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Theorem islmhmd 19811
 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 3186 . . 3 (𝜑 → ∀𝑥𝑁𝑦𝑋 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
73, 4, 63jca 1125 . 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 19799 . 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 1341 1 (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  Scalarcsca 16568   ·𝑠 cvsca 16569   GrpHom cghm 18355  LModclmod 19634   LMHom clmhm 19791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-lmhm 19794 This theorem is referenced by:  0lmhm  19812  idlmhm  19813  invlmhm  19814  lmhmco  19815  lmhmplusg  19816  lmhmvsca  19817  lmhmf1o  19818  reslmhm2  19825  reslmhm2b  19826  pwsdiaglmhm  19829  pwssplit3  19833  frlmup1  20494  quslmhm  30961  frlmsnic  39391
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