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Theorem lmhmima 18868
 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 18852 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3 lmhmlmod1 18854 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
43adantr 480 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑆 ∈ LMod)
5 simpr 476 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑈𝑋)
6 lmhmima.x . . . . 5 𝑋 = (LSubSp‘𝑆)
76lsssubg 18778 . . . 4 ((𝑆 ∈ LMod ∧ 𝑈𝑋) → 𝑈 ∈ (SubGrp‘𝑆))
84, 5, 7syl2anc 691 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑈 ∈ (SubGrp‘𝑆))
9 ghmima 17504 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
102, 8, 9syl2anc 691 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
11 eqid 2610 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
12 eqid 2610 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
1311, 12lmhmf 18855 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1413adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
15 ffn 5958 . . . . . . . 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 18758 . . . . . . . 8 (𝑈𝑋𝑈 ⊆ (Base‘𝑆))
185, 17syl 17 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑈 ⊆ (Base‘𝑆))
19 fvelimab 6163 . . . . . . 7 ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆)) → (𝑏 ∈ (𝐹𝑈) ↔ ∃𝑐𝑈 (𝐹𝑐) = 𝑏))
2016, 18, 19syl2anc 691 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝑏 ∈ (𝐹𝑈) ↔ ∃𝑐𝑈 (𝐹𝑐) = 𝑏))
2120adantr 480 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (𝑏 ∈ (𝐹𝑈) ↔ ∃𝑐𝑈 (𝐹𝑐) = 𝑏))
22 simpll 786 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
23 eqid 2610 . . . . . . . . . . . . . . . 16 (Scalar‘𝑆) = (Scalar‘𝑆)
24 eqid 2610 . . . . . . . . . . . . . . . 16 (Scalar‘𝑇) = (Scalar‘𝑇)
2523, 24lmhmsca 18851 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
2625adantr 480 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (Scalar‘𝑇) = (Scalar‘𝑆))
2726fveq2d 6107 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
2827eleq2d 2673 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
2928biimpa 500 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
3029adantrr 749 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
3118sselda 3568 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑐𝑈) → 𝑐 ∈ (Base‘𝑆))
3231adantrl 748 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑐 ∈ (Base‘𝑆))
33 eqid 2610 . . . . . . . . . . 11 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
34 eqid 2610 . . . . . . . . . . 11 ( ·𝑠𝑆) = ( ·𝑠𝑆)
35 eqid 2610 . . . . . . . . . . 11 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3623, 33, 11, 34, 35lmhmlin 18856 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) = (𝑎( ·𝑠𝑇)(𝐹𝑐)))
3722, 30, 32, 36syl3anc 1318 . . . . . . . . 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 788 . . . . . . . . . . 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 480 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑆 ∈ LMod)
42 simprr 792 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → 𝑐𝑈)
4323, 34, 33, 6lssvscl 18776 . . . . . . . . . . 11 (((𝑆 ∈ LMod ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑐𝑈)) → (𝑎( ·𝑠𝑆)𝑐) ∈ 𝑈)
4441, 39, 30, 42, 43syl22anc 1319 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝑎( ·𝑠𝑆)𝑐) ∈ 𝑈)
45 fnfvima 6400 . . . . . . . . . 10 ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆) ∧ (𝑎( ·𝑠𝑆)𝑐) ∈ 𝑈) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) ∈ (𝐹𝑈))
4638, 40, 44, 45syl3anc 1318 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑐)) ∈ (𝐹𝑈))
4737, 46eqeltrrd 2689 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑐𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑐)) ∈ (𝐹𝑈))
4847anassrs 678 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) ∧ 𝑐𝑈) → (𝑎( ·𝑠𝑇)(𝐹𝑐)) ∈ (𝐹𝑈))
49 oveq2 6557 . . . . . . . 8 ((𝐹𝑐) = 𝑏 → (𝑎( ·𝑠𝑇)(𝐹𝑐)) = (𝑎( ·𝑠𝑇)𝑏))
5049eleq1d 2672 . . . . . . 7 ((𝐹𝑐) = 𝑏 → ((𝑎( ·𝑠𝑇)(𝐹𝑐)) ∈ (𝐹𝑈) ↔ (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5148, 50syl5ibcom 234 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) ∧ 𝑐𝑈) → ((𝐹𝑐) = 𝑏 → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5251rexlimdva 3013 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (∃𝑐𝑈 (𝐹𝑐) = 𝑏 → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5321, 52sylbid 229 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → (𝑏 ∈ (𝐹𝑈) → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈)))
5453impr 647 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈))
5554ralrimivva 2954 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → ∀𝑎 ∈ (Base‘(Scalar‘𝑇))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑇)𝑏) ∈ (𝐹𝑈))
56 lmhmlmod2 18853 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
5756adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → 𝑇 ∈ LMod)
58 eqid 2610 . . . 4 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
59 lmhmima.y . . . 4 𝑌 = (LSubSp‘𝑇)
6024, 58, 12, 35, 59islss4 18783 . . 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 959 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  SubGrpcsubg 17411   GrpHom cghm 17480  LModclmod 18686  LSubSpclss 18753   LMHom clmhm 18840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lmhm 18843 This theorem is referenced by:  lmhmlsp  18870  lmhmrnlss  18871
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