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Theorem lspfval 18736
Description: The span function for a left vector space (or a left module). (df-span 27354 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 3180 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6084 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lspval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2657 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4108 . . . . 5 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fveq2 6084 . . . . . . . 8 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
8 lspval.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
97, 8syl6eqr 2657 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
10 rabeq 3161 . . . . . . 7 ((LSubSp‘𝑤) = 𝑆 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
119, 10syl 17 . . . . . 6 (𝑤 = 𝑊 → {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
1211inteqd 4405 . . . . 5 (𝑤 = 𝑊 {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡} = {𝑡𝑆𝑠𝑡})
136, 12mpteq12dv 4653 . . . 4 (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
14 df-lsp 18735 . . . 4 LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))
15 fvex 6094 . . . . . . 7 (Base‘𝑊) ∈ V
164, 15eqeltri 2679 . . . . . 6 𝑉 ∈ V
1716pwex 4765 . . . . 5 𝒫 𝑉 ∈ V
1817mptex 6364 . . . 4 (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}) ∈ V
1913, 14, 18fvmpt 6172 . . 3 (𝑊 ∈ V → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
202, 19syl 17 . 2 (𝑊𝑋 → (LSpan‘𝑊) = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
211, 20syl5eq 2651 1 (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  wss 3535  𝒫 cpw 4103   cint 4400  cmpt 4633  cfv 5786  Basecbs 15637  LSubSpclss 18695  LSpanclspn 18734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-lsp 18735
This theorem is referenced by:  lspf  18737  lspval  18738  00lsp  18744  mrclsp  18752  lsppropd  18781
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