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Theorem lss1 20549
Description: The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lssss.v 𝑉 = (Baseβ€˜π‘Š)
lssss.s 𝑆 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
lss1 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝑆)

Proof of Theorem lss1
Dummy variables π‘Ž 𝑏 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2734 . 2 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š))
2 eqidd 2734 . 2 (π‘Š ∈ LMod β†’ (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
3 lssss.v . . 3 𝑉 = (Baseβ€˜π‘Š)
43a1i 11 . 2 (π‘Š ∈ LMod β†’ 𝑉 = (Baseβ€˜π‘Š))
5 eqidd 2734 . 2 (π‘Š ∈ LMod β†’ (+gβ€˜π‘Š) = (+gβ€˜π‘Š))
6 eqidd 2734 . 2 (π‘Š ∈ LMod β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š))
7 lssss.s . . 3 𝑆 = (LSubSpβ€˜π‘Š)
87a1i 11 . 2 (π‘Š ∈ LMod β†’ 𝑆 = (LSubSpβ€˜π‘Š))
9 ssidd 4006 . 2 (π‘Š ∈ LMod β†’ 𝑉 βŠ† 𝑉)
103lmodbn0 20482 . 2 (π‘Š ∈ LMod β†’ 𝑉 β‰  βˆ…)
11 simpl 484 . . 3 ((π‘Š ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ π‘Š ∈ LMod)
12 eqid 2733 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
13 eqid 2733 . . . . 5 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
14 eqid 2733 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
153, 12, 13, 14lmodvscl 20489 . . . 4 ((π‘Š ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Ž ∈ 𝑉) β†’ (π‘₯( ·𝑠 β€˜π‘Š)π‘Ž) ∈ 𝑉)
16153adant3r3 1185 . . 3 ((π‘Š ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ (π‘₯( ·𝑠 β€˜π‘Š)π‘Ž) ∈ 𝑉)
17 simpr3 1197 . . 3 ((π‘Š ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ 𝑏 ∈ 𝑉)
18 eqid 2733 . . . 4 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
193, 18lmodvacl 20486 . . 3 ((π‘Š ∈ LMod ∧ (π‘₯( ·𝑠 β€˜π‘Š)π‘Ž) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) β†’ ((π‘₯( ·𝑠 β€˜π‘Š)π‘Ž)(+gβ€˜π‘Š)𝑏) ∈ 𝑉)
2011, 16, 17, 19syl3anc 1372 . 2 ((π‘Š ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ ((π‘₯( ·𝑠 β€˜π‘Š)π‘Ž)(+gβ€˜π‘Š)𝑏) ∈ 𝑉)
211, 2, 4, 5, 6, 8, 9, 10, 20islssd 20546 1 (π‘Š ∈ LMod β†’ 𝑉 ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200   ·𝑠 cvsca 17201  LModclmod 20471  LSubSpclss 20542
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-lmod 20473  df-lss 20543
This theorem is referenced by:  lssuni  20550  islss3  20570  lssmre  20577  lspf  20585  lspval  20586  lmhmrnlss  20661  lidl1  20845  isphld  21207  ocv1  21232  aspval  21427  islshpcv  37923  dochexmidlem8  40338  hdmaprnlem4N  40724  lnmfg  41824
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