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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) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)

Proof of Theorem lmhmpreima
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef 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)
43lsssubg 20848 . . . 4 ((𝑇 ∈ LMod ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
52, 4sylan 578 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6 ghmpreima 19199 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
71, 5, 6syl2an2r 683 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
8 lmhmlmod1 20925 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
98ad2antrr 724 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑆 ∈ LMod)
10 simprl 769 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
11 cnvimass 6090 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
12 eqid 2728 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
13 eqid 2728 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
1412, 13lmhmf 20926 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1514adantr 479 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1611, 15fssdm 6747 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ⊆ (Base‘𝑆))
1716sselda 3982 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑏 ∈ (𝐹𝑈)) → 𝑏 ∈ (Base‘𝑆))
1817adantrl 714 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (Base‘𝑆))
19 eqid 2728 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
20 eqid 2728 . . . . . 6 ( ·𝑠𝑆) = ( ·𝑠𝑆)
21 eqid 2728 . . . . . 6 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2212, 19, 20, 21lmodvscl 20768 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
239, 10, 18, 22syl3anc 1368 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
24 simpll 765 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
25 eqid 2728 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2619, 21, 12, 20, 25lmhmlin 20927 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
2724, 10, 18, 26syl3anc 1368 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
282ad2antrr 724 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑇 ∈ LMod)
29 simplr 767 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑈𝑌)
30 eqid 2728 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
3119, 30lmhmsca 20922 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
3231adantr 479 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
3332fveq2d 6906 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
3433eleq2d 2815 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
3534biimpar 476 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3635adantrr 715 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3715ffund 6731 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → Fun 𝐹)
38 simprr 771 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (𝐹𝑈))
39 fvimacnvi 7066 . . . . . . 7 ((Fun 𝐹𝑏 ∈ (𝐹𝑈)) → (𝐹𝑏) ∈ 𝑈)
4037, 38, 39syl2an2r 683 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹𝑏) ∈ 𝑈)
41 eqid 2728 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
4230, 25, 41, 3lssvscl 20846 . . . . . 6 (((𝑇 ∈ LMod ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹𝑏) ∈ 𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4328, 29, 36, 40, 42syl22anc 837 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4427, 43eqeltrd 2829 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)
45 ffn 6727 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
46 elpreima 7072 . . . . . 6 (𝐹 Fn (Base‘𝑆) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
4715, 45, 463syl 18 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
4847adantr 479 . . . 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)))
4923, 44, 48mpbir2and 711 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
5049ralrimivva 3198 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
518adantr 479 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑆 ∈ LMod)
52 lmhmima.x . . . 4 𝑋 = (LSubSp‘𝑆)
5319, 21, 12, 20, 52islss4 20853 . . 3 (𝑆 ∈ LMod → ((𝐹𝑈) ∈ 𝑋 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))))
5451, 53syl 17 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ((𝐹𝑈) ∈ 𝑋 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))))
557, 50, 54mpbir2and 711 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)
Colors of variables: wff setvar class
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
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