Theorem lmhmima 21046

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 21030	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod1 21032	. . . 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 20955	. . . 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 19255	. . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
11	9, 10	lmhmf 21033	. . . . . . . . 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 6736	. . . . . . . 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 20934	. . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
21		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
22	20, 21	lmhmsca 21029	. . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
25	24	eleq2d 2827	. . . . . . . . . . . 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 3983	. . . . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31		eqid 2737	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
32		eqid 2737	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
33	20, 30, 9, 31, 32	lmhmlin 21034	. . . . . . . . . 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 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 20953	. . . . . . . . . . 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 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 2842	. . . . . . . 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 2826	. . . . . . 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 3155	. . . . 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 3202	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑇)𝑏) ∈ (𝐹 “ 𝑈))
54		lmhmlmod2 21031	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
55	54	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → 𝑇 ∈ LMod)
56		eqid 2737	. . . 4 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
57		lmhmima.y	. . . 4 ⊢ 𝑌 = (LSubSp‘𝑇)
58	21, 56, 10, 32, 57	islss4 20960	. . 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 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951   “ cima 5688   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·_𝑠 cvsca 17301  SubGrpcsubg 19138   GrpHom cghm 19230  LModclmod 20858  LSubSpclss 20929   LMHom clmhm 21018

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-mgp 20138  df-ur 20179  df-ring 20232  df-lmod 20860  df-lss 20930  df-lmhm 21021

This theorem is referenced by:  lmhmlsp  21048  lmhmrnlss  21049