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Theorem slmdcmn 32620
Description: A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
slmdcmn (π‘Š ∈ SLMod β†’ π‘Š ∈ CMnd)

Proof of Theorem slmdcmn
Dummy variables π‘₯ 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 eqid 2730 . . 3 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
3 eqid 2730 . . 3 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
4 eqid 2730 . . 3 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 eqid 2730 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
6 eqid 2730 . . 3 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
7 eqid 2730 . . 3 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
8 eqid 2730 . . 3 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
9 eqid 2730 . . 3 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
10 eqid 2730 . . 3 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 32617 . 2 (π‘Š ∈ SLMod ↔ (π‘Š ∈ CMnd ∧ (Scalarβ€˜π‘Š) ∈ SRing ∧ βˆ€π‘€ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))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€˜(Scalarβ€˜π‘Š))𝑧)( ·𝑠 β€˜π‘Š)𝑦) = ((𝑀( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)𝑦))) ∧ (((𝑀(.rβ€˜(Scalarβ€˜π‘Š))𝑧)( ·𝑠 β€˜π‘Š)𝑦) = (𝑀( ·𝑠 β€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)𝑦)) ∧ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑦) = 𝑦 ∧ ((0gβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑦) = (0gβ€˜π‘Š)))))
1211simp1bi 1143 1 (π‘Š ∈ SLMod β†’ π‘Š ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  β€˜cfv 6542  (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 32615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-slmd 32616
This theorem is referenced by:  slmdmnd  32621  gsumvsca1  32641  gsumvsca2  32642
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