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

Theorem frlmval 21170
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 3462 . . 3 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 elex 3462 . . 3 (𝐼 ∈ π‘Š β†’ 𝐼 ∈ V)
4 id 22 . . . . 5 (π‘Ÿ = 𝑅 β†’ π‘Ÿ = 𝑅)
5 fveq2 6843 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (ringLModβ€˜π‘Ÿ) = (ringLModβ€˜π‘…))
65sneqd 4599 . . . . . 6 (π‘Ÿ = 𝑅 β†’ {(ringLModβ€˜π‘Ÿ)} = {(ringLModβ€˜π‘…)})
76xpeq2d 5664 . . . . 5 (π‘Ÿ = 𝑅 β†’ (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)}) = (𝑖 Γ— {(ringLModβ€˜π‘…)}))
84, 7oveq12d 7376 . . . 4 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)})) = (𝑅 βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘…)})))
9 xpeq1 5648 . . . . 5 (𝑖 = 𝐼 β†’ (𝑖 Γ— {(ringLModβ€˜π‘…)}) = (𝐼 Γ— {(ringLModβ€˜π‘…)}))
109oveq2d 7374 . . . 4 (𝑖 = 𝐼 β†’ (𝑅 βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘…)})) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
11 df-frlm 21169 . . . 4 freeLMod = (π‘Ÿ ∈ V, 𝑖 ∈ V ↦ (π‘Ÿ βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)})))
12 ovex 7391 . . . 4 (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})) ∈ V
138, 10, 11, 12ovmpo 7516 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) β†’ (𝑅 freeLMod 𝐼) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
142, 3, 13syl2an 597 . 2 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘Š) β†’ (𝑅 freeLMod 𝐼) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
151, 14eqtrid 2785 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444  {csn 4587   Γ— cxp 5632  β€˜cfv 6497  (class class class)co 7358  ringLModcrglmod 20646   βŠ•m cdsmm 21153   freeLMod cfrlm 21168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frlm 21169
This theorem is referenced by:  frlmlmod  21171  frlmpws  21172  frlmlss  21173  frlmpwsfi  21174  frlmbas  21177
  Copyright terms: Public domain W3C validator