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Theorem lspfval 18794
 Description: The span function for a left vector space (or a left module). (df-span 27552 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 (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
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.
StepHypRef Expression
1 lspval.n . 2 𝑁 = (LSpan‘𝑊)
2 elex 3185 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lspval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4113 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
97, 8syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10 rabeq 3166 . . . . . . 7 ((LSubSp‘𝑤) = 𝑆 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
119, 10syl 17 . . . . . 6 (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
1211inteqd 4415 . . . . 5 (𝑤 = 𝑊 {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
136, 12mpteq12dv 4663 . . . 4 (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
14 df-lsp 18793 . . . 4 LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))
15 fvex 6113 . . . . . . 7 (Base‘𝑊) ∈ V
164, 15eqeltri 2684 . . . . . 6 𝑉 ∈ V
1716pwex 4774 . . . . 5 𝒫 𝑉 ∈ V
1817mptex 6390 . . . 4 (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}) ∈ V
1913, 14, 18fvmpt 6191 . . 3 (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
202, 19syl 17 . 2 (𝑊𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
211, 20syl5eq 2656 1 (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∩ cint 4410   ↦ cmpt 4643  ‘cfv 5804  Basecbs 15695  LSubSpclss 18753  LSpanclspn 18792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-lsp 18793 This theorem is referenced by:  lspf  18795  lspval  18796  00lsp  18802  mrclsp  18810  lsppropd  18839
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