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Theorem slmdsrg 32610
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
slmdsrg (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
Proof of Theorem slmdsrg
Dummy variables π‘₯ 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2732 . . 3 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
3 eqid 2732 . . 3 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
4 eqid 2732 . . 3 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 slmdsrg.1 . . 3 𝐹 = (Scalarβ€˜π‘Š)
6 eqid 2732 . . 3 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
7 eqid 2732 . . 3 (+gβ€˜πΉ) = (+gβ€˜πΉ)
8 eqid 2732 . . 3 (.rβ€˜πΉ) = (.rβ€˜πΉ)
9 eqid 2732 . . 3 (1rβ€˜πΉ) = (1rβ€˜πΉ)
10 eqid 2732 . . 3 (0gβ€˜πΉ) = (0gβ€˜πΉ)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 32605 . 2 (π‘Š ∈ SLMod ↔ (π‘Š ∈ CMnd ∧ 𝐹 ∈ SRing ∧ βˆ€π‘€ ∈ (Baseβ€˜πΉ)βˆ€π‘§ ∈ (Baseβ€˜πΉ)βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š) ∧ (𝑧( ·𝑠 β€˜π‘Š)(𝑦(+gβ€˜π‘Š)π‘₯)) = ((𝑧( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)π‘₯)) ∧ ((𝑀(+gβ€˜πΉ)𝑧)( ·𝑠 β€˜π‘Š)𝑦) = ((𝑀( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)𝑦))) ∧ (((𝑀(.rβ€˜πΉ)𝑧)( ·𝑠 β€˜π‘Š)𝑦) = (𝑀( ·𝑠 β€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)𝑦)) ∧ ((1rβ€˜πΉ)( ·𝑠 β€˜π‘Š)𝑦) = 𝑦 ∧ ((0gβ€˜πΉ)( ·𝑠 β€˜π‘Š)𝑦) = (0gβ€˜π‘Š)))))
1211simp2bi 1146 1 (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389  CMndccmn 19689  1rcur 20075  SRingcsrg 20080  SLModcslmd 32603
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-slmd 32604
This theorem is referenced by:  slmdacl  32612  slmdmcl  32613  slmdsn0  32614  slmd0cl  32621  slmd1cl  32622  slmdvs0  32628  gsumvsca2  32630
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