Theorem frlmval 21788

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 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))

Proof of Theorem frlmval

Dummy variables 𝑟 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmval.f	. 2 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
2		elex 3474	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3474	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
5		fveq2 6862	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
6	5	sneqd 4591	. . . . . 6 ⊢ (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
7	6	xpeq2d 5673	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)}))
8	4, 7	oveq12d 7409	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})))
9		xpeq1 5657	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)}))
10	9	oveq2d 7407	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
11		df-frlm 21787	. . . 4 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
12		ovex 7424	. . . 4 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ V
13	8, 10, 11, 12	ovmpo 7551	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
14	2, 3, 13	syl2an 605	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
15	1, 14	eqtrid 2808	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {csn 4579   × cxp 5641  ‘cfv 6516  (class class class)co 7391  ringLModcrglmod 21227   ⊕_m cdsmm 21771   freeLMod cfrlm 21786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-frlm 21787

This theorem is referenced by:  frlmlmod  21789  frlmpws  21790  frlmlss  21791  frlmpwsfi  21792  frlmbas  21795