Theorem lmhmpreima 19048

Description: The inverse 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
lmhmpreima	⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋)

Proof of Theorem lmhmpreima

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

Step	Hyp	Ref	Expression
1		lmghm 19031	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2	1	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3		lmhmlmod2 19032	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
4		lmhmima.y	. . . . 5 ⊢ 𝑌 = (LSubSp‘𝑇)
5	4	lsssubg 18957	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6	3, 5	sylan 488	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
7		ghmpreima 17682	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
8	2, 6, 7	syl2anc 693	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
9		lmhmlmod1 19033	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	ad2antrr 762	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑆 ∈ LMod)
11		simprl 794	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
12		cnvimass 5485	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
13		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘𝑇) = (Base‘𝑇)
15	13, 14	lmhmf 19034	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16	15	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
17		fdm 6051	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → dom 𝐹 = (Base‘𝑆))
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → dom 𝐹 = (Base‘𝑆))
19	12, 18	syl5sseq 3653	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ⊆ (Base‘𝑆))
20	19	sselda 3603	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → 𝑏 ∈ (Base‘𝑆))
21	20	adantrl 752	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (Base‘𝑆))
22		eqid 2622	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
23		eqid 2622	. . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
24		eqid 2622	. . . . . 6 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
25	13, 22, 23, 24	lmodvscl 18880	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
26	10, 11, 21, 25	syl3anc 1326	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
27		simpll 790	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
28		eqid 2622	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
29	22, 24, 13, 23, 28	lmhmlin 19035	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
30	27, 11, 21, 29	syl3anc 1326	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
31	3	ad2antrr 762	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑇 ∈ LMod)
32		simplr 792	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑈 ∈ 𝑌)
33		eqid 2622	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
34	22, 33	lmhmsca 19030	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
35	34	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
36	35	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
37	36	eleq2d 2687	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
38	37	biimpar 502	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
39	38	adantrr 753	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
40		ffun 6048	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Fun 𝐹)
41	16, 40	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → Fun 𝐹)
42	41	adantr 481	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → Fun 𝐹)
43		simprr 796	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (◡𝐹 “ 𝑈))
44		fvimacnvi 6331	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → (𝐹‘𝑏) ∈ 𝑈)
45	42, 43, 44	syl2anc 693	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘𝑏) ∈ 𝑈)
46		eqid 2622	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
47	33, 28, 46, 4	lssvscl 18955	. . . . . 6 ⊢ (((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹‘𝑏) ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
48	31, 32, 39, 45, 47	syl22anc 1327	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
49	30, 48	eqeltrd 2701	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)
50		ffn 6045	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
51		elpreima 6337	. . . . . 6 ⊢ (𝐹 Fn (Base‘𝑆) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
52	16, 50, 51	3syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
53	52	adantr 481	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
54	26, 49, 53	mpbir2and 957	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
55	54	ralrimivva 2971	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
56	9	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑆 ∈ LMod)
57		lmhmima.x	. . . 4 ⊢ 𝑋 = (LSubSp‘𝑆)
58	22, 24, 13, 23, 57	islss4 18962	. . 3 ⊢ (𝑆 ∈ LMod → ((◡𝐹 “ 𝑈) ∈ 𝑋 ↔ ((◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))))
59	56, 58	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ((◡𝐹 “ 𝑈) ∈ 𝑋 ↔ ((◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))))
60	8, 55, 59	mpbir2and 957	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 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  lmhmkerlss  19051  lnmepi  37655