MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlmval Structured version   Visualization version   GIF version

Theorem rlmval 19954
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.
StepHypRef Expression
1 fveq2 6652 . . . 4 (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2 fveq2 6652 . . . 4 (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
31, 2fveq12d 6659 . . 3 (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4 df-rgmod 19936 . . 3 ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5 fvex 6665 . . 3 ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
63, 4, 5fvmpt 6750 . 2 (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7 0fv 6691 . . . 4 (∅‘(Base‘𝑊)) = ∅
87eqcomi 2831 . . 3 ∅ = (∅‘(Base‘𝑊))
9 fvprc 6645 . . 3 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10 fvprc 6645 . . . 4 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
1110fveq1d 6654 . . 3 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
128, 9, 113eqtr4a 2883 . 2 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
136, 12pm2.61i 185 1 (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265  cfv 6334  Basecbs 16474  subringAlg csra 19931  ringLModcrglmod 19932
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-rgmod 19936
This theorem is referenced by:  rlmval2  19957  rlmbas  19958  rlmplusg  19959  rlm0  19960  rlmmulr  19962  rlmsca  19963  rlmsca2  19964  rlmvsca  19965  rlmtopn  19966  rlmds  19967  rlmlmod  19968  frlmip  20465  rlmassa  20555  rlmnlm  23292  rlmbn  23963  rrxprds  23991  rgmoddim  31065  extdgid  31107
  Copyright terms: Public domain W3C validator