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Theorem frlmval 21786
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 3499 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3499 . . 3 (𝐼𝑊𝐼 ∈ V)
4 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
5 fveq2 6907 . . . . . . 7 (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
65sneqd 4643 . . . . . 6 (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
76xpeq2d 5719 . . . . 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 5703 . . . . 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 21785 . . . 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 2787 1 ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631   × cxp 5687  cfv 6563  (class class class)co 7431  ringLModcrglmod 21189  m cdsmm 21769   freeLMod cfrlm 21784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frlm 21785
This theorem is referenced by:  frlmlmod  21787  frlmpws  21788  frlmlss  21789  frlmpwsfi  21790  frlmbas  21793
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