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Theorem reslmhm2b 18875
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 2610 . . 3 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2610 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
3 eqid 2610 . . 3 ( ·𝑠𝑈) = ( ·𝑠𝑈)
4 eqid 2610 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
5 eqid 2610 . . 3 (Scalar‘𝑈) = (Scalar‘𝑈)
6 eqid 2610 . . 3 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
7 lmhmlmod1 18854 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
87adantl 481 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
9 simpl1 1057 . . . 4 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑇 ∈ LMod)
10 simpl2 1058 . . . 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 18781 . . . 4 ((𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝑈 ∈ LMod)
149, 10, 13syl2anc 691 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝑈 ∈ LMod)
15 eqid 2610 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
1611, 15resssca 15854 . . . . 5 (𝑋𝐿 → (Scalar‘𝑇) = (Scalar‘𝑈))
17163ad2ant2 1076 . . . 4 ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (Scalar‘𝑇) = (Scalar‘𝑈))
184, 15lmhmsca 18851 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
1917, 18sylan9req 2665 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (Scalar‘𝑈) = (Scalar‘𝑆))
20 lmghm 18852 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2112lsssubg 18778 . . . . . 6 ((𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝑋 ∈ (SubGrp‘𝑇))
2211resghm2b 17501 . . . . . 6 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
2321, 22stoic3 1692 . . . . 5 ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
2423biimpa 500 . . . 4 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
2520, 24sylan2 490 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑈))
26 eqid 2610 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
274, 6, 1, 2, 26lmhmlin 18856 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
28273expb 1258 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠𝑆)𝑦)) = (𝑥( ·𝑠𝑇)(𝐹𝑦)))
2928adantll 746 . . . 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 1094 . . . . 5 ((((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑋𝐿)
3111, 26ressvsca 15855 . . . . . 6 (𝑋𝐿 → ( ·𝑠𝑇) = ( ·𝑠𝑈))
3231oveqd 6566 . . . . 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 2644 . . 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 18860 . 2 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
36 simpr 476 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑈))
37 simpl1 1057 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑇 ∈ LMod)
38 simpl2 1058 . . 3 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝑋𝐿)
3911, 12reslmhm2 18874 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))
4036, 37, 38, 39syl3anc 1318 . 2 (((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 LMHom 𝑈)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
4135, 40impbida 873 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540  ran crn 5039  cfv 5804  (class class class)co 6549  Basecbs 15695  s cress 15696  Scalarcsca 15771   ·𝑠 cvsca 15772  SubGrpcsubg 17411   GrpHom cghm 17480  LModclmod 18686  LSubSpclss 18753   LMHom clmhm 18840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-sca 15784  df-vsca 15785  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lmhm 18843
This theorem is referenced by:  pj1lmhm2  18922  frlmsplit2  19931
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