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Theorem lspfval 18913
Description: The span function for a left vector space (or a left module). (df-span 28056 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 3202 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6158 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lspval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2673 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4141 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fveq2 6158 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
97, 8syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10 rabeq 3183 . . . . . . 7 ((LSubSp‘𝑤) = 𝑆 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
119, 10syl 17 . . . . . 6 (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
1211inteqd 4452 . . . . 5 (𝑤 = 𝑊 {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
136, 12mpteq12dv 4703 . . . 4 (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
14 df-lsp 18912 . . . 4 LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))
15 fvex 6168 . . . . . . 7 (Base‘𝑊) ∈ V
164, 15eqeltri 2694 . . . . . 6 𝑉 ∈ V
1716pwex 4818 . . . . 5 𝒫 𝑉 ∈ V
1817mptex 6451 . . . 4 (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}) ∈ V
1913, 14, 18fvmpt 6249 . . 3 (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
202, 19syl 17 . 2 (𝑊𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
211, 20syl5eq 2667 1 (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  wss 3560  𝒫 cpw 4136   cint 4447  cmpt 4683  cfv 5857  Basecbs 15800  LSubSpclss 18872  LSpanclspn 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-lsp 18912
This theorem is referenced by:  lspf  18914  lspval  18915  00lsp  18921  mrclsp  18929  lsppropd  18958
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