Theorem rlmval 21118

Description: Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Assertion

Ref	Expression
rlmval	⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

Proof of Theorem rlmval

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6817	. . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2		fveq2 6817	. . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
3	1, 2	fveq12d 6824	. . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4		df-rgmod 21101	. . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5		fvex 6830	. . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
6	3, 4, 5	fvmpt 6924	. 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7		0fv 6858	. . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅
8	7	eqcomi 2739	. . 3 ⊢ ∅ = (∅‘(Base‘𝑊))
9		fvprc 6809	. . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10		fvprc 6809	. . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
11	10	fveq1d 6819	. . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
12	8, 9, 11	3eqtr4a 2791	. 2 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
13	6, 12	pm2.61i 182	1 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2110  Vcvv 3434  ∅c0 4281  ‘cfv 6477  Basecbs 17112  subringAlg csra 21098  ringLModcrglmod 21099

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

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-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-rgmod 21101

This theorem is referenced by:  rlmval2  21119  rlmbas  21120  rlmplusg  21121  rlm0  21122  rlmmulr  21124  rlmsca  21125  rlmsca2  21126  rlmvsca  21127  rlmtopn  21128  rlmds  21129  rlmlmod  21130  frlmip  21708  rlmassa  21801  rlmnlm  24596  rlmbn  25281  rrxprds  25309  rlmdim  33612  rgmoddimOLD  33613  extdgid  33663