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Theorem islmhm 18941
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 18936 . . 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 6830 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
3 oveq12 6614 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
4 fvex 6160 . . . . . . . 8 (Scalar‘𝑠) ∈ V
54a1i 11 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Scalar‘𝑠) ∈ V)
6 simplr 791 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑡 = 𝑇)
76fveq2d 6154 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = (Scalar‘𝑇))
8 islmhm.l . . . . . . . . . 10 𝐿 = (Scalar‘𝑇)
97, 8syl6eqr 2678 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = 𝐿)
10 simpr 477 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑠))
11 simpll 789 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑠 = 𝑆)
1211fveq2d 6154 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑠) = (Scalar‘𝑆))
1310, 12eqtrd 2660 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑆))
14 islmhm.k . . . . . . . . . 10 𝐾 = (Scalar‘𝑆)
1513, 14syl6eqr 2678 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = 𝐾)
169, 15eqeq12d 2641 . . . . . . . 8 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((Scalar‘𝑡) = 𝑤𝐿 = 𝐾))
1715fveq2d 6154 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = (Base‘𝐾))
18 islmhm.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1917, 18syl6eqr 2678 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = 𝐵)
2011fveq2d 6154 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = (Base‘𝑆))
21 islmhm.e . . . . . . . . . . 11 𝐸 = (Base‘𝑆)
2220, 21syl6eqr 2678 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = 𝐸)
2311fveq2d 6154 . . . . . . . . . . . . . 14 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑠) = ( ·𝑠𝑆))
24 islmhm.m . . . . . . . . . . . . . 14 · = ( ·𝑠𝑆)
2523, 24syl6eqr 2678 . . . . . . . . . . . . 13 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑠) = · )
2625oveqd 6622 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠𝑠)𝑦) = (𝑥 · 𝑦))
2726fveq2d 6154 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
286fveq2d 6154 . . . . . . . . . . . . 13 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑡) = ( ·𝑠𝑇))
29 islmhm.n . . . . . . . . . . . . 13 × = ( ·𝑠𝑇)
3028, 29syl6eqr 2678 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑡) = × )
3130oveqd 6622 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠𝑡)(𝑓𝑦)) = (𝑥 × (𝑓𝑦)))
3227, 31eqeq12d 2641 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))))
3322, 32raleqbidv 3146 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))))
3419, 33raleqbidv 3146 . . . . . . . 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))))
3516, 34anbi12d 746 . . . . . . 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)))))
365, 35sbcied 3459 . . . . . 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)))))
373, 36rabeqbidv 3186 . . . . 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)))})
38 ovex 6633 . . . . . 6 (𝑆 GrpHom 𝑇) ∈ V
3938rabex 4778 . . . . 5 {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))} ∈ V
4037, 1, 39ovmpt2a 6745 . . . 4 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 LMHom 𝑇) = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))})
4140eleq2d 2689 . . 3 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))}))
42 fveq1 6149 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
43 fveq1 6149 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
4443oveq2d 6621 . . . . . . . 8 (𝑓 = 𝐹 → (𝑥 × (𝑓𝑦)) = (𝑥 × (𝐹𝑦)))
4542, 44eqeq12d 2641 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
46452ralbidv 2988 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
4746anbi2d 739 . . . . 5 (𝑓 = 𝐹 → ((𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
4847elrab 3351 . . . 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)))))
49 3anass 1040 . . . 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)))))
5048, 49bitr4i 267 . . 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))))
5141, 50syl6bb 276 . 2 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
522, 51biadan2 673 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  [wsbc 3422  cfv 5850  (class class class)co 6605  Basecbs 15776  Scalarcsca 15860   ·𝑠 cvsca 15861   GrpHom cghm 17573  LModclmod 18779   LMHom clmhm 18933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-lmhm 18936
This theorem is referenced by:  islmhm3  18942  lmhmlem  18943  lmhmlin  18949  islmhmd  18953  reslmhm  18966  lmhmpropd  18987
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