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Theorem lspfval 19332
 Description: The span function for a left vector space (or a left module). (df-span 28723 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 3429 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6433 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lspval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2879 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4383 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fveq2 6433 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
97, 8syl6eqr 2879 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10 rabeq 3405 . . . . . . 7 ((LSubSp‘𝑤) = 𝑆 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
119, 10syl 17 . . . . . 6 (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
1211inteqd 4702 . . . . 5 (𝑤 = 𝑊 {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
136, 12mpteq12dv 4956 . . . 4 (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
14 df-lsp 19331 . . . 4 LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))
154fvexi 6447 . . . . . 6 𝑉 ∈ V
1615pwex 5080 . . . . 5 𝒫 𝑉 ∈ V
1716mptex 6742 . . . 4 (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}) ∈ V
1813, 14, 17fvmpt 6529 . . 3 (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
192, 18syl 17 . 2 (𝑊𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
201, 19syl5eq 2873 1 (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  {crab 3121  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4378  ∩ cint 4697   ↦ cmpt 4952  ‘cfv 6123  Basecbs 16222  LSubSpclss 19288  LSpanclspn 19330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-lsp 19331 This theorem is referenced by:  lspf  19333  lspval  19334  00lsp  19340  mrclsp  19348  lsppropd  19377
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