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Theorem rlmval 19185
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 6189 . . . 4 (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊))
2 fveq2 6189 . . . 4 (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊))
31, 2fveq12d 6195 . . 3 (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
4 df-rgmod 19167 . . 3 ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎)))
5 fvex 6199 . . 3 ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V
63, 4, 5fvmpt 6280 . 2 (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
7 0fv 6225 . . . 4 (∅‘(Base‘𝑊)) = ∅
87eqcomi 2630 . . 3 ∅ = (∅‘(Base‘𝑊))
9 fvprc 6183 . . 3 𝑊 ∈ V → (ringLMod‘𝑊) = ∅)
10 fvprc 6183 . . . 4 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅)
1110fveq1d 6191 . . 3 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊)))
128, 9, 113eqtr4a 2681 . 2 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
136, 12pm2.61i 176 1 (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1482  wcel 1989  Vcvv 3198  c0 3913  cfv 5886  Basecbs 15851  subringAlg csra 19162  ringLModcrglmod 19163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-rgmod 19167
This theorem is referenced by:  rlmval2  19188  rlmbas  19189  rlmplusg  19190  rlm0  19191  rlmmulr  19193  rlmsca  19194  rlmsca2  19195  rlmvsca  19196  rlmtopn  19197  rlmds  19198  rlmlmod  19199  rlmassa  19320  frlmip  20111  rlmnlm  22486  rlmbn  23151  rrxprds  23171
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