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Theorem lmhmpreima 21065
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 21048 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 lmhmlmod2 21049 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3 lmhmima.y . . . . 5 𝑌 = (LSubSp‘𝑇)
43lsssubg 20973 . . . 4 ((𝑇 ∈ LMod ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
52, 4sylan 580 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6 ghmpreima 19269 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
71, 5, 6syl2an2r 685 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
8 lmhmlmod1 21050 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
98ad2antrr 726 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑆 ∈ LMod)
10 simprl 771 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
11 cnvimass 6102 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
12 eqid 2735 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
13 eqid 2735 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
1412, 13lmhmf 21051 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1514adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1611, 15fssdm 6756 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ⊆ (Base‘𝑆))
1716sselda 3995 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑏 ∈ (𝐹𝑈)) → 𝑏 ∈ (Base‘𝑆))
1817adantrl 716 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (Base‘𝑆))
19 eqid 2735 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
20 eqid 2735 . . . . . 6 ( ·𝑠𝑆) = ( ·𝑠𝑆)
21 eqid 2735 . . . . . 6 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2212, 19, 20, 21lmodvscl 20893 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
239, 10, 18, 22syl3anc 1370 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
24 simpll 767 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
25 eqid 2735 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2619, 21, 12, 20, 25lmhmlin 21052 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
2724, 10, 18, 26syl3anc 1370 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
282ad2antrr 726 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑇 ∈ LMod)
29 simplr 769 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑈𝑌)
30 eqid 2735 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
3119, 30lmhmsca 21047 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
3231adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
3332fveq2d 6911 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
3433eleq2d 2825 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
3534biimpar 477 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3635adantrr 717 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3715ffund 6741 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → Fun 𝐹)
38 simprr 773 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (𝐹𝑈))
39 fvimacnvi 7072 . . . . . . 7 ((Fun 𝐹𝑏 ∈ (𝐹𝑈)) → (𝐹𝑏) ∈ 𝑈)
4037, 38, 39syl2an2r 685 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹𝑏) ∈ 𝑈)
41 eqid 2735 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
4230, 25, 41, 3lssvscl 20971 . . . . . 6 (((𝑇 ∈ LMod ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹𝑏) ∈ 𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4328, 29, 36, 40, 42syl22anc 839 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4427, 43eqeltrd 2839 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)
45 ffn 6737 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
46 elpreima 7078 . . . . . 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 480 . . . 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 713 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
5049ralrimivva 3200 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
518adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑆 ∈ LMod)
52 lmhmima.x . . . 4 𝑋 = (LSubSp‘𝑆)
5319, 21, 12, 20, 52islss4 20978 . . 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 713 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  ccnv 5688  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  SubGrpcsubg 19151   GrpHom cghm 19243  LModclmod 20875  LSubSpclss 20947   LMHom clmhm 21036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877  df-lss 20948  df-lmhm 21039
This theorem is referenced by:  lmhmlsp  21066  lmhmkerlss  21068  lnmepi  43074
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