Theorem slmdsrg 33701

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 2760	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2760	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		eqid 2760	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2760	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdsrg.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
6		eqid 2760	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		eqid 2760	. . 3 ⊢ (+g‘𝐹) = (+g‘𝐹)
8		eqid 2760	. . 3 ⊢ (.r‘𝐹) = (.r‘𝐹)
9		eqid 2760	. . 3 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2760	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	isslmd 33696	. 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 1164	1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  +_gcplusg 17389  ._rcmulr 17390  Scalarcsca 17392   ·_𝑠 cvsca 17393  0_gc0g 17571  CMndccmn 19955  1_rcur 20368  SRingcsrg 20373  SLModcslmd 33694

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-ov 7411  df-slmd 33695

This theorem is used by:  slmdacl  33703  slmdmcl  33704  slmdsn0  33705  slmd0cl  33712  slmd1cl  33713  slmdvs0  33719  gsumvsca2  33721