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Theorem islmhm3 19792
Description: Property of a module homomorphism, similar to ismhm 17950. (Contributed by Stefan O'Rear, 7-Mar-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
islmhm3 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (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 islmhm3
StepHypRef Expression
1 islmhm.k . . 3 𝐾 = (Scalar‘𝑆)
2 islmhm.l . . 3 𝐿 = (Scalar‘𝑇)
3 islmhm.b . . 3 𝐵 = (Base‘𝐾)
4 islmhm.e . . 3 𝐸 = (Base‘𝑆)
5 islmhm.m . . 3 · = ( ·𝑠𝑆)
6 islmhm.n . . 3 × = ( ·𝑠𝑇)
71, 2, 3, 4, 5, 6islmhm 19791 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
87baib 538 1 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  cfv 6348  (class class class)co 7148  Basecbs 16475  Scalarcsca 16560   ·𝑠 cvsca 16561   GrpHom cghm 18347  LModclmod 19626   LMHom clmhm 19783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-lmhm 19786
This theorem is referenced by:  islmhm2  19802  pj1lmhm  19864
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