MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspss Structured version   Visualization version   GIF version
Theorem lspss 20587
Description: Span preserves subset ordering. (spanss 30588 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lspss.v 𝑉 = (Baseβ€˜π‘Š)
lspss.n 𝑁 = (LSpanβ€˜π‘Š)
Assertion
Ref Expression
lspss ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) βŠ† (π‘β€˜π‘ˆ))
Proof of Theorem lspss
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 simpl3 1193 . . . . 5 (((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) ∧ 𝑑 ∈ (LSubSpβ€˜π‘Š)) β†’ 𝑇 βŠ† π‘ˆ)
2 sstr2 3988 . . . . 5 (𝑇 βŠ† π‘ˆ β†’ (π‘ˆ βŠ† 𝑑 β†’ 𝑇 βŠ† 𝑑))
31, 2syl 17 . . . 4 (((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) ∧ 𝑑 ∈ (LSubSpβ€˜π‘Š)) β†’ (π‘ˆ βŠ† 𝑑 β†’ 𝑇 βŠ† 𝑑))
43ss2rabdv 4072 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑} βŠ† {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑})
5 intss 4972 . . 3 ({𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑} βŠ† {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑} β†’ ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑} βŠ† ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑})
64, 5syl 17 . 2 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑} βŠ† ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑})
7 simp1 1136 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ π‘Š ∈ LMod)
8 simp3 1138 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ 𝑇 βŠ† π‘ˆ)
9 simp2 1137 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ π‘ˆ βŠ† 𝑉)
108, 9sstrd 3991 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ 𝑇 βŠ† 𝑉)
11 lspss.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
12 eqid 2732 . . . 4 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
13 lspss.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
1411, 12, 13lspval 20578 . . 3 ((π‘Š ∈ LMod ∧ 𝑇 βŠ† 𝑉) β†’ (π‘β€˜π‘‡) = ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑})
157, 10, 14syl2anc 584 . 2 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) = ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑})
1611, 12, 13lspval 20578 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑})
17163adant3 1132 . 2 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑})
186, 15, 173sstr4d 4028 1 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) βŠ† (π‘β€˜π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3947  βˆ© cint 4949  β€˜cfv 6540  Basecbs 17140  LModclmod 20463  LSubSpclss 20534  LSpanclspn 20574
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  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 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-lmod 20465  df-lss 20535  df-lsp 20575
This theorem is referenced by:  lspun  20590  lspssp  20591  lspprid1  20600  lbspss  20685  lspsolvlem  20747  lspsolv  20748  lsppratlem3  20754  lbsextlem2  20764  lbsextlem3  20765  lbsextlem4  20766  lindfrn  21367  f1lindf  21368  mxidlprm  32574  idlsrgmulrss1  32613  idlsrgmulrss2  32614  lindsunlem  32697  dimkerim  32700  lindsadd  36469  lssats  37870  lpssat  37871  lssatle  37873  lssat  37874  dvhdimlem  40303  dvh3dim3N  40308  mapdindp2  40580  lspindp5  40629
  Copyright terms: Public domain W3C validator