Theorem slmdsrg 33176

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 2731	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2731	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		eqid 2731	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2731	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdsrg.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
6		eqid 2731	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		eqid 2731	. . 3 ⊢ (+g‘𝐹) = (+g‘𝐹)
8		eqid 2731	. . 3 ⊢ (.r‘𝐹) = (.r‘𝐹)
9		eqid 2731	. . 3 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2731	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	isslmd 33171	. 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 1146	1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343  CMndccmn 19692  1_rcur 20099  SRingcsrg 20104  SLModcslmd 33169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-slmd 33170

This theorem is referenced by:  slmdacl  33178  slmdmcl  33179  slmdsn0  33180  slmd0cl  33187  slmd1cl  33188  slmdvs0  33194  gsumvsca2  33196