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Theorem islmhm 19313
Description: Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
islmhm.k 𝐾 = (Scalar‘𝑆)
islmhm.l 𝐿 = (Scalar‘𝑇)
islmhm.b 𝐵 = (Base‘𝐾)
islmhm.e 𝐸 = (Base‘𝑆)
islmhm.m · = ( ·𝑠𝑆)
islmhm.n × = ( ·𝑠𝑇)
Assertion
Ref Expression
islmhm (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
Distinct variable groups:   𝑥,𝐵   𝑦,𝐸   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦   𝑥,𝑇,𝑦
Allowed substitution hints:   𝐵(𝑦)   · (𝑥,𝑦)   × (𝑥,𝑦)   𝐸(𝑥)   𝐾(𝑥,𝑦)   𝐿(𝑥,𝑦)

Proof of Theorem islmhm
Dummy variables 𝑓 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmhm 19308 . . 3 LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)))})
21elmpt2cl 7078 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
3 oveq12 6855 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
4 fvexd 6394 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Scalar‘𝑠) ∈ V)
5 simplr 785 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑡 = 𝑇)
65fveq2d 6383 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = (Scalar‘𝑇))
7 islmhm.l . . . . . . . . . 10 𝐿 = (Scalar‘𝑇)
86, 7syl6eqr 2817 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = 𝐿)
9 simpr 477 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑠))
10 simpll 783 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑠 = 𝑆)
1110fveq2d 6383 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑠) = (Scalar‘𝑆))
129, 11eqtrd 2799 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑆))
13 islmhm.k . . . . . . . . . 10 𝐾 = (Scalar‘𝑆)
1412, 13syl6eqr 2817 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = 𝐾)
158, 14eqeq12d 2780 . . . . . . . 8 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((Scalar‘𝑡) = 𝑤𝐿 = 𝐾))
1614fveq2d 6383 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = (Base‘𝐾))
17 islmhm.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1816, 17syl6eqr 2817 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = 𝐵)
1910fveq2d 6383 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = (Base‘𝑆))
20 islmhm.e . . . . . . . . . . 11 𝐸 = (Base‘𝑆)
2119, 20syl6eqr 2817 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = 𝐸)
2210fveq2d 6383 . . . . . . . . . . . . . 14 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑠) = ( ·𝑠𝑆))
23 islmhm.m . . . . . . . . . . . . . 14 · = ( ·𝑠𝑆)
2422, 23syl6eqr 2817 . . . . . . . . . . . . 13 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑠) = · )
2524oveqd 6863 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠𝑠)𝑦) = (𝑥 · 𝑦))
2625fveq2d 6383 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
275fveq2d 6383 . . . . . . . . . . . . 13 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑡) = ( ·𝑠𝑇))
28 islmhm.n . . . . . . . . . . . . 13 × = ( ·𝑠𝑇)
2927, 28syl6eqr 2817 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑡) = × )
3029oveqd 6863 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠𝑡)(𝑓𝑦)) = (𝑥 × (𝑓𝑦)))
3126, 30eqeq12d 2780 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))))
3221, 31raleqbidv 3300 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))))
3318, 32raleqbidv 3300 . . . . . . . 8 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))))
3415, 33anbi12d 624 . . . . . . 7 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))))
354, 34sbcied 3635 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ([(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))))
363, 35rabeqbidv 3344 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)))} = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))})
37 ovex 6878 . . . . . 6 (𝑆 GrpHom 𝑇) ∈ V
3837rabex 4975 . . . . 5 {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))} ∈ V
3936, 1, 38ovmpt2a 6993 . . . 4 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 LMHom 𝑇) = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))})
4039eleq2d 2830 . . 3 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))}))
41 fveq1 6378 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
42 fveq1 6378 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
4342oveq2d 6862 . . . . . . . 8 (𝑓 = 𝐹 → (𝑥 × (𝑓𝑦)) = (𝑥 × (𝐹𝑦)))
4441, 43eqeq12d 2780 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
45442ralbidv 3136 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
4645anbi2d 622 . . . . 5 (𝑓 = 𝐹 → ((𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
4746elrab 3521 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
48 3anass 1116 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
4947, 48bitr4i 269 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
5040, 49syl6bb 278 . 2 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
512, 50biadan2 853 1 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  [wsbc 3598  cfv 6070  (class class class)co 6846  Basecbs 16144  Scalarcsca 16231   ·𝑠 cvsca 16232   GrpHom cghm 17935  LModclmod 19146   LMHom clmhm 19305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-lmhm 19308
This theorem is referenced by:  islmhm3  19314  lmhmlem  19315  lmhmlin  19321  islmhmd  19325  reslmhm  19338  lmhmpropd  19359
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