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Theorem lspfval 19720
Description: The span function for a left vector space (or a left module). (df-span 29070 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 3489 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6643 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lspval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2874 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4531 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fveq2 6643 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
97, 8syl6eqr 2874 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
109rabeqdv 3461 . . . . . 6 (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
1110inteqd 4854 . . . . 5 (𝑤 = 𝑊 {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
126, 11mpteq12dv 5124 . . . 4 (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
13 df-lsp 19719 . . . 4 LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))
144fvexi 6657 . . . . . 6 𝑉 ∈ V
1514pwex 5254 . . . . 5 𝒫 𝑉 ∈ V
1615mptex 6959 . . . 4 (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}) ∈ V
1712, 13, 16fvmpt 6741 . . 3 (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
182, 17syl 17 . 2 (𝑊𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
191, 18syl5eq 2868 1 (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3130  Vcvv 3471  wss 3910  𝒫 cpw 4512   cint 4849  cmpt 5119  cfv 6328  Basecbs 16461  LSubSpclss 19678  LSpanclspn 19718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-lsp 19719
This theorem is referenced by:  lspf  19721  lspval  19722  00lsp  19728  mrclsp  19736  lsppropd  19765
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