Theorem rlmval 21323

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 6881	. . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2		fveq2 6881	. . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
3	1, 2	fveq12d 6888	. . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4		df-rgmod 21306	. . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5		fvex 6894	. . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
6	3, 4, 5	fvmpt 6989	. 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7		0fv 6922	. . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅
8	7	eqcomi 2771	. . 3 ⊢ ∅ = (∅‘(Base‘𝑊))
9		fvprc 6873	. . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10		fvprc 6873	. . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
11	10	fveq1d 6883	. . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
12	8, 9, 11	3eqtr4a 2823	. 2 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
13	6, 12	pm2.61i 184	1 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  ‘cfv 6536  Basecbs 17275  subringAlg csra 21303  ringLModcrglmod 21304

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-rgmod 21306

This theorem is used by:  rlmval2  21324  rlmbas  21325  rlmplusg  21326  rlm0  21327  rlmmulr  21329  rlmsca  21330  rlmsca2  21331  rlmvsca  21332  rlmtopn  21333  rlmds  21334  rlmlmod  21335  frlmip  21939  rlmassa  22031  rlmnlm  24856  rlmbn  25531  rrxprds  25559  rlmdim  34009  extdgid  34059