Theorem lmhmpreima 20509

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 20492	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod2 20493	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3		lmhmima.y	. . . . 5 ⊢ 𝑌 = (LSubSp‘𝑇)
4	3	lsssubg 20418	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
5	2, 4	sylan 580	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6		ghmpreima 19030	. . 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 20494	. . . . . 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 6033	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
12		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
14	12, 13	lmhmf 20495	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
15	14	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16	11, 15	fssdm 6688	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ⊆ (Base‘𝑆))
17	16	sselda 3944	. . . . . 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 2736	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
20		eqid 2736	. . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
21		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
22	12, 19, 20, 21	lmodvscl 20339	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
23	9, 10, 18, 22	syl3anc 1371	. . . 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 2736	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
26	19, 21, 12, 20, 25	lmhmlin 20496	. . . . . 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 1371	. . . . 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 2736	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
31	19, 30	lmhmsca 20491	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
32	31	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
33	32	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
34	33	eleq2d 2823	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
35	34	biimpar 478	. . . . . . 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 6672	. . . . . . 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 7002	. . . . . . 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 2736	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
42	30, 25, 41, 3	lssvscl 20416	. . . . . 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 2838	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)
45		ffn 6668	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
46		elpreima 7008	. . . . . 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 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)))
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 3197	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
51	8	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑆 ∈ LMod)
52		lmhmima.x	. . . 4 ⊢ 𝑋 = (LSubSp‘𝑆)
53	19, 21, 12, 20, 52	islss4 20423	. . 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ^◡ccnv 5632   “ cima 5636  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  Scalarcsca 17136   ·_𝑠 cvsca 17137  SubGrpcsubg 18922   GrpHom cghm 19005  LModclmod 20322  LSubSpclss 20392   LMHom clmhm 20480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-ghm 19006  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-lss 20393  df-lmhm 20483

This theorem is referenced by:  lmhmlsp  20510  lmhmkerlss  20512  lnmepi  41398