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Theorem reslmhm 20117
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 20098 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 reslmhm.r . . . 4 𝑅 = (𝑆s 𝑋)
3 reslmhm.u . . . 4 𝑈 = (LSubSp‘𝑆)
42, 3lsslmod 20025 . . 3 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
51, 4sylan 583 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
6 lmhmlmod2 20097 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
76adantr 484 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑇 ∈ LMod)
8 lmghm 20096 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
93lsssubg 20022 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
101, 9sylan 583 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
112resghm 18666 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
128, 10, 11syl2an2r 685 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
13 eqid 2738 . . . . 5 (Scalar‘𝑆) = (Scalar‘𝑆)
14 eqid 2738 . . . . 5 (Scalar‘𝑇) = (Scalar‘𝑇)
1513, 14lmhmsca 20095 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
162, 13resssca 16904 . . . 4 (𝑋𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
1715, 16sylan9eq 2799 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
18 simpll 767 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
19 simprl 771 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
20 eqid 2738 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
2120, 3lssss 20001 . . . . . . . . . 10 (𝑋𝑈𝑋 ⊆ (Base‘𝑆))
2221adantl 485 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ⊆ (Base‘𝑆))
2322adantr 484 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆))
242, 20ressbas2 16819 . . . . . . . . . . . 12 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
2522, 24syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 = (Base‘𝑅))
2625eleq2d 2824 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑅)))
2726biimpar 481 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏𝑋)
2827adantrl 716 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏𝑋)
2923, 28sseldd 3917 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2738 . . . . . . . 8 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31 eqid 2738 . . . . . . . 8 ( ·𝑠𝑆) = ( ·𝑠𝑆)
32 eqid 2738 . . . . . . . 8 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3313, 30, 20, 31, 32lmhmlin 20100 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3418, 19, 29, 33syl3anc 1373 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
351adantr 484 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑆 ∈ LMod)
3635adantr 484 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ LMod)
37 simplr 769 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋𝑈)
3813, 31, 30, 3lssvscl 20020 . . . . . . . 8 (((𝑆 ∈ LMod ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏𝑋)) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
3936, 37, 19, 28, 38syl22anc 839 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
4039fvresd 6756 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠𝑆)𝑏)))
41 fvres 6755 . . . . . . . 8 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4241oveq2d 7248 . . . . . . 7 (𝑏𝑋 → (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
4328, 42syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
4434, 40, 433eqtr4d 2788 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4544ralrimivva 3113 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4616adantl 485 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅))
4746fveq2d 6740 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
482, 31ressvsca 16905 . . . . . . . . 9 (𝑋𝑈 → ( ·𝑠𝑆) = ( ·𝑠𝑅))
4948adantl 485 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5049oveqd 7249 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑎( ·𝑠𝑆)𝑏) = (𝑎( ·𝑠𝑅)𝑏))
5150fveqeq2d 6744 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5251ralbidv 3119 . . . . 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))))
5347, 52raleqbidv 3326 . . . 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))))
5445, 53mpbid 235 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
5512, 17, 543jca 1130 . 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))))
56 eqid 2738 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
57 eqid 2738 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58 eqid 2738 . . 3 (Base‘𝑅) = (Base‘𝑅)
59 eqid 2738 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
6056, 14, 57, 58, 59, 32islmhm 20092 . 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)))))
615, 7, 55, 60syl21anbrc 1346 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2111  wral 3062  wss 3881  cres 5568  cfv 6398  (class class class)co 7232  Basecbs 16788  s cress 16812  Scalarcsca 16833   ·𝑠 cvsca 16834  SubGrpcsubg 18565   GrpHom cghm 18647  LModclmod 19927  LSubSpclss 19996   LMHom clmhm 20084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542  ax-cnex 10810  ax-resscn 10811  ax-1cn 10812  ax-icn 10813  ax-addcl 10814  ax-addrcl 10815  ax-mulcl 10816  ax-mulrcl 10817  ax-mulcom 10818  ax-addass 10819  ax-mulass 10820  ax-distr 10821  ax-i2m1 10822  ax-1ne0 10823  ax-1rid 10824  ax-rnegex 10825  ax-rrecex 10826  ax-cnre 10827  ax-pre-lttri 10828  ax-pre-lttrn 10829  ax-pre-ltadd 10830  ax-pre-mulgt0 10831
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-lim 6236  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-riota 7189  df-ov 7235  df-oprab 7236  df-mpo 7237  df-om 7664  df-1st 7780  df-2nd 7781  df-wrecs 8068  df-recs 8129  df-rdg 8167  df-er 8412  df-en 8648  df-dom 8649  df-sdom 8650  df-pnf 10894  df-mnf 10895  df-xr 10896  df-ltxr 10897  df-le 10898  df-sub 11089  df-neg 11090  df-nn 11856  df-2 11918  df-3 11919  df-4 11920  df-5 11921  df-6 11922  df-sets 16745  df-slot 16763  df-ndx 16773  df-base 16789  df-ress 16813  df-plusg 16843  df-sca 16846  df-vsca 16847  df-0g 16974  df-mgm 18142  df-sgrp 18191  df-mnd 18202  df-grp 18396  df-minusg 18397  df-sbg 18398  df-subg 18568  df-ghm 18648  df-mgp 19533  df-ur 19545  df-ring 19592  df-lmod 19929  df-lss 19997  df-lmhm 20087
This theorem is referenced by:  frlmsplit2  20763  dimkerim  31449  lmhmlnmsplit  40648  pwssplit4  40650
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