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Theorem reslmhm 19418
 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 19399 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 reslmhm.r . . . . 5 𝑅 = (𝑆s 𝑋)
3 reslmhm.u . . . . 5 𝑈 = (LSubSp‘𝑆)
42, 3lsslmod 19326 . . . 4 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
51, 4sylan 575 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
6 lmhmlmod2 19398 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
76adantr 474 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑇 ∈ LMod)
85, 7jca 507 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑅 ∈ LMod ∧ 𝑇 ∈ LMod))
9 lmghm 19397 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
109adantr 474 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
113lsssubg 19323 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
121, 11sylan 575 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
132resghm 18034 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
1410, 12, 13syl2anc 579 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
15 eqid 2825 . . . . 5 (Scalar‘𝑆) = (Scalar‘𝑆)
16 eqid 2825 . . . . 5 (Scalar‘𝑇) = (Scalar‘𝑇)
1715, 16lmhmsca 19396 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
182, 15resssca 16397 . . . 4 (𝑋𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
1917, 18sylan9eq 2881 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
20 simpll 783 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21 simprl 787 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
22 eqid 2825 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
2322, 3lssss 19300 . . . . . . . . . 10 (𝑋𝑈𝑋 ⊆ (Base‘𝑆))
2423adantl 475 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ⊆ (Base‘𝑆))
2524adantr 474 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆))
262, 22ressbas2 16301 . . . . . . . . . . . 12 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
2724, 26syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 = (Base‘𝑅))
2827eleq2d 2892 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑅)))
2928biimpar 471 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏𝑋)
3029adantrl 707 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏𝑋)
3125, 30sseldd 3828 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
32 eqid 2825 . . . . . . . 8 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
33 eqid 2825 . . . . . . . 8 ( ·𝑠𝑆) = ( ·𝑠𝑆)
34 eqid 2825 . . . . . . . 8 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3515, 32, 22, 33, 34lmhmlin 19401 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3620, 21, 31, 35syl3anc 1494 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
371adantr 474 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑆 ∈ LMod)
3837adantr 474 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ LMod)
39 simplr 785 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋𝑈)
4015, 33, 32, 3lssvscl 19321 . . . . . . . 8 (((𝑆 ∈ LMod ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏𝑋)) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
4138, 39, 21, 30, 40syl22anc 872 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
42 fvres 6456 . . . . . . 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 6456 . . . . . . . 8 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4544oveq2d 6926 . . . . . . 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 2871 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4847ralrimivva 3180 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4918adantl 475 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅))
5049fveq2d 6441 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
512, 33ressvsca 16398 . . . . . . . . 9 (𝑋𝑈 → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5251adantl 475 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5352oveqd 6927 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑎( ·𝑠𝑆)𝑏) = (𝑎( ·𝑠𝑅)𝑏))
5453fveqeq2d 6445 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5554ralbidv 3195 . . . . 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))))
5650, 55raleqbidv 3364 . . . 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))))
5748, 56mpbid 224 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
5814, 19, 573jca 1162 . 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))))
59 eqid 2825 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
60 eqid 2825 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
61 eqid 2825 . . 3 (Base‘𝑅) = (Base‘𝑅)
62 eqid 2825 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
6359, 16, 60, 61, 62, 34islmhm 19393 . 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)))))
648, 58, 63sylanbrc 578 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117   ⊆ wss 3798   ↾ cres 5348  ‘cfv 6127  (class class class)co 6910  Basecbs 16229   ↾s cress 16230  Scalarcsca 16315   ·𝑠 cvsca 16316  SubGrpcsubg 17946   GrpHom cghm 18015  LModclmod 19226  LSubSpclss 19295   LMHom clmhm 19385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-plusg 16325  df-sca 16328  df-vsca 16329  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-sbg 17788  df-subg 17949  df-ghm 18016  df-mgp 18851  df-ur 18863  df-ring 18910  df-lmod 19228  df-lss 19296  df-lmhm 19388 This theorem is referenced by:  frlmsplit2  20486  lmhmlnmsplit  38495  pwssplit4  38497
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