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Theorem reslmhm2b 19802
Description: Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Hypotheses
Ref Expression
reslmhm2.u 𝑈 = (𝑇s 𝑋)
reslmhm2.l 𝐿 = (LSubSp‘𝑇)
Assertion
Ref Expression
reslmhm2b ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))

Proof of Theorem reslmhm2b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2820 . . 3 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2820 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
3 eqid 2820 . . 3 ( ·𝑠𝑈) = ( ·𝑠𝑈)
4 eqid 2820 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
5 eqid 2820 . . 3 (Scalar‘𝑈) = (Scalar‘𝑈)
6 eqid 2820 . . 3 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
7 lmhmlmod1 19781 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
87adantl 484 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
9 simpl1 1187 . . . 4 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ LMod)
10 simpl2 1188 . . . 4 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑋𝐿)
11 reslmhm2.u . . . . 5 𝑈 = (𝑇s 𝑋)
12 reslmhm2.l . . . . 5 𝐿 = (LSubSp‘𝑇)
1311, 12lsslmod 19708 . . . 4 ((𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝑈 ∈ LMod)
149, 10, 13syl2anc 586 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑈 ∈ LMod)
15 eqid 2820 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
1611, 15resssca 16629 . . . . 5 (𝑋𝐿 → (Scalar‘𝑇) = (Scalar‘𝑈))
17163ad2ant2 1130 . . . 4 ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (Scalar‘𝑇) = (Scalar‘𝑈))
184, 15lmhmsca 19778 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1917, 18sylan9req 2876 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (Scalar‘𝑈) = (Scalar‘𝑆))
20 lmghm 19779 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2112lsssubg 19705 . . . . . 6 ((𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝑋 ∈ (SubGrp‘𝑇))
2211resghm2b 18355 . . . . . 6 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
2321, 22stoic3 1777 . . . . 5 ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
2423biimpa 479 . . . 4 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
2520, 24sylan2 594 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
26 eqid 2820 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
274, 6, 1, 2, 26lmhmlin 19783 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
28273expb 1116 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
2928adantll 712 . . . 4 ((((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
30 simpll2 1209 . . . . 5 ((((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑋𝐿)
3111, 26ressvsca 16630 . . . . . 6 (𝑋𝐿 → ( ·𝑠𝑇) = ( ·𝑠𝑈))
3231oveqd 7150 . . . . 5 (𝑋𝐿 → (𝑥( ·𝑠𝑇)(𝐹𝑦)) = (𝑥( ·𝑠𝑈)(𝐹𝑦)))
3330, 32syl 17 . . . 4 ((((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑇)(𝐹𝑦)) = (𝑥( ·𝑠𝑈)(𝐹𝑦)))
3429, 33eqtrd 2855 . . 3 ((((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑈)(𝐹𝑦)))
351, 2, 3, 4, 5, 6, 8, 14, 19, 25, 34islmhmd 19787 . 2 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
36 simpr 487 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
37 simpl1 1187 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑇 ∈ LMod)
38 simpl2 1188 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑋𝐿)
3911, 12reslmhm2 19801 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))
4036, 37, 38, 39syl3anc 1367 . 2 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
4135, 40impbida 799 1 ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wss 3913  ran crn 5532  cfv 6331  (class class class)co 7133  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 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  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-mhm 17935  df-submnd 17936  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:  pj1lmhm2  19849  frlmsplit2  20893  dimkerim  31034
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