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Theorem islmhmd 20627
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 3201 . . 3 (𝜑 → ∀𝑥𝑁𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
73, 4, 63jca 1129 . 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 20615 . 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 1345 1 (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  cfv 6535  (class class class)co 7396  Basecbs 17131  Scalarcsca 17187   ·𝑠 cvsca 17188   GrpHom cghm 19074  LModclmod 20448   LMHom clmhm 20607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-lmhm 20610
This theorem is referenced by:  0lmhm  20628  idlmhm  20629  invlmhm  20630  lmhmco  20631  lmhmplusg  20632  lmhmvsca  20633  lmhmf1o  20634  reslmhm2  20641  reslmhm2b  20642  pwsdiaglmhm  20645  pwssplit3  20649  frlmup1  21326  quslmhm  32432  lmhmqusker  32489  frlmsnic  41014
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