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Theorem reslmhm 21069
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 21050 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 reslmhm.r . . . 4 𝑅 = (𝑆s 𝑋)
3 reslmhm.u . . . 4 𝑈 = (LSubSp‘𝑆)
42, 3lsslmod 20976 . . 3 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
51, 4sylan 580 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑅 ∈ LMod)
6 lmhmlmod2 21049 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
76adantr 480 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑇 ∈ LMod)
8 lmghm 21048 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
93lsssubg 20973 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
101, 9sylan 580 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ∈ (SubGrp‘𝑆))
112resghm 19263 . . . 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 2735 . . . . 5 (Scalar‘𝑆) = (Scalar‘𝑆)
14 eqid 2735 . . . . 5 (Scalar‘𝑇) = (Scalar‘𝑇)
1513, 14lmhmsca 21047 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
162, 13resssca 17389 . . . 4 (𝑋𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅))
1715, 16sylan9eq 2795 . . 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 2735 . . . . . . . . . . 11 (Base‘𝑆) = (Base‘𝑆)
2120, 3lssss 20952 . . . . . . . . . 10 (𝑋𝑈𝑋 ⊆ (Base‘𝑆))
2221adantl 481 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 ⊆ (Base‘𝑆))
2322adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆))
242, 20ressbas2 17283 . . . . . . . . . . . 12 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅))
2522, 24syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑋 = (Base‘𝑅))
2625eleq2d 2825 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑏𝑋𝑏 ∈ (Base‘𝑅)))
2726biimpar 477 . . . . . . . . 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 3996 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆))
30 eqid 2735 . . . . . . . 8 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31 eqid 2735 . . . . . . . 8 ( ·𝑠𝑆) = ( ·𝑠𝑆)
32 eqid 2735 . . . . . . . 8 ( ·𝑠𝑇) = ( ·𝑠𝑇)
3313, 30, 20, 31, 32lmhmlin 21052 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
3418, 19, 29, 33syl3anc 1370 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
351adantr 480 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → 𝑆 ∈ LMod)
3635adantr 480 . . . . . . . 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 20971 . . . . . . . 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 6927 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠𝑆)𝑏)))
41 fvres 6926 . . . . . . . 8 (𝑏𝑋 → ((𝐹𝑋)‘𝑏) = (𝐹𝑏))
4241oveq2d 7447 . . . . . . 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 2785 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4544ralrimivva 3200 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
4616adantl 481 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅))
4746fveq2d 6911 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑅)))
482, 31ressvsca 17390 . . . . . . . . 9 (𝑋𝑈 → ( ·𝑠𝑆) = ( ·𝑠𝑅))
4948adantl 481 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ( ·𝑠𝑆) = ( ·𝑠𝑅))
5049oveqd 7448 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝑎( ·𝑠𝑆)𝑏) = (𝑎( ·𝑠𝑅)𝑏))
5150fveqeq2d 6915 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (((𝐹𝑋)‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)) ↔ ((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏))))
5251ralbidv 3176 . . . . 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 3344 . . . 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 232 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹𝑋)‘(𝑎( ·𝑠𝑅)𝑏)) = (𝑎( ·𝑠𝑇)((𝐹𝑋)‘𝑏)))
5512, 17, 543jca 1127 . 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 2735 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
57 eqid 2735 . . 3 (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅))
58 eqid 2735 . . 3 (Base‘𝑅) = (Base‘𝑅)
59 eqid 2735 . . 3 ( ·𝑠𝑅) = ( ·𝑠𝑅)
6056, 14, 57, 58, 59, 32islmhm 21044 . 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 1343 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  cres 5691  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Scalarcsca 17301   ·𝑠 cvsca 17302  SubGrpcsubg 19151   GrpHom cghm 19243  LModclmod 20875  LSubSpclss 20947   LMHom clmhm 21036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-sca 17314  df-vsca 17315  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877  df-lss 20948  df-lmhm 21039
This theorem is referenced by:  frlmsplit2  21811  dimkerim  33655  lmhmlnmsplit  43076  pwssplit4  43078
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