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Theorem islmhmd 21056
Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Hypotheses
Ref Expression
islmhmd.x 𝑋 = (Base‘𝑆)
islmhmd.a · = ( ·𝑠𝑆)
islmhmd.b × = ( ·𝑠𝑇)
islmhmd.k 𝐾 = (Scalar‘𝑆)
islmhmd.j 𝐽 = (Scalar‘𝑇)
islmhmd.n 𝑁 = (Base‘𝐾)
islmhmd.s (𝜑𝑆 ∈ LMod)
islmhmd.t (𝜑𝑇 ∈ LMod)
islmhmd.c (𝜑𝐽 = 𝐾)
islmhmd.f (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))
islmhmd.l ((𝜑 ∧ (𝑥𝑁𝑦𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
Assertion
Ref Expression
islmhmd (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝑋,𝑦   𝑥,𝐽,𝑦   𝑥,𝑁,𝑦   𝑥,𝐾,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem islmhmd
StepHypRef Expression
1 islmhmd.s . 2 (𝜑𝑆 ∈ LMod)
2 islmhmd.t . 2 (𝜑𝑇 ∈ LMod)
3 islmhmd.f . . 3 (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))
4 islmhmd.c . . 3 (𝜑𝐽 = 𝐾)
5 islmhmd.l . . . 4 ((𝜑 ∧ (𝑥𝑁𝑦𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
65ralrimivva 3200 . . 3 (𝜑 → ∀𝑥𝑁𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
73, 4, 63jca 1127 . 2 (𝜑 → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥𝑁𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
8 islmhmd.k . . 3 𝐾 = (Scalar‘𝑆)
9 islmhmd.j . . 3 𝐽 = (Scalar‘𝑇)
10 islmhmd.n . . 3 𝑁 = (Base‘𝐾)
11 islmhmd.x . . 3 𝑋 = (Base‘𝑆)
12 islmhmd.a . . 3 · = ( ·𝑠𝑆)
13 islmhmd.b . . 3 × = ( ·𝑠𝑇)
148, 9, 10, 11, 12, 13islmhm 21044 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐽 = 𝐾 ∧ ∀𝑥𝑁𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
151, 2, 7, 14syl21anbrc 1343 1 (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302   GrpHom cghm 19243  LModclmod 20875   LMHom clmhm 21036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-lmhm 21039
This theorem is referenced by:  0lmhm  21057  idlmhm  21058  invlmhm  21059  lmhmco  21060  lmhmplusg  21061  lmhmvsca  21062  lmhmf1o  21063  reslmhm2  21070  reslmhm2b  21071  pwsdiaglmhm  21074  pwssplit3  21078  frlmup1  21836  imaslmhm  33365  quslmhm  33367  lmhmqusker  33425  frlmsnic  42527
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