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Theorem lmhmpreima 19271
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) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)

Proof of Theorem lmhmpreima
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 19254 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21adantr 472 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3 lmhmlmod2 19255 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
4 lmhmima.y . . . . 5 𝑌 = (LSubSp‘𝑇)
54lsssubg 19180 . . . 4 ((𝑇 ∈ LMod ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
63, 5sylan 489 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
7 ghmpreima 17904 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
82, 6, 7syl2anc 696 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
9 lmhmlmod1 19256 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109ad2antrr 764 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑆 ∈ LMod)
11 simprl 811 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
12 cnvimass 5644 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
13 eqid 2761 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2761 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
1513, 14lmhmf 19257 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1615adantr 472 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
17 fdm 6213 . . . . . . . . 9 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → dom 𝐹 = (Base‘𝑆))
1816, 17syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → dom 𝐹 = (Base‘𝑆))
1912, 18syl5sseq 3795 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ⊆ (Base‘𝑆))
2019sselda 3745 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑏 ∈ (𝐹𝑈)) → 𝑏 ∈ (Base‘𝑆))
2120adantrl 754 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (Base‘𝑆))
22 eqid 2761 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
23 eqid 2761 . . . . . 6 ( ·𝑠𝑆) = ( ·𝑠𝑆)
24 eqid 2761 . . . . . 6 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2513, 22, 23, 24lmodvscl 19103 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
2610, 11, 21, 25syl3anc 1477 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
27 simpll 807 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
28 eqid 2761 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2922, 24, 13, 23, 28lmhmlin 19258 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3027, 11, 21, 29syl3anc 1477 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
313ad2antrr 764 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑇 ∈ LMod)
32 simplr 809 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑈𝑌)
33 eqid 2761 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
3422, 33lmhmsca 19253 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
3534adantr 472 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
3635fveq2d 6358 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
3736eleq2d 2826 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
3837biimpar 503 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3938adantrr 755 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
40 ffun 6210 . . . . . . . . 9 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Fun 𝐹)
4116, 40syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → Fun 𝐹)
4241adantr 472 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → Fun 𝐹)
43 simprr 813 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (𝐹𝑈))
44 fvimacnvi 6496 . . . . . . 7 ((Fun 𝐹𝑏 ∈ (𝐹𝑈)) → (𝐹𝑏) ∈ 𝑈)
4542, 43, 44syl2anc 696 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹𝑏) ∈ 𝑈)
46 eqid 2761 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
4733, 28, 46, 4lssvscl 19178 . . . . . 6 (((𝑇 ∈ LMod ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹𝑏) ∈ 𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4831, 32, 39, 45, 47syl22anc 1478 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4930, 48eqeltrd 2840 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)
50 ffn 6207 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
51 elpreima 6502 . . . . . 6 (𝐹 Fn (Base‘𝑆) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
5216, 50, 513syl 18 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
5352adantr 472 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
5426, 49, 53mpbir2and 995 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
5554ralrimivva 3110 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
569adantr 472 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑆 ∈ LMod)
57 lmhmima.x . . . 4 𝑋 = (LSubSp‘𝑆)
5822, 24, 13, 23, 57islss4 19185 . . 3 (𝑆 ∈ LMod → ((𝐹𝑈) ∈ 𝑋 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))))
5956, 58syl 17 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ((𝐹𝑈) ∈ 𝑋 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))))
608, 55, 59mpbir2and 995 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  wral 3051  ccnv 5266  dom cdm 5267  cima 5270  Fun wfun 6044   Fn wfn 6045  wf 6046  cfv 6050  (class class class)co 6815  Basecbs 16080  Scalarcsca 16167   ·𝑠 cvsca 16168  SubGrpcsubg 17810   GrpHom cghm 17879  LModclmod 19086  LSubSpclss 19155   LMHom clmhm 19242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-ress 16088  df-plusg 16177  df-0g 16325  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-grp 17647  df-minusg 17648  df-sbg 17649  df-subg 17813  df-ghm 17880  df-mgp 18711  df-ur 18723  df-ring 18770  df-lmod 19088  df-lss 19156  df-lmhm 19245
This theorem is referenced by:  lmhmlsp  19272  lmhmkerlss  19274  lnmepi  38176
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