Theorem lmhmima 19047

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

Hypotheses

Ref	Expression
lmhmima.x	|- X = ( LSubSp ` S )
lmhmima.y	|- Y = ( LSubSp ` T )

Assertion

Ref	Expression
lmhmima	|- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( F " U ) e. Y )

Proof of Theorem lmhmima

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

Step	Hyp	Ref	Expression
1		lmghm 19031	. . . 4 |- ( F e. ( S LMHom T ) -> F e. ( S GrpHom T ) )
2	1	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> F e. ( S GrpHom T ) )
3		lmhmlmod1 19033	. . . . 5 |- ( F e. ( S LMHom T ) -> S e. LMod )
4	3	adantr 481	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> S e. LMod )
5		simpr 477	. . . 4 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> U e. X )
6		lmhmima.x	. . . . 5 |- X = ( LSubSp ` S )
7	6	lsssubg 18957	. . . 4 |- ( ( S e. LMod /\ U e. X ) -> U e. (SubGrp ` S ) )
8	4, 5, 7	syl2anc 693	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> U e. (SubGrp ` S ) )
9		ghmima 17681	. . 3 |- ( ( F e. ( S GrpHom T ) /\ U e. (SubGrp ` S ) ) -> ( F " U ) e. (SubGrp ` T ) )
10	2, 8, 9	syl2anc 693	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( F " U ) e. (SubGrp ` T ) )
11		eqid 2622	. . . . . . . . . 10 |- ( Base ` S ) = ( Base ` S )
12		eqid 2622	. . . . . . . . . 10 |- ( Base ` T ) = ( Base ` T )
13	11, 12	lmhmf 19034	. . . . . . . . 9 |- ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
14	13	adantr 481	. . . . . . . 8 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> F : ( Base ` S ) --> ( Base ` T ) )
15		ffn 6045	. . . . . . . 8 |- ( F : ( Base ` S ) --> ( Base ` T ) -> F Fn ( Base ` S ) )
16	14, 15	syl 17	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> F Fn ( Base ` S ) )
17	11, 6	lssss 18937	. . . . . . . 8 |- ( U e. X -> U C_ ( Base ` S ) )
18	5, 17	syl 17	. . . . . . 7 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> U C_ ( Base ` S ) )
19		fvelimab 6253	. . . . . . 7 |- ( ( F Fn ( Base ` S ) /\ U C_ ( Base ` S ) ) -> ( b e. ( F " U ) <-> E. c e. U ( F ` c ) = b ) )
20	16, 18, 19	syl2anc 693	. . . . . 6 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( b e. ( F " U ) <-> E. c e. U ( F ` c ) = b ) )
21	20	adantr 481	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ a e. ( Base ` (Scalar ` T ) ) ) -> ( b e. ( F " U ) <-> E. c e. U ( F ` c ) = b ) )
22		simpll 790	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> F e. ( S LMHom T ) )
23		eqid 2622	. . . . . . . . . . . . . . . 16 |- (Scalar ` S ) = (Scalar ` S )
24		eqid 2622	. . . . . . . . . . . . . . . 16 |- (Scalar ` T ) = (Scalar ` T )
25	23, 24	lmhmsca 19030	. . . . . . . . . . . . . . 15 |- ( F e. ( S LMHom T ) -> (Scalar ` T ) = (Scalar ` S ) )
26	25	adantr 481	. . . . . . . . . . . . . 14 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> (Scalar ` T ) = (Scalar ` S ) )
27	26	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( Base ` (Scalar ` T ) ) = ( Base ` (Scalar ` S ) ) )
28	27	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( a e. ( Base ` (Scalar ` T ) ) <-> a e. ( Base ` (Scalar ` S ) ) ) )
29	28	biimpa 501	. . . . . . . . . . 11 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ a e. ( Base ` (Scalar ` T ) ) ) -> a e. ( Base ` (Scalar ` S ) ) )
30	29	adantrr 753	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> a e. ( Base ` (Scalar ` S ) ) )
31	18	sselda 3603	. . . . . . . . . . 11 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ c e. U ) -> c e. ( Base ` S ) )
32	31	adantrl 752	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> c e. ( Base ` S ) )
33		eqid 2622	. . . . . . . . . . 11 |- ( Base ` (Scalar ` S ) ) = ( Base ` (Scalar ` S ) )
34		eqid 2622	. . . . . . . . . . 11 |- ( .s ` S ) = ( .s ` S )
35		eqid 2622	. . . . . . . . . . 11 |- ( .s ` T ) = ( .s ` T )
