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Theorem slmdsrg 30888
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 2801 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2801 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2801 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2801 . . 3 (0g𝑊) = (0g𝑊)
5 slmdsrg.1 . . 3 𝐹 = (Scalar‘𝑊)
6 eqid 2801 . . 3 (Base‘𝐹) = (Base‘𝐹)
7 eqid 2801 . . 3 (+g𝐹) = (+g𝐹)
8 eqid 2801 . . 3 (.r𝐹) = (.r𝐹)
9 eqid 2801 . . 3 (1r𝐹) = (1r𝐹)
10 eqid 2801 . . 3 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 30883 . 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 1143 1 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  0gc0g 16708  CMndccmn 18901  1rcur 19247  SRingcsrg 19251  SLModcslmd 30881
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-slmd 30882
This theorem is referenced by:  slmdacl  30890  slmdmcl  30891  slmdsn0  30892  slmd0cl  30899  slmd1cl  30900  slmdvs0  30906  gsumvsca2  30908
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