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Theorem frlmval 21638
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 3487 . . 3 (𝑅 ∈ 𝑉 β†’ 𝑅 ∈ V)
3 elex 3487 . . 3 (𝐼 ∈ π‘Š β†’ 𝐼 ∈ V)
4 id 22 . . . . 5 (π‘Ÿ = 𝑅 β†’ π‘Ÿ = 𝑅)
5 fveq2 6884 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (ringLModβ€˜π‘Ÿ) = (ringLModβ€˜π‘…))
65sneqd 4635 . . . . . 6 (π‘Ÿ = 𝑅 β†’ {(ringLModβ€˜π‘Ÿ)} = {(ringLModβ€˜π‘…)})
76xpeq2d 5699 . . . . 5 (π‘Ÿ = 𝑅 β†’ (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)}) = (𝑖 Γ— {(ringLModβ€˜π‘…)}))
84, 7oveq12d 7422 . . . 4 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)})) = (𝑅 βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘…)})))
9 xpeq1 5683 . . . . 5 (𝑖 = 𝐼 β†’ (𝑖 Γ— {(ringLModβ€˜π‘…)}) = (𝐼 Γ— {(ringLModβ€˜π‘…)}))
109oveq2d 7420 . . . 4 (𝑖 = 𝐼 β†’ (𝑅 βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘…)})) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
11 df-frlm 21637 . . . 4 freeLMod = (π‘Ÿ ∈ V, 𝑖 ∈ V ↦ (π‘Ÿ βŠ•m (𝑖 Γ— {(ringLModβ€˜π‘Ÿ)})))
12 ovex 7437 . . . 4 (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})) ∈ V
138, 10, 11, 12ovmpo 7563 . . 3 ((𝑅 ∈ V ∧ 𝐼 ∈ V) β†’ (𝑅 freeLMod 𝐼) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
142, 3, 13syl2an 595 . 2 ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘Š) β†’ (𝑅 freeLMod 𝐼) = (𝑅 βŠ•m (𝐼 Γ— {(ringLModβ€˜π‘…)})))
151, 14eqtrid 2778 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  {csn 4623   Γ— cxp 5667  β€˜cfv 6536  (class class class)co 7404  ringLModcrglmod 21017   βŠ•m cdsmm 21621   freeLMod cfrlm 21636
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-frlm 21637
This theorem is referenced by:  frlmlmod  21639  frlmpws  21640  frlmlss  21641  frlmpwsfi  21642  frlmbas  21645
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