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Theorem slmdsrg 30762
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 2818 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2818 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2818 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2818 . . 3 (0g𝑊) = (0g𝑊)
5 slmdsrg.1 . . 3 𝐹 = (Scalar‘𝑊)
6 eqid 2818 . . 3 (Base‘𝐹) = (Base‘𝐹)
7 eqid 2818 . . 3 (+g𝐹) = (+g𝐹)
8 eqid 2818 . . 3 (.r𝐹) = (.r𝐹)
9 eqid 2818 . . 3 (1r𝐹) = (1r𝐹)
10 eqid 2818 . . 3 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 30757 . 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 1138 1 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701  CMndccmn 18835  1rcur 19180  SRingcsrg 19184  SLModcslmd 30755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-slmd 30756
This theorem is referenced by:  slmdacl  30764  slmdmcl  30765  slmdsn0  30766  slmd0cl  30773  slmd1cl  30774  slmdvs0  30780  gsumvsca2  30782
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