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Theorem frlmval 21768
Description: Value of the "free module" function. (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 3501 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3501 . . 3 (𝐼𝑊𝐼 ∈ V)
4 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
5 fveq2 6906 . . . . . . 7 (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
65sneqd 4638 . . . . . 6 (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
76xpeq2d 5715 . . . . 5 (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)}))
84, 7oveq12d 7449 . . . 4 (𝑟 = 𝑅 → (𝑟m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅m (𝑖 × {(ringLMod‘𝑅)})))
9 xpeq1 5699 . . . . 5 (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)}))
109oveq2d 7447 . . . 4 (𝑖 = 𝐼 → (𝑅m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
11 df-frlm 21767 . . . 4 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
12 ovex 7464 . . . 4 (𝑅m (𝐼 × {(ringLMod‘𝑅)})) ∈ V
138, 10, 11, 12ovmpo 7593 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
142, 3, 13syl2an 596 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 freeLMod 𝐼) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
151, 14eqtrid 2789 1 ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626   × cxp 5683  cfv 6561  (class class class)co 7431  ringLModcrglmod 21171  m cdsmm 21751   freeLMod cfrlm 21766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frlm 21767
This theorem is referenced by:  frlmlmod  21769  frlmpws  21770  frlmlss  21771  frlmpwsfi  21772  frlmbas  21775
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