Theorem frlmval 21703

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 3461	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3461	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
5		fveq2 6834	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
6	5	sneqd 4592	. . . . . 6 ⊢ (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
7	6	xpeq2d 5654	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)}))
8	4, 7	oveq12d 7376	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})))
9		xpeq1 5638	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)}))
10	9	oveq2d 7374	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
11		df-frlm 21702	. . . 4 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
12		ovex 7391	. . . 4 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ V
13	8, 10, 11, 12	ovmpo 7518	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
14	2, 3, 13	syl2an 596	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
15	1, 14	eqtrid 2783	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  {csn 4580   × cxp 5622  ‘cfv 6492  (class class class)co 7358  ringLModcrglmod 21124   ⊕_m cdsmm 21686   freeLMod cfrlm 21701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frlm 21702

This theorem is referenced by:  frlmlmod  21704  frlmpws  21705  frlmlss  21706  frlmpwsfi  21707  frlmbas  21710