .s ` S ) b ) e. X )
41	38, 39, 21, 30, 40	syl22anc 1327	. . . . . . 7 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> ( a ( .s ` S ) b ) e. X )
42		fvres 6207	. . . . . . 7 |- ( ( a ( .s ` S ) b ) e. X -> ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( F ` ( a ( .s ` S ) b ) ) )
43	41, 42	syl 17	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( F ` ( a ( .s ` S ) b ) ) )
44		fvres 6207	. . . . . . . 8 |- ( b e. X -> ( ( F |` X ) ` b ) = ( F ` b ) )
45	44	oveq2d 6666	. . . . . . 7 |- ( b e. X -> ( a ( .s ` T ) ( ( F |` X ) ` b ) ) = ( a ( .s ` T ) ( F ` b ) ) )
46	30, 45	syl 17	. . . . . 6 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> ( a ( .s ` T ) ( ( F |` X ) ` b ) ) = ( a ( .s ` T ) ( F ` b ) ) )
47	36, 43, 46	3eqtr4d 2666	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ X e. U ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ b e. ( Base ` R ) ) ) -> ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) )
48	47	ralrimivva 2971	. . . 4 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> A. a e. ( Base ` (Scalar ` S ) ) A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) )
49	18	adantl 482	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> (Scalar ` S ) = (Scalar ` R ) )
50	49	fveq2d 6195	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` R ) ) )
51	2, 33	ressvsca 16032	. . . . . . . . . 10 |- ( X e. U -> ( .s ` S ) = ( .s ` R ) )
52	51	adantl 482	. . . . . . . . 9 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( .s ` S ) = ( .s ` R ) )
53	52	oveqd 6667	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( a ( .s ` S ) b ) = ( a ( .s ` R ) b ) )
54	53	fveq2d 6195	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( ( F |` X ) ` ( a ( .s ` R ) b ) ) )
55	54	eqeq1d 2624	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) <-> ( ( F |` X ) ` ( a ( .s ` R ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) ) )
56	55	ralbidv 2986	. . . . 5 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) <-> A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` R ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) ) )
57	50, 56	raleqbidv 3152	. . . 4 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( A. a e. ( Base ` (Scalar ` S ) ) A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) <-> A. a e. ( Base ` (Scalar ` R ) ) A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` R ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) ) )
58	48, 57	mpbid 222	. . 3 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> A. a e. ( Base ` (Scalar ` R ) ) A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` R ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) )
59	14, 19, 58	3jca 1242	. 2 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( ( F |` X ) e. ( R GrpHom T ) /\ (Scalar ` T ) = (Scalar ` R ) /\ A. a e. ( Base ` (Scalar ` R ) ) A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` R ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) ) )
60		eqid 2622	. . 3 |- (Scalar ` R ) = (Scalar ` R )
61		eqid 2622	. . 3 |- ( Base ` (Scalar ` R ) ) = ( Base ` (Scalar ` R ) )
62		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
63		eqid 2622	. . 3 |- ( .s ` R ) = ( .s ` R )
64	60, 16, 61, 62, 63, 34	islmhm 19027	. 2 |- ( ( F |` X ) e. ( R LMHom T ) <-> ( ( R e. LMod /\ T e. LMod ) /\ ( ( F |` X ) e. ( R GrpHom T ) /\ (Scalar ` T ) = (Scalar ` R ) /\ A. a e. ( Base ` (Scalar ` R ) ) A. b e. ( Base ` R ) ( ( F |` X ) ` ( a ( .s ` R ) b ) ) = ( a ( .s ` T ) ( ( F |` X ) ` b ) ) ) ) )
65	8, 59, 64	sylanbrc 698	1 |- ( ( F e. ( S LMHom T ) /\ X e. U ) -> ( F |` X ) e. ( R LMHom T ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   .scvsca 15945  SubGrpcsubg 17588   GrpHom cghm 17657   LModclmod 18863   LSubSpclss 18932   LMHom clmhm 19019

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  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-rmo 2920  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lmhm 19022

This theorem is referenced by:  frlmsplit2  20112  lmhmlnmsplit  37657  pwssplit4  37659