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Theorem slmdsrg 31460
Description: The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypothesis
Ref Expression
slmdsrg.1 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
slmdsrg (𝑊 ∈ SLMod → 𝐹 ∈ SRing)

Proof of Theorem slmdsrg
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2738 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2738 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2738 . . 3 (0g𝑊) = (0g𝑊)
5 slmdsrg.1 . . 3 𝐹 = (Scalar‘𝑊)
6 eqid 2738 . . 3 (Base‘𝐹) = (Base‘𝐹)
7 eqid 2738 . . 3 (+g𝐹) = (+g𝐹)
8 eqid 2738 . . 3 (.r𝐹) = (.r𝐹)
9 eqid 2738 . . 3 (1r𝐹) = (1r𝐹)
10 eqid 2738 . . 3 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 31455 . 2 (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑤 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝑧( ·𝑠𝑊)(𝑦(+g𝑊)𝑥)) = ((𝑧( ·𝑠𝑊)𝑦)(+g𝑊)(𝑧( ·𝑠𝑊)𝑥)) ∧ ((𝑤(+g𝐹)𝑧)( ·𝑠𝑊)𝑦) = ((𝑤( ·𝑠𝑊)𝑦)(+g𝑊)(𝑧( ·𝑠𝑊)𝑦))) ∧ (((𝑤(.r𝐹)𝑧)( ·𝑠𝑊)𝑦) = (𝑤( ·𝑠𝑊)(𝑧( ·𝑠𝑊)𝑦)) ∧ ((1r𝐹)( ·𝑠𝑊)𝑦) = 𝑦 ∧ ((0g𝐹)( ·𝑠𝑊)𝑦) = (0g𝑊)))))
1211simp2bi 1145 1 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  Scalarcsca 16965   ·𝑠 cvsca 16966  0gc0g 17150  CMndccmn 19386  1rcur 19737  SRingcsrg 19741  SLModcslmd 31453
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-slmd 31454
This theorem is referenced by:  slmdacl  31462  slmdmcl  31463  slmdsn0  31464  slmd0cl  31471  slmd1cl  31472  slmdvs0  31478  gsumvsca2  31480
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