Theorem frlmval 20092

Description: Value of the free module. (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 3212	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3212	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
5		fveq2 6191	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅))
6	5	sneqd 4189	. . . . . 6 ⊢ (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)})
7	6	xpeq2d 5139	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)}))
8	4, 7	oveq12d 6668	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})))
9		xpeq1 5128	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)}))
10	9	oveq2d 6666	. . . 4 ⊢ (𝑖 = 𝐼 → (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
11		df-frlm 20091	. . . 4 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
12		ovex 6678	. . . 4 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ V
13	8, 10, 11, 12	ovmpt2 6796	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
14	2, 3, 13	syl2an 494	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
15	1, 14	syl5eq 2668	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177   × cxp 5112  ‘cfv 5888  (class class class)co 6650  ringLModcrglmod 19169   ⊕_m cdsmm 20075   freeLMod cfrlm 20090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-frlm 20091

This theorem is referenced by:  frlmlmod  20093  frlmpws  20094  frlmlss  20095  frlmpwsfi  20096  frlmbas  20099