Theorem rlmval 21265

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 6867	. . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2		fveq2 6867	. . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
3	1, 2	fveq12d 6874	. . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4		df-rgmod 21248	. . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5		fvex 6880	. . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
6	3, 4, 5	fvmpt 6975	. 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7		0fv 6908	. . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅
8	7	eqcomi 2772	. . 3 ⊢ ∅ = (∅‘(Base‘𝑊))
9		fvprc 6859	. . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10		fvprc 6859	. . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
11	10	fveq1d 6869	. . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
12	8, 9, 11	3eqtr4a 2824	. 2 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
13	6, 12	pm2.61i 183	1 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

Syntax hints:  ¬ wn 3   = wceq 1561   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  ‘cfv 6521  Basecbs 17255  subringAlg csra 21245  ringLModcrglmod 21246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-rgmod 21248

This theorem is referenced by:  rlmval2  21266  rlmbas  21267  rlmplusg  21268  rlm0  21269  rlmmulr  21271  rlmsca  21272  rlmsca2  21273  rlmvsca  21274  rlmtopn  21275  rlmds  21276  rlmlmod  21277  frlmip  21837  rlmassa  21929  rlmnlm  24755  rlmbn  25430  rrxprds  25458  rlmdim  33909  extdgid  33959