Theorem lspfval 20879

Description: The span function for a left vector space (or a left module). (df-span 31238 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lspval.v	⊢ 𝑉 = (Base‘𝑊)
lspval.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspval.n	⊢ 𝑁 = (LSpan‘𝑊)

Assertion

Ref	Expression
lspfval	⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))

Distinct variable groups:   𝑡,𝑠,𝑆   𝑉,𝑠,𝑡   𝑊,𝑠

Allowed substitution hints:   𝑁(𝑡,𝑠)   𝑊(𝑡)   𝑋(𝑡,𝑠)

Proof of Theorem lspfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspval.n	. 2 ⊢ 𝑁 = (LSpan‘𝑊)
2		elex 3468	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6858	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lspval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2782	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4580	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fveq2 6858	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8		lspval.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊)
9	7, 8	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10	9	rabeqdv 3421	. . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
11	10	inteqd 4915	. . . . 5 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
12	6, 11	mpteq12dv 5194	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
13		df-lsp 20878	. . . 4 ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
14	4	fvexi 6872	. . . . . 6 ⊢ 𝑉 ∈ V
15	14	pwex 5335	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
16	15	mptex 7197	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) ∈ V
17	12, 13, 16	fvmpt 6968	. . 3 ⊢ (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
18	2, 17	syl 17	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
19	1, 18	eqtrid 2776	1 ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∩ cint 4910   ↦ cmpt 5188  ‘cfv 6511  Basecbs 17179  LSubSpclss 20837  LSpanclspn 20877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-lsp 20878

This theorem is referenced by:  lspf  20880  lspval  20881  00lsp  20887  mrclsp  20895  lsppropd  20925