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Theorem reslmhm 19800
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 19781 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 reslmhm.r . . . 4 𝑅 = (𝑆s 𝑋)
3 reslmhm.u . . . 4 𝑈 = (LSubSp‘𝑆)
42, 3lsslmod 19708 . . 3 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
51, 4sylan 583 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
6 lmhmlmod2 19780 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
76adantr 484 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑇 ∈ LMod)
8 lmghm 19779 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
93lsssubg 19705 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
101, 9sylan 583 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
112resghm 18353 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
128, 10, 11syl2an2r 684 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 GrpHom 𝑇))
13 eqid 2821 . . . . 5 (Scalar‘𝑆) = (Scalar‘𝑆)
14 eqid 2821 . . . . 5 (Scalar‘𝑇) = (Scalar‘𝑇)
1513, 14lmhmsca 19778 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
162, 13resssca 16629 . . . 4 (𝑋𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
1715, 16sylan9eq 2876 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅))
18 simpll 766 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
19 simprl 770 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
20 eqid 2821 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
2120, 3lssss 19684 . . . . . . . . . 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 16534 . . . . . . . . . . . 12 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
2522, 24syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 = (Base‘𝑅))
2625eleq2d 2897 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑅)))
2726biimpar 481 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏𝑋)
2827adantrl 715 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏𝑋)
2923, 28sseldd 3944 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2821 . . . . . . . 8 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31 eqid 2821 . . . . . . . 8 ( ·𝑠𝑆) = ( ·𝑠𝑆)
32 eqid 2821 . . . . . . . 8 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3313, 30, 20, 31, 32lmhmlin 19783 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3418, 19, 29, 33syl3anc 1368 . . . . . 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 768 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋𝑈)
3813, 31, 30, 3lssvscl 19703 . . . . . . . 8 (((𝑆 ∈ LMod ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏𝑋)) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
3936, 37, 19, 28, 38syl22anc 837 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠𝑆)𝑏) ∈ 𝑋)
4039fvresd 6663 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠𝑆)𝑏)))
41 fvres 6662 . . . . . . . 8 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4241oveq2d 7146 . . . . . . 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 2866 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4544ralrimivva 3179 . . . 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 6647 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
482, 31ressvsca 16630 . . . . . . . . 9 (𝑋𝑈 → ( ·𝑠𝑆) = ( ·𝑠𝑅))
4948adantl 485 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5049oveqd 7147 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑎( ·𝑠𝑆)𝑏) = (𝑎( ·𝑠𝑅)𝑏))
5150fveqeq2d 6651 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5251ralbidv 3185 . . . . 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 3386 . . . 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 1125 . 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 2821 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
57 eqid 2821 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58 eqid 2821 . . 3 (Base‘𝑅) = (Base‘𝑅)
59 eqid 2821 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
6056, 14, 57, 58, 59, 32islmhm 19775 . 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 1341 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  wss 3910  cres 5530  cfv 6328  (class class class)co 7130  Basecbs 16462  s cress 16463  Scalarcsca 16547   ·𝑠 cvsca 16548  SubGrpcsubg 18252   GrpHom cghm 18334  LModclmod 19610  LSubSpclss 19679   LMHom clmhm 19767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-sca 16560  df-vsca 16561  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-minusg 18086  df-sbg 18087  df-subg 18255  df-ghm 18335  df-mgp 19219  df-ur 19231  df-ring 19278  df-lmod 19612  df-lss 19680  df-lmhm 19770
This theorem is referenced by:  frlmsplit2  20893  dimkerim  31034  lmhmlnmsplit  39842  pwssplit4  39844
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