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Theorem lmhmima 18961
Description: The 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
lmhmima ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝑌)

Proof of Theorem lmhmima
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 18945 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3 lmhmlmod1 18947 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
43adantr 481 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑆 ∈ LMod)
5 simpr 477 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑈𝑋)
6 lmhmima.x . . . . 5 𝑋 = (LSubSp‘𝑆)
76lsssubg 18871 . . . 4 ((𝑆 ∈ LMod ∧ 𝑈𝑋) → 𝑈 ∈ (SubGrp‘𝑆))
84, 5, 7syl2anc 692 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑈 ∈ (SubGrp‘𝑆))
9 ghmima 17597 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
102, 8, 9syl2anc 692 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
11 eqid 2626 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
12 eqid 2626 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
1311, 12lmhmf 18948 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1413adantr 481 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
15 ffn 6004 . . . . . . . 8 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
1614, 15syl 17 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝐹 Fn (Base‘𝑆))
1711, 6lssss 18851 . . . . . . . 8 (𝑈𝑋𝑈 ⊆ (Base‘𝑆))
185, 17syl 17 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑈 ⊆ (Base‘𝑆))
19 fvelimab 6211 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆)) → (𝑏 ∈ (𝐹𝑈) ↔ ∃𝑐𝑈 (𝐹𝑐) = 𝑏))
2016, 18, 19syl2anc 692 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝑏 ∈ (𝐹𝑈) ↔ ∃𝑐𝑈 (𝐹𝑐) = 𝑏))
2120adantr 481 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (𝑏 ∈ (𝐹𝑈) ↔ ∃𝑐𝑈 (𝐹𝑐) = 𝑏))
22 simpll 789 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
23 eqid 2626 . . . . . . . . . . . . . . . 16 (Scalar‘𝑆) = (Scalar‘𝑆)
24 eqid 2626 . . . . . . . . . . . . . . . 16 (Scalar‘𝑇) = (Scalar‘𝑇)
2523, 24lmhmsca 18944 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
2625adantr 481 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (Scalar‘𝑇) = (Scalar‘𝑆))
2726fveq2d 6154 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
2827eleq2d 2689 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
2928biimpa 501 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
3029adantrr 752 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
3118sselda 3588 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑐𝑈) → 𝑐 ∈ (Base‘𝑆))
3231adantrl 751 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑐 ∈ (Base‘𝑆))
33 eqid 2626 . . . . . . . . . . 11 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
34 eqid 2626 . . . . . . . . . . 11 ( ·𝑠𝑆) = ( ·𝑠𝑆)
35 eqid 2626 . . . . . . . . . . 11 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3623, 33, 11, 34, 35lmhmlin 18949 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) = (𝑎( ·𝑠𝑇)(𝐹𝑐)))
3722, 30, 32, 36syl3anc 1323 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) = (𝑎( ·𝑠𝑇)(𝐹𝑐)))
3822, 13, 153syl 18 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝐹 Fn (Base‘𝑆))
39 simplr 791 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑈𝑋)
4039, 17syl 17 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑈 ⊆ (Base‘𝑆))
414adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑆 ∈ LMod)
42 simprr 795 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑐𝑈)
4323, 34, 33, 6lssvscl 18869 . . . . . . . . . . 11 (((𝑆 ∈ LMod ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐𝑈)) → (𝑎( ·𝑠𝑆)𝑐) ∈ 𝑈)
4441, 39, 30, 42, 43syl22anc 1324 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝑎( ·𝑠𝑆)𝑐) ∈ 𝑈)
45 fnfvima 6451 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆) ∧ (𝑎( ·𝑠𝑆)𝑐) ∈ 𝑈) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) ∈ (𝐹𝑈))
4638, 40, 44, 45syl3anc 1323 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) ∈ (𝐹𝑈))
4737, 46eqeltrrd 2705 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑐)) ∈ (𝐹𝑈))
4847anassrs 679 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) ∧ 𝑐𝑈) → (𝑎( ·𝑠𝑇)(𝐹𝑐)) ∈ (𝐹𝑈))
49 oveq2 6613 . . . . . . . 8 ((𝐹𝑐) = 𝑏 → (𝑎( ·𝑠𝑇)(𝐹𝑐)) = (𝑎( ·𝑠𝑇)𝑏))
5049eleq1d 2688 . . . . . . 7 ((𝐹𝑐) = 𝑏 → ((𝑎( ·𝑠𝑇)(𝐹𝑐)) ∈ (𝐹𝑈) ↔ (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5148, 50syl5ibcom 235 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) ∧ 𝑐𝑈) → ((𝐹𝑐) = 𝑏 → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5251rexlimdva 3029 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (∃𝑐𝑈 (𝐹𝑐) = 𝑏 → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5321, 52sylbid 230 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (𝑏 ∈ (𝐹𝑈) → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5453impr 648 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈))
5554ralrimivva 2970 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈))
56 lmhmlmod2 18946 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
5756adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑇 ∈ LMod)
58 eqid 2626 . . . 4 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
59 lmhmima.y . . . 4 𝑌 = (LSubSp‘𝑇)
6024, 58, 12, 35, 59islss4 18876 . . 3 (𝑇 ∈ LMod → ((𝐹𝑈) ∈ 𝑌 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈))))
6157, 60syl 17 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → ((𝐹𝑈) ∈ 𝑌 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈))))
6210, 55, 61mpbir2and 956 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560  cima 5082   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861  SubGrpcsubg 17504   GrpHom cghm 17573  LModclmod 18779  LSubSpclss 18846   LMHom clmhm 18933
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-subg 17507  df-ghm 17574  df-mgp 18406  df-ur 18418  df-ring 18465  df-lmod 18781  df-lss 18847  df-lmhm 18936
This theorem is referenced by:  lmhmlsp  18963  lmhmrnlss  18964
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