Theorem lspfval 20864

Description: The span function for a left vector space (or a left module). (df-span 31139 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 3492	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6902	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lspval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2786	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4623	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fveq2 6902	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8		lspval.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊)
9	7, 8	eqtr4di 2786	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10	9	rabeqdv 3446	. . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
11	10	inteqd 4958	. . . . 5 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
12	6, 11	mpteq12dv 5243	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
13		df-lsp 20863	. . . 4 ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
14	4	fvexi 6916	. . . . . 6 ⊢ 𝑉 ∈ V
15	14	pwex 5384	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
16	15	mptex 7241	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) ∈ V
17	12, 13, 16	fvmpt 7010	. . 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 2780	1 ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473   ⊆ wss 3949  𝒫 cpw 4606  ∩ cint 4953   ↦ cmpt 5235  ‘cfv 6553  Basecbs 17187  LSubSpclss 20822  LSpanclspn 20862

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-lsp 20863

This theorem is referenced by:  lspf  20865  lspval  20866  00lsp  20872  mrclsp  20880  lsppropd  20910