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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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