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Theorem slmdsrg 29587
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 2621 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . 3 (+g𝑊) = (+g𝑊)
3 eqid 2621 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
4 eqid 2621 . . 3 (0g𝑊) = (0g𝑊)
5 slmdsrg.1 . . 3 𝐹 = (Scalar‘𝑊)
6 eqid 2621 . . 3 (Base‘𝐹) = (Base‘𝐹)
7 eqid 2621 . . 3 (+g𝐹) = (+g𝐹)
8 eqid 2621 . . 3 (.r𝐹) = (.r𝐹)
9 eqid 2621 . . 3 (1r𝐹) = (1r𝐹)
10 eqid 2621 . . 3 (0g𝐹) = (0g𝐹)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10isslmd 29582 . 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 1075 1 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  0gc0g 16040  CMndccmn 18133  1rcur 18441  SRingcsrg 18445  SLModcslmd 29580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-slmd 29581
This theorem is referenced by:  slmdacl  29589  slmdmcl  29590  slmdsn0  29591  slmd0cl  29598  slmd1cl  29599  slmdvs0  29605  gsumvsca2  29610
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