Theorem lspfval 14536

Description: The span function for a left vector space (or a left module). (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		df-lsp 14535	. . 3 ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
3		fveq2 5670	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lspval.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2283	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 3674	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fveq2 5670	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8		lspval.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊)
9	7, 8	eqtr4di 2283	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10	9	rabeqdv 2807	. . . . 5 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
11	10	inteqd 3954	. . . 4 ⊢ (𝑤 = 𝑊 → ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})
12	6, 11	mpteq12dv 4192	. . 3 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
13		elex 2825	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
14		basfn 13271	. . . . . . 7 ⊢ Base Fn V
15		funfvex 5687	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
16	15	funfni 5458	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
17	14, 13, 16	sylancr 414	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (Base‘𝑊) ∈ V)
18	4, 17	eqeltrid 2319	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → 𝑉 ∈ V)
19	18	pwexd 4294	. . . 4 ⊢ (𝑊 ∈ 𝑋 → 𝒫 𝑉 ∈ V)
20	19	mptexd 5913	. . 3 ⊢ (𝑊 ∈ 𝑋 → (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}) ∈ V)
21	2, 12, 13, 20	fvmptd3 5771	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))
22	1, 21	eqtrid 2277	1 ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡}))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  {crab 2524  Vcvv 2813   ⊆ wss 3211  𝒫 cpw 3669  ∩ cint 3949   ↦ cmpt 4171   Fn wfn 5347  ‘cfv 5352  Basecbs 13212  LSubSpclss 14500  LSpanclspn 14534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-lsp 14535

This theorem is referenced by:  lspf  14537  lspval  14538  lspex  14543  lsppropd  14580