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Theorem lspfval 20449
Description: The span function for a left vector space (or a left module). (df-span 30293 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lspval.v 𝑉 = (Baseβ€˜π‘Š)
lspval.s 𝑆 = (LSubSpβ€˜π‘Š)
lspval.n 𝑁 = (LSpanβ€˜π‘Š)
Assertion
Ref Expression
lspfval (π‘Š ∈ 𝑋 β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
Distinct variable groups:   𝑑,𝑠,𝑆   𝑉,𝑠,𝑑   π‘Š,𝑠
Allowed substitution hints:   𝑁(𝑑,𝑠)   π‘Š(𝑑)   𝑋(𝑑,𝑠)

Proof of Theorem lspfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lspval.n . 2 𝑁 = (LSpanβ€˜π‘Š)
2 elex 3462 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6843 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 lspval.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2791 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
65pweqd 4578 . . . . 5 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
7 fveq2 6843 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = (LSubSpβ€˜π‘Š))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSpβ€˜π‘Š)
97, 8eqtr4di 2791 . . . . . . 7 (𝑀 = π‘Š β†’ (LSubSpβ€˜π‘€) = 𝑆)
109rabeqdv 3421 . . . . . 6 (𝑀 = π‘Š β†’ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
1110inteqd 4913 . . . . 5 (𝑀 = π‘Š β†’ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
126, 11mpteq12dv 5197 . . . 4 (𝑀 = π‘Š β†’ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
13 df-lsp 20448 . . . 4 LSpan = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}))
144fvexi 6857 . . . . . 6 𝑉 ∈ V
1514pwex 5336 . . . . 5 𝒫 𝑉 ∈ V
1615mptex 7174 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}) ∈ V
1712, 13, 16fvmpt 6949 . . 3 (π‘Š ∈ V β†’ (LSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
182, 17syl 17 . 2 (π‘Š ∈ 𝑋 β†’ (LSpanβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
191, 18eqtrid 2785 1 (π‘Š ∈ 𝑋 β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444   βŠ† wss 3911  π’« cpw 4561  βˆ© cint 4908   ↦ cmpt 5189  β€˜cfv 6497  Basecbs 17088  LSubSpclss 20407  LSpanclspn 20447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-lsp 20448
This theorem is referenced by:  lspf  20450  lspval  20451  00lsp  20457  mrclsp  20465  lsppropd  20494
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