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Theorem frlmval 20140
Description: Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypothesis
Ref Expression
frlmval.f 𝐹 = (𝑅 freeLMod 𝐼)
Assertion
Ref Expression
frlmval ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))

Proof of Theorem frlmval
Dummy variables 𝑟 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmval.f . 2 𝐹 = (𝑅 freeLMod 𝐼)
2 elex 3243 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3243 . . 3 (𝐼𝑊𝐼 ∈ V)
4 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
5 fveq2 6229 . . . . . . 7 (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
65sneqd 4222 . . . . . 6 (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
76xpeq2d 5173 . . . . 5 (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)}))
84, 7oveq12d 6708 . . . 4 (𝑟 = 𝑅 → (𝑟m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅m (𝑖 × {(ringLMod‘𝑅)})))
9 xpeq1 5157 . . . . 5 (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)}))
109oveq2d 6706 . . . 4 (𝑖 = 𝐼 → (𝑅m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
11 df-frlm 20139 . . . 4 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
12 ovex 6718 . . . 4 (𝑅m (𝐼 × {(ringLMod‘𝑅)})) ∈ V
138, 10, 11, 12ovmpt2 6838 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
142, 3, 13syl2an 493 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 freeLMod 𝐼) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
151, 14syl5eq 2697 1 ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210   × cxp 5141  cfv 5926  (class class class)co 6690  ringLModcrglmod 19217  m cdsmm 20123   freeLMod cfrlm 20138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-frlm 20139
This theorem is referenced by:  frlmlmod  20141  frlmpws  20142  frlmlss  20143  frlmpwsfi  20144  frlmbas  20147
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