Theorem lmhmima 20974

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 20958	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod1 20960	. . . 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 20883	. . . 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 19142	. . 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 2730	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	lmhmf 20961	. . . . . . . . 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 6647	. . . . . . . 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 20862	. . . . . . . 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 6890	. . . . . 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
21		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
22	20, 21	lmhmsca 20957	. . . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
25	24	eleq2d 2815	. . . . . . . . . . . 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 3932	. . . . . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2730	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2730	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	20, 30, 9, 31, 32	lmhmlin 20962	. . . . . . . . . 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 20881	. . . . . . . . . . 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 7162	. . . . . . . . . 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 2830	. . . . . . . 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 7349	. . . . . . . 8 ⊢ ((𝐹‘𝑐) = 𝑏 → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)) = (𝑎( ·𝑠 ‘𝑇)𝑏))
48	47	eleq1d 2814	. . . . . . 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 3131	. . . . 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 3173	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))
54		lmhmlmod2 20959	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
55	54	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑇 ∈ LMod)
56		eqid 2730	. . . 4 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
57		lmhmima.y	. . . 4 ⊢ 𝑌 = (LSubSp‘𝑇)
58	21, 56, 10, 32, 57	islss4 20888	. . 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 1541   ∈ wcel 2110  ∀wral 3045  ∃wrex 3054   ⊆ wss 3900   “ cima 5617   Fn wfn 6472  ⟶wf 6473  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  Scalarcsca 17156   ·_𝑠 cvsca 17157  SubGrpcsubg 19025   GrpHom cghm 19117  LModclmod 20786  LSubSpclss 20857   LMHom clmhm 20946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-0g 17337  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-grp 18841  df-minusg 18842  df-sbg 18843  df-subg 19028  df-ghm 19118  df-mgp 20052  df-ur 20093  df-ring 20146  df-lmod 20788  df-lss 20858  df-lmhm 20949

This theorem is referenced by:  lmhmlsp  20976  lmhmrnlss  20977