Theorem rlmval 21199

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 6905	. . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2		fveq2 6905	. . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
3	1, 2	fveq12d 6912	. . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4		df-rgmod 21174	. . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5		fvex 6918	. . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
6	3, 4, 5	fvmpt 7015	. 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7		0fv 6949	. . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅
8	7	eqcomi 2745	. . 3 ⊢ ∅ = (∅‘(Base‘𝑊))
9		fvprc 6897	. . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10		fvprc 6897	. . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
11	10	fveq1d 6907	. . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
12	8, 9, 11	3eqtr4a 2802	. 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 1539   ∈ wcel 2107  Vcvv 3479  ∅c0 4332  ‘cfv 6560  Basecbs 17248  subringAlg csra 21171  ringLModcrglmod 21172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-rgmod 21174

This theorem is referenced by:  rlmval2  21200  rlmbas  21201  rlmplusg  21202  rlm0  21203  rlmmulr  21205  rlmsca  21206  rlmsca2  21207  rlmvsca  21208  rlmtopn  21209  rlmds  21210  rlmlmod  21211  frlmip  21799  rlmassa  21892  rlmnlm  24710  rlmbn  25396  rrxprds  25424  rlmdim  33661  rgmoddimOLD  33662  extdgid  33712