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Theorem lmhmpreima 18967
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 18950 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3 lmhmlmod2 18951 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
4 lmhmima.y . . . . 5 𝑌 = (LSubSp‘𝑇)
54lsssubg 18876 . . . 4 ((𝑇 ∈ LMod ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
63, 5sylan 488 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
7 ghmpreima 17603 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
82, 6, 7syl2anc 692 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
9 lmhmlmod1 18952 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109ad2antrr 761 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑆 ∈ LMod)
11 simprl 793 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
12 cnvimass 5444 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
13 eqid 2621 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2621 . . . . . . . . . . 11 (Base‘𝑇) = (Base‘𝑇)
1513, 14lmhmf 18953 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1615adantr 481 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
17 fdm 6008 . . . . . . . . 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 3632 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ⊆ (Base‘𝑆))
2019sselda 3583 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑏 ∈ (𝐹𝑈)) → 𝑏 ∈ (Base‘𝑆))
2120adantrl 751 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (Base‘𝑆))
22 eqid 2621 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
23 eqid 2621 . . . . . 6 ( ·𝑠𝑆) = ( ·𝑠𝑆)
24 eqid 2621 . . . . . 6 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2513, 22, 23, 24lmodvscl 18801 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
2610, 11, 21, 25syl3anc 1323 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
27 simpll 789 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
28 eqid 2621 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2922, 24, 13, 23, 28lmhmlin 18954 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3027, 11, 21, 29syl3anc 1323 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
313ad2antrr 761 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑇 ∈ LMod)
32 simplr 791 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑈𝑌)
33 eqid 2621 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
3422, 33lmhmsca 18949 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
3534adantr 481 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
3635fveq2d 6152 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
3736eleq2d 2684 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
3837biimpar 502 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3938adantrr 752 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
40 ffun 6005 . . . . . . . . 9 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Fun 𝐹)
4116, 40syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → Fun 𝐹)
4241adantr 481 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → Fun 𝐹)
43 simprr 795 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (𝐹𝑈))
44 fvimacnvi 6287 . . . . . . 7 ((Fun 𝐹𝑏 ∈ (𝐹𝑈)) → (𝐹𝑏) ∈ 𝑈)
4542, 43, 44syl2anc 692 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹𝑏) ∈ 𝑈)
46 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
4733, 28, 46, 4lssvscl 18874 . . . . . 6 (((𝑇 ∈ LMod ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹𝑏) ∈ 𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4831, 32, 39, 45, 47syl22anc 1324 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4930, 48eqeltrd 2698 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)
50 ffn 6002 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
51 elpreima 6293 . . . . . 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 481 . . . 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 956 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
5554ralrimivva 2965 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
569adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑆 ∈ LMod)
57 lmhmima.x . . . 4 𝑋 = (LSubSp‘𝑆)
5822, 24, 13, 23, 57islss4 18881 . . 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 956 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  ccnv 5073  dom cdm 5074  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865   ·𝑠 cvsca 15866  SubGrpcsubg 17509   GrpHom cghm 17578  LModclmod 18784  LSubSpclss 18851   LMHom clmhm 18938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-ghm 17579  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-lss 18852  df-lmhm 18941
This theorem is referenced by:  lmhmlsp  18968  lmhmkerlss  18970  lnmepi  37135
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