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Theorem frlmval 20735
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 3439 . . 3 (𝑅𝑉𝑅 ∈ V)
3 elex 3439 . . 3 (𝐼𝑊𝐼 ∈ V)
4 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
5 fveq2 6736 . . . . . . 7 (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
65sneqd 4568 . . . . . 6 (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
76xpeq2d 5596 . . . . 5 (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)}))
84, 7oveq12d 7250 . . . 4 (𝑟 = 𝑅 → (𝑟m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅m (𝑖 × {(ringLMod‘𝑅)})))
9 xpeq1 5580 . . . . 5 (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)}))
109oveq2d 7248 . . . 4 (𝑖 = 𝐼 → (𝑅m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
11 df-frlm 20734 . . . 4 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
12 ovex 7265 . . . 4 (𝑅m (𝐼 × {(ringLMod‘𝑅)})) ∈ V
138, 10, 11, 12ovmpo 7388 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
142, 3, 13syl2an 599 . 2 ((𝑅𝑉𝐼𝑊) → (𝑅 freeLMod 𝐼) = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
151, 14eqtrid 2790 1 ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  Vcvv 3421  {csn 4556   × cxp 5564  cfv 6398  (class class class)co 7232  ringLModcrglmod 20231  m cdsmm 20718   freeLMod cfrlm 20733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-iota 6356  df-fun 6400  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-frlm 20734
This theorem is referenced by:  frlmlmod  20736  frlmpws  20737  frlmlss  20738  frlmpwsfi  20739  frlmbas  20742
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