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Theorem lspss 20739
Description: Span preserves subset ordering. (spanss 30856 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 3989 . . . . 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 4073 . . 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 4973 . . 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 3992 . . 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 20730 . . 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 20730 . . 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 4029 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 3948  βˆ© cint 4950  β€˜cfv 6543  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 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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  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:  lspun  20742  lspssp  20743  lspprid1  20752  lbspss  20837  lspsolvlem  20900  lspsolv  20901  lsppratlem3  20907  lbsextlem2  20917  lbsextlem3  20918  lbsextlem4  20919  lindfrn  21595  f1lindf  21596  mxidlprm  32848  idlsrgmