MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frlmval Structured version   Visualization version   GIF version

Theorem frlmval 21689
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 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘Š) β†’ 𝐹 = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))

Proof of Theorem frlmval
Dummy variables π‘Ÿ 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmval.f . 2 𝐹 = (𝑅 freeLMod 𝐼)
2 elex 3492 . . 3 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 elex 3492 . . 3 (𝐼 ∈ π‘Š β†’ 𝐼 ∈ V)
4 id 22 . . . . 5 (π‘Ÿ = 𝑅 β†’ π‘Ÿ = 𝑅)
5 fveq2 6902 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (ringLModβ€˜π‘Ÿ) = (ringLModβ€˜π‘…))
65sneqd 4644 . . . . . 6 (π‘Ÿ = 𝑅 β†’ {(ringLModβ€˜π‘Ÿ)} = {(ringLModβ€˜π‘…)})
76xpeq2d 5712 . . . . 5 (π‘Ÿ = 𝑅 β†’ (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)}) = (𝑖 Γ— {(ringLModβ€˜π‘…)}))
84, 7oveq12d 7444 . . . 4 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)})) = (𝑅 βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘…)})))
9 xpeq1 5696 . . . . 5 (𝑖 = 𝐼 β†’ (𝑖 Γ— {(ringLModβ€˜π‘…)}) = (𝐼 Γ— {(ringLModβ€˜π‘…)}))
109oveq2d 7442 . . . 4 (𝑖 = 𝐼 β†’ (𝑅 βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘…)})) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
11 df-frlm 21688 . . . 4 freeLMod = (π‘Ÿ ∈ V, 𝑖 ∈ V ↦ (π‘Ÿ βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)})))
12 ovex 7459 . . . 4 (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})) ∈ V
138, 10, 11, 12ovmpo 7587 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) β†’ (𝑅 freeLMod 𝐼) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
142, 3, 13syl2an 594 . 2 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘Š) β†’ (𝑅 freeLMod 𝐼) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
151, 14eqtrid 2780 1 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘Š) β†’ 𝐹 = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473  {csn 4632   Γ— cxp 5680  β€˜cfv 6553  (class class class)co 7426  ringLModcrglmod 21064   βŠ•m cdsmm 21672   freeLMod cfrlm 21687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-frlm 21688
This theorem is referenced by:  frlmlmod  21690  frlmpws  21691  frlmlss  21692  frlmpwsfi  21693  frlmbas  21696
  Copyright terms: Public domain W3C validator