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Theorem lspval 20730
Description: The span of a set of vectors (in a left module). (spanval 30853 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
lspval ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
Distinct variable groups:   𝑑,𝑆   𝑑,π‘ˆ   𝑑,𝑉
Allowed substitution hints:   𝑁(𝑑)   π‘Š(𝑑)
Proof of Theorem lspval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 lspval.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
2 lspval.s . . . . 5 𝑆 = (LSubSpβ€˜π‘Š)
3 lspval.n . . . . 5 𝑁 = (LSpanβ€˜π‘Š)
41, 2, 3lspfval 20728 . . . 4 (π‘Š ∈ LMod β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
54fveq1d 6892 . . 3 (π‘Š ∈ LMod β†’ (π‘β€˜π‘ˆ) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})β€˜π‘ˆ))
65adantr 479 . 2 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})β€˜π‘ˆ))
7 eqid 2730 . . 3 (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}) = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})
8 sseq1 4006 . . . . 5 (𝑠 = π‘ˆ β†’ (𝑠 βŠ† 𝑑 ↔ π‘ˆ βŠ† 𝑑))
98rabbidv 3438 . . . 4 (𝑠 = π‘ˆ β†’ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑} = {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
109inteqd 4954 . . 3 (𝑠 = π‘ˆ β†’ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑} = ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
11 simpr 483 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ π‘ˆ βŠ† 𝑉)
121fvexi 6904 . . . . 5 𝑉 ∈ V
1312elpw2 5344 . . . 4 (π‘ˆ ∈ 𝒫 𝑉 ↔ π‘ˆ βŠ† 𝑉)
1411, 13sylibr 233 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ π‘ˆ ∈ 𝒫 𝑉)
151, 2lss1 20693 . . . . 5 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝑆)
16 sseq2 4007 . . . . . 6 (𝑑 = 𝑉 β†’ (π‘ˆ βŠ† 𝑑 ↔ π‘ˆ βŠ† 𝑉))
1716rspcev 3611 . . . . 5 ((𝑉 ∈ 𝑆 ∧ π‘ˆ βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ 𝑆 π‘ˆ βŠ† 𝑑)
1815, 17sylan 578 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ βˆƒπ‘‘ ∈ 𝑆 π‘ˆ βŠ† 𝑑)
19 intexrab 5339 . . . 4 (βˆƒπ‘‘ ∈ 𝑆 π‘ˆ βŠ† 𝑑 ↔ ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑} ∈ V)
2018, 19sylib 217 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑} ∈ V)
217, 10, 14, 20fvmptd3 7020 . 2 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ ((𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑})β€˜π‘ˆ) = ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
226, 21eqtrd 2770 1 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  {crab 3430  Vcvv 3472   βŠ† wss 3947  π’« cpw 4601  βˆ© cint 4949   ↦ cmpt 5230  β€˜cfv 6542  Basecbs 17148  LModclmod 20614  LSubSpclss 20686  LSpanclspn 20726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-lmod 20616  df-lss 20687  df-lsp 20727
This theorem is referenced by:  lspid  20737  lspss  20739  lspssid  20740  dochspss  40552  lcosslsp  47206
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