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Theorem reslmhm 18873
 Description: Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
reslmhm.u 𝑈 = (LSubSp‘𝑆)
reslmhm.r 𝑅 = (𝑆s 𝑋)
Assertion
Ref Expression
reslmhm ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))

Proof of Theorem reslmhm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlmod1 18854 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 reslmhm.r . . . . 5 𝑅 = (𝑆s 𝑋)
3 reslmhm.u . . . . 5 𝑈 = (LSubSp‘𝑆)
42, 3lsslmod 18781 . . . 4 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
51, 4sylan 487 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
6 lmhmlmod2 18853 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
76adantr 480 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑇 ∈ LMod)
85, 7jca 553 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑅 ∈ LMod ∧ 𝑇 ∈ LMod))
9 lmghm 18852 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
109adantr 480 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
113lsssubg 18778 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
121, 11sylan 487 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
132resghm 17499 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
1410, 12, 13syl2anc 691 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
15 eqid 2610 . . . . 5 (Scalar‘𝑆) = (Scalar‘𝑆)
16 eqid 2610 . . . . 5 (Scalar‘𝑇) = (Scalar‘𝑇)
1715, 16lmhmsca 18851 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
182, 15resssca 15854 . . . 4 (𝑋𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
1917, 18sylan9eq 2664 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
20 simpll 786 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21 simprl 790 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
22 eqid 2610 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
2322, 3lssss 18758 . . . . . . . . . 10 (𝑋𝑈𝑋 ⊆ (Base‘𝑆))
2423adantl 481 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ⊆ (Base‘𝑆))
2524adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆))
262, 22ressbas2 15758 . . . . . . . . . . . 12 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
2724, 26syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 = (Base‘𝑅))
2827eleq2d 2673 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑅)))
2928biimpar 501 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏𝑋)
3029adantrl 748 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏𝑋)
3125, 30sseldd 3569 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
32 eqid 2610 . . . . . . . 8 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
33 eqid 2610 . . . . . . . 8 ( ·𝑠𝑆) = ( ·𝑠𝑆)
34 eqid 2610 . . . . . . . 8 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3515, 32, 22, 33, 34lmhmlin 18856 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3620, 21, 31, 35syl3anc 1318 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
371adantr 480 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑆 ∈ LMod)
3837adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ LMod)
39 simplr 788 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋𝑈)
4015, 33, 32, 3lssvscl 18776 . . . . . . . 8 (((𝑆 ∈ LMod ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏𝑋)) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
4138, 39, 21, 30, 40syl22anc 1319 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
42 fvres 6117 . . . . . . 7 ((𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋 → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠𝑆)𝑏)))
4341, 42syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠𝑆)𝑏)))
44 fvres 6117 . . . . . . . 8 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4544oveq2d 6565 . . . . . . 7 (𝑏𝑋 → (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
4630, 45syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
4736, 43, 463eqtr4d 2654 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4847ralrimivva 2954 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4918adantl 481 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅))
5049fveq2d 6107 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
512, 33ressvsca 15855 . . . . . . . . . 10 (𝑋𝑈 → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5251adantl 481 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5352oveqd 6566 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑎( ·𝑠𝑆)𝑏) = (𝑎( ·𝑠𝑅)𝑏))
5453fveq2d 6107 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = ((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)))
5554eqeq1d 2612 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5655ralbidv 2969 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5750, 56raleqbidv 3129 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5848, 57mpbid 221 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
5914, 19, 583jca 1235 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ((𝐹𝑋) ∈ (𝑅 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑅) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
60 eqid 2610 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
61 eqid 2610 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
62 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
63 eqid 2610 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
6460, 16, 61, 62, 63, 34islmhm 18848 . 2 ((𝐹𝑋) ∈ (𝑅 LMHom 𝑇) ↔ ((𝑅 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ ((𝐹𝑋) ∈ (𝑅 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑅) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))))
658, 59, 64sylanbrc 695 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  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-3 10957  df-4 10958  df-5 10959  df-6 10960  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-sca 15784  df-vsca 15785  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:  frlmsplit2  19931  lmhmlnmsplit  36675  pwssplit4  36677
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