36	23, 33, 11, 34, 35	lmhmlin 19035	. . . . . . . . . 10 |- ( ( F e. ( S LMHom T ) /\ a e. ( Base ` (Scalar ` S ) ) /\ c e. ( Base ` S ) ) -> ( F ` ( a ( .s ` S ) c ) ) = ( a ( .s ` T ) ( F ` c ) ) )
37	22, 30, 32, 36	syl3anc 1326	. . . . . . . . 9 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> ( F ` ( a ( .s ` S ) c ) ) = ( a ( .s ` T ) ( F ` c ) ) )
38	22, 13, 15	3syl 18	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> F Fn ( Base ` S ) )
39		simplr 792	. . . . . . . . . . 11 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> U e. X )
40	39, 17	syl 17	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> U C_ ( Base ` S ) )
41	4	adantr 481	. . . . . . . . . . 11 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> S e. LMod )
42		simprr 796	. . . . . . . . . . 11 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> c e. U )
43	23, 34, 33, 6	lssvscl 18955	. . . . . . . . . . 11 |- ( ( ( S e. LMod /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` S ) ) /\ c e. U ) ) -> ( a ( .s ` S ) c ) e. U )
44	41, 39, 30, 42, 43	syl22anc 1327	. . . . . . . . . 10 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> ( a ( .s ` S ) c ) e. U )
45		fnfvima 6496	. . . . . . . . . 10 |- ( ( F Fn ( Base ` S ) /\ U C_ ( Base ` S ) /\ ( a ( .s ` S ) c ) e. U ) -> ( F ` ( a ( .s ` S ) c ) ) e. ( F " U ) )
46	38, 40, 44, 45	syl3anc 1326	. . . . . . . . 9 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> ( F ` ( a ( .s ` S ) c ) ) e. ( F " U ) )
47	37, 46	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ c e. U ) ) -> ( a ( .s ` T ) ( F ` c ) ) e. ( F " U ) )
48	47	anassrs 680	. . . . . . 7 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ a e. ( Base ` (Scalar ` T ) ) ) /\ c e. U ) -> ( a ( .s ` T ) ( F ` c ) ) e. ( F " U ) )
49		oveq2 6658	. . . . . . . 8 |- ( ( F ` c ) = b -> ( a ( .s ` T ) ( F ` c ) ) = ( a ( .s ` T ) b ) )
50	49	eleq1d 2686	. . . . . . 7 |- ( ( F ` c ) = b -> ( ( a ( .s ` T ) ( F ` c ) ) e. ( F " U ) <-> ( a ( .s ` T ) b ) e. ( F " U ) ) )
51	48, 50	syl5ibcom 235	. . . . . 6 |- ( ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ a e. ( Base ` (Scalar ` T ) ) ) /\ c e. U ) -> ( ( F ` c ) = b -> ( a ( .s ` T ) b ) e. ( F " U ) ) )
52	51	rexlimdva 3031	. . . . 5 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ a e. ( Base ` (Scalar ` T ) ) ) -> ( E. c e. U ( F ` c ) = b -> ( a ( .s ` T ) b ) e. ( F " U ) ) )
53	21, 52	sylbid 230	. . . 4 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ a e. ( Base ` (Scalar ` T ) ) ) -> ( b e. ( F " U ) -> ( a ( .s ` T ) b ) e. ( F " U ) ) )
54	53	impr 649	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ U e. X ) /\ ( a e. ( Base ` (Scalar ` T ) ) /\ b e. ( F " U ) ) ) -> ( a ( .s ` T ) b ) e. ( F " U ) )
55	54	ralrimivva 2971	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> A. a e. ( Base ` (Scalar ` T ) ) A. b e. ( F " U ) ( a ( .s ` T ) b ) e. ( F " U ) )
56		lmhmlmod2 19032	. . . 4 |- ( F e. ( S LMHom T ) -> T e. LMod )
57	56	adantr 481	. . 3 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> T e. LMod )
58		eqid 2622	. . . 4 |- ( Base ` (Scalar ` T ) ) = ( Base ` (Scalar ` T ) )
59		lmhmima.y	. . . 4 |- Y = ( LSubSp ` T )
60	24, 58, 12, 35, 59	islss4 18962	. . 3 |- ( T e. LMod -> ( ( F " U ) e. Y <-> ( ( F " U ) e. (SubGrp ` T ) /\ A. a e. ( Base ` (Scalar ` T ) ) A. b e. ( F " U ) ( a ( .s ` T ) b ) e. ( F " U ) ) ) )
61	57, 60	syl 17	. 2 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( ( F " U ) e. Y <-> ( ( F " U ) e. (SubGrp ` T ) /\ A. a e. ( Base ` (Scalar ` T ) ) A. b e. ( F " U ) ( a ( .s ` T ) b ) e. ( F " U ) ) ) )
62	10, 55, 61	mpbir2and 957	1 |- ( ( F e. ( S LMHom T ) /\ U e. X ) -> ( F " U ) e. Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  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:  lmhmlsp  19049  lmhmrnlss  19050