Theorem slmdsrg 33287

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 2737	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		eqid 2737	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		slmdsrg.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
6		eqid 2737	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		eqid 2737	. . 3 ⊢ (+g‘𝐹) = (+g‘𝐹)
8		eqid 2737	. . 3 ⊢ (.r‘𝐹) = (.r‘𝐹)
9		eqid 2737	. . 3 ⊢ (1r‘𝐹) = (1r‘𝐹)
10		eqid 2737	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	isslmd 33282	. 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 1147	1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  +_gcplusg 17215  ._rcmulr 17216  Scalarcsca 17218   ·_𝑠 cvsca 17219  0_gc0g 17397  CMndccmn 19750  1_rcur 20157  SRingcsrg 20162  SLModcslmd 33280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6450  df-fv 6502  df-ov 7365  df-slmd 33281

This theorem is referenced by:  slmdacl  33289  slmdmcl  33290  slmdsn0  33291  slmd0cl  33298  slmd1cl  33299  slmdvs0  33305  gsumvsca2  33307