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Theorem islmhm 19229
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 19224 . . 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 7041 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
3 oveq12 6822 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
4 fvexd 6364 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Scalar‘𝑠) ∈ V)
5 simplr 809 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑡 = 𝑇)
65fveq2d 6356 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = (Scalar‘𝑇))
7 islmhm.l . . . . . . . . . 10 𝐿 = (Scalar‘𝑇)
86, 7syl6eqr 2812 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = 𝐿)
9 simpr 479 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑠))
10 simpll 807 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑠 = 𝑆)
1110fveq2d 6356 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑠) = (Scalar‘𝑆))
129, 11eqtrd 2794 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑆))
13 islmhm.k . . . . . . . . . 10 𝐾 = (Scalar‘𝑆)
1412, 13syl6eqr 2812 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = 𝐾)
158, 14eqeq12d 2775 . . . . . . . 8 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((Scalar‘𝑡) = 𝑤𝐿 = 𝐾))
1614fveq2d 6356 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = (Base‘𝐾))
17 islmhm.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1816, 17syl6eqr 2812 . . . . . . . . 9 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = 𝐵)
1910fveq2d 6356 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = (Base‘𝑆))
20 islmhm.e . . . . . . . . . . 11 𝐸 = (Base‘𝑆)
2119, 20syl6eqr 2812 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = 𝐸)
2210fveq2d 6356 . . . . . . . . . . . . . 14 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑠) = ( ·𝑠𝑆))
23 islmhm.m . . . . . . . . . . . . . 14 · = ( ·𝑠𝑆)
2422, 23syl6eqr 2812 . . . . . . . . . . . . 13 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑠) = · )
2524oveqd 6830 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠𝑠)𝑦) = (𝑥 · 𝑦))
2625fveq2d 6356 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
275fveq2d 6356 . . . . . . . . . . . . 13 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑡) = ( ·𝑠𝑇))
28 islmhm.n . . . . . . . . . . . . 13 × = ( ·𝑠𝑇)
2927, 28syl6eqr 2812 . . . . . . . . . . . 12 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠𝑡) = × )
3029oveqd 6830 . . . . . . . . . . 11 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠𝑡)(𝑓𝑦)) = (𝑥 × (𝑓𝑦)))
3126, 30eqeq12d 2775 . . . . . . . . . 10 (((𝑠 = 𝑆𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))))
3221, 31raleqbidv 3291 . . . . . . . . 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 3291 . . . . . . . 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 749 . . . . . . 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 3613 . . . . . 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 3335 . . . . 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 6841 . . . . . 6 (𝑆 GrpHom 𝑇) ∈ V
3837rabex 4964 . . . . 5 {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))} ∈ V
3936, 1, 38ovmpt2a 6956 . . . 4 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 LMHom 𝑇) = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))})
4039eleq2d 2825 . . 3 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)))}))
41 fveq1 6351 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
42 fveq1 6351 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
4342oveq2d 6829 . . . . . . . 8 (𝑓 = 𝐹 → (𝑥 × (𝑓𝑦)) = (𝑥 × (𝐹𝑦)))
4441, 43eqeq12d 2775 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
45442ralbidv 3127 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
4645anbi2d 742 . . . . 5 (𝑓 = 𝐹 → ((𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
4746elrab 3504 . . . 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 1081 . . . 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 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))))
5040, 49syl6bb 276 . 2 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
512, 50biadan2 677 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  [wsbc 3576  cfv 6049  (class class class)co 6813  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147   GrpHom cghm 17858  LModclmod 19065   LMHom clmhm 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-lmhm 19224
This theorem is referenced by:  islmhm3  19230  lmhmlem  19231  lmhmlin  19237  islmhmd  19241  reslmhm  19254  lmhmpropd  19275
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