Theorem slmdsrg 31362

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.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2738	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		eqid 2738	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2738	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdsrg.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
6		eqid 2738	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		eqid 2738	. . 3 ⊢ (+g‘𝐹) = (+g‘𝐹)
8		eqid 2738	. . 3 ⊢ (.r‘𝐹) = (.r‘𝐹)
9		eqid 2738	. . 3 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2738	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	isslmd 31357	. 2 ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑤 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝑧( ·𝑠 ‘𝑊)(𝑦(+g‘𝑊)𝑥)) = ((𝑧( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑥)) ∧ ((𝑤(+g‘𝐹)𝑧)( ·𝑠 ‘𝑊)𝑦) = ((𝑤( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦))) ∧ (((𝑤(.r‘𝐹)𝑧)( ·𝑠 ‘𝑊)𝑦) = (𝑤( ·𝑠 ‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝑊)𝑦) = 𝑦 ∧ ((0g‘𝐹)( ·𝑠 ‘𝑊)𝑦) = (0g‘𝑊)))))
12	11	simp2bi 1144	1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  ._rcmulr 16889  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  CMndccmn 19301  1_rcur 19652  SRingcsrg 19656  SLModcslmd 31355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-slmd 31356

This theorem is referenced by:  slmdacl  31364  slmdmcl  31365  slmdsn0  31366  slmd0cl  31373  slmd1cl  31374  slmdvs0  31380  gsumvsca2  31382