Theorem lmhmima 20969

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 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (𝐹 “ 𝑈) ∈ 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 20953	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod1 20955	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
3		simpr 484	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑈 ∈ 𝑋)
4		lmhmima.x	. . . . 5 ⊢ 𝑋 = (LSubSp‘𝑆)
5	4	lsssubg 20878	. . . 4 ⊢ ((𝑆 ∈ LMod ∧ 𝑈 ∈ 𝑋) → 𝑈 ∈ (SubGrp‘𝑆))
6	2, 3, 5	syl2an2r 685	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑈 ∈ (SubGrp‘𝑆))
7		ghmima 19134	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
8	1, 6, 7	syl2an2r 685	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇))
9		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	lmhmf 20956	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
13		ffn 6656	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
14	12, 13	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝐹 Fn (Base‘𝑆))
15	9, 4	lssss 20857	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑋 → 𝑈 ⊆ (Base‘𝑆))
16	3, 15	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑈 ⊆ (Base‘𝑆))
17	14, 16	fvelimabd 6900	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (𝑏 ∈ (𝐹 “ 𝑈) ↔ ∃𝑐 ∈ 𝑈 (𝐹‘𝑐) = 𝑏))
18	17	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (𝑏 ∈ (𝐹 “ 𝑈) ↔ ∃𝑐 ∈ 𝑈 (𝐹‘𝑐) = 𝑏))
19		simpll 766	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
21		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
22	20, 21	lmhmsca 20952	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
23	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (Scalar‘𝑇) = (Scalar‘𝑆))
24	23	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
25	24	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
26	25	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
27	26	adantrr 717	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
28	16	sselda 3937	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑐 ∈ 𝑈) → 𝑐 ∈ (Base‘𝑆))
29	28	adantrl 716	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑐 ∈ (Base‘𝑆))
30		eqid 2729	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2729	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2729	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	20, 30, 9, 31, 32	lmhmlin 20957	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑐)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)))
34	19, 27, 29, 33	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑐)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)))
35	19, 11, 13	3syl 18	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝐹 Fn (Base‘𝑆))
36		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑈 ∈ 𝑋)
37	36, 15	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝑆))
38	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑆 ∈ LMod)
39	38	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑆 ∈ LMod)
40		simprr 772	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑐 ∈ 𝑈)
41	20, 31, 30, 4	lssvscl 20876	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ LMod ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐 ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑆)𝑐) ∈ 𝑈)
42	39, 36, 27, 40, 41	syl22anc 838	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑆)𝑐) ∈ 𝑈)
43		fnfvima 7173	. . . . . . . . . 10 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆) ∧ (𝑎( ·𝑠 ‘𝑆)𝑐) ∈ 𝑈) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑐)) ∈ (𝐹 “ 𝑈))
44	35, 37, 42, 43	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑐)) ∈ (𝐹 “ 𝑈))
45	34, 44	eqeltrrd 2829	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)) ∈ (𝐹 “ 𝑈))
46	45	anassrs 467	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) ∧ 𝑐 ∈ 𝑈) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)) ∈ (𝐹 “ 𝑈))
47		oveq2 7361	. . . . . . . 8 ⊢ ((𝐹‘𝑐) = 𝑏 → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)) = (𝑎( ·𝑠 ‘𝑇)𝑏))
48	47	eleq1d 2813	. . . . . . 7 ⊢ ((𝐹‘𝑐) = 𝑏 → ((𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)) ∈ (𝐹 “ 𝑈) ↔ (𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈)))
49	46, 48	syl5ibcom 245	. . . . . 6 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) ∧ 𝑐 ∈ 𝑈) → ((𝐹‘𝑐) = 𝑏 → (𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈)))
50	49	rexlimdva 3130	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (∃𝑐 ∈ 𝑈 (𝐹‘𝑐) = 𝑏 → (𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈)))
51	18, 50	sylbid 240	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (𝑏 ∈ (𝐹 “ 𝑈) → (𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈)))
52	51	impr 454	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ (𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))
53	52	ralrimivva 3172	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))
54		lmhmlmod2 20954	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
55	54	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑇 ∈ LMod)
56		eqid 2729	. . . 4 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
57		lmhmima.y	. . . 4 ⊢ 𝑌 = (LSubSp‘𝑇)
58	21, 56, 10, 32, 57	islss4 20883	. . 3 ⊢ (𝑇 ∈ LMod → ((𝐹 “ 𝑈) ∈ 𝑌 ↔ ((𝐹 “ 𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))))
59	55, 58	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → ((𝐹 “ 𝑈) ∈ 𝑌 ↔ ((𝐹 “ 𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))))
60	8, 53, 59	mpbir2and 713	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (𝐹 “ 𝑈) ∈ 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3905   “ cima 5626   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Scalarcsca 17182   ·_𝑠 cvsca 17183  SubGrpcsubg 19017   GrpHom cghm 19109  LModclmod 20781  LSubSpclss 20852   LMHom clmhm 20941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-minusg 18834  df-sbg 18835  df-subg 19020  df-ghm 19110  df-mgp 20044  df-ur 20085  df-ring 20138  df-lmod 20783  df-lss 20853  df-lmhm 20944

This theorem is referenced by:  lmhmlsp  20971  lmhmrnlss  20972