Theorem lmhmpreima 20940

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 20923	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod2 20924	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3		lmhmima.y	. . . . 5 ⊢ 𝑌 = (LSubSp‘𝑇)
4	3	lsssubg 20848	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
5	2, 4	sylan 578	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6		ghmpreima 19199	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
7	1, 5, 6	syl2an2r 683	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
8		lmhmlmod1 20925	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
9	8	ad2antrr 724	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑆 ∈ LMod)
10		simprl 769	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
11		cnvimass 6090	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
12		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
14	12, 13	lmhmf 20926	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
15	14	adantr 479	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16	11, 15	fssdm 6747	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ⊆ (Base‘𝑆))
17	16	sselda 3982	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → 𝑏 ∈ (Base‘𝑆))
18	17	adantrl 714	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (Base‘𝑆))
19		eqid 2728	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
20		eqid 2728	. . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
21		eqid 2728	. . . . . 6 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
22	12, 19, 20, 21	lmodvscl 20768	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
23	9, 10, 18, 22	syl3anc 1368	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
24		simpll 765	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
25		eqid 2728	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
26	19, 21, 12, 20, 25	lmhmlin 20927	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
27	24, 10, 18, 26	syl3anc 1368	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
28	2	ad2antrr 724	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑇 ∈ LMod)
29		simplr 767	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑈 ∈ 𝑌)
30		eqid 2728	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
31	19, 30	lmhmsca 20922	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
32	31	adantr 479	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
33	32	fveq2d 6906	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
34	33	eleq2d 2815	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
35	34	biimpar 476	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
36	35	adantrr 715	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
37	15	ffund 6731	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → Fun 𝐹)
38		simprr 771	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (◡𝐹 “ 𝑈))
39		fvimacnvi 7066	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → (𝐹‘𝑏) ∈ 𝑈)
40	37, 38, 39	syl2an2r 683	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘𝑏) ∈ 𝑈)
41		eqid 2728	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
42	30, 25, 41, 3	lssvscl 20846	. . . . . 6 ⊢ (((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹‘𝑏) ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
43	28, 29, 36, 40, 42	syl22anc 837	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
44	27, 43	eqeltrd 2829	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)
45		ffn 6727	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
46		elpreima 7072	. . . . . 6 ⊢ (𝐹 Fn (Base‘𝑆) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
47	15, 45, 46	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)))
48	47	adantr 479	. . . 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)))
49	23, 44, 48	mpbir2and 711	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
50	49	ralrimivva 3198	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
51	8	adantr 479	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑆 ∈ LMod)
52		lmhmima.x	. . . 4 ⊢ 𝑋 = (LSubSp‘𝑆)
53	19, 21, 12, 20, 52	islss4 20853	. . 3 ⊢ (𝑆 ∈ LMod → ((◡𝐹 “ 𝑈) ∈ 𝑋 ↔ ((◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))))
54	51, 53	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))))
55	7, 50, 54	mpbir2and 711	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ^◡ccnv 5681   “ cima 5685  Fun wfun 6547   Fn wfn 6548  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  Scalarcsca 17243   ·_𝑠 cvsca 17244  SubGrpcsubg 19082   GrpHom cghm 19174  LModclmod 20750  LSubSpclss 20822   LMHom clmhm 20911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-ghm 19175  df-mgp 20082  df-ur 20129  df-ring 20182  df-lmod 20752  df-lss 20823  df-lmhm 20914

This theorem is referenced by:  lmhmlsp  20941  lmhmkerlss  20943  lnmepi  42540