Theorem rlmval 21160

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 6844	. . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2		fveq2 6844	. . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
3	1, 2	fveq12d 6851	. . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4		df-rgmod 21143	. . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5		fvex 6857	. . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
6	3, 4, 5	fvmpt 6951	. 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7		0fv 6885	. . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅
8	7	eqcomi 2746	. . 3 ⊢ ∅ = (∅‘(Base‘𝑊))
9		fvprc 6836	. . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10		fvprc 6836	. . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
11	10	fveq1d 6846	. . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
12	8, 9, 11	3eqtr4a 2798	. 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 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ‘cfv 6502  Basecbs 17150  subringAlg csra 21140  ringLModcrglmod 21141

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6458  df-fun 6504  df-fv 6510  df-rgmod 21143

This theorem is referenced by:  rlmval2  21161  rlmbas  21162  rlmplusg  21163  rlm0  21164  rlmmulr  21166  rlmsca  21167  rlmsca2  21168  rlmvsca  21169  rlmtopn  21170  rlmds  21171  rlmlmod  21172  frlmip  21750  rlmassa  21843  rlmnlm  24649  rlmbn  25334  rrxprds  25362  rlmdim  33793  rgmoddimOLD  33794  extdgid  33844