Theorem lmhmima 21064

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 21048	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod1 21050	. . . 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 20973	. . . 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 19268	. . 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 2735	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	lmhmf 21051	. . . . . . . . 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 6737	. . . . . . . 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 20952	. . . . . . . 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 6982	. . . . . 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 767	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
21		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
22	20, 21	lmhmsca 21047	. . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
25	24	eleq2d 2825	. . . . . . . . . . . 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 3995	. . . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2735	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2735	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	20, 30, 9, 31, 32	lmhmlin 21052	. . . . . . . . . 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 1370	. . . . . . . . 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 769	. . . . . . . . . . 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 773	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → 𝑐 ∈ 𝑈)
41	20, 31, 30, 4	lssvscl 20971	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ LMod ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐 ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑆)𝑐) ∈ 𝑈)
42	39, 36, 27, 40, 41	syl22anc 839	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑆)𝑐) ∈ 𝑈)
43		fnfvima 7253	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐 ∈ 𝑈)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑐)) ∈ (𝐹 “ 𝑈))
45	34, 44	eqeltrrd 2840	. . . . . . . 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 7439	. . . . . . . 8 ⊢ ((𝐹‘𝑐) = 𝑏 → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑐)) = (𝑎( ·𝑠 ‘𝑇)𝑏))
48	47	eleq1d 2824	. . . . . . 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 3153	. . . . 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 3200	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))
54		lmhmlmod2 21049	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
55	54	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑇 ∈ LMod)
56		eqid 2735	. . . 4 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
57		lmhmima.y	. . . 4 ⊢ 𝑌 = (LSubSp‘𝑇)
58	21, 56, 10, 32, 57	islss4 20978	. . 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963   “ cima 5692   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·_𝑠 cvsca 17302  SubGrpcsubg 19151   GrpHom cghm 19243  LModclmod 20875  LSubSpclss 20947   LMHom clmhm 21036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877  df-lss 20948  df-lmhm 21039

This theorem is referenced by:  lmhmlsp  21066  lmhmrnlss  21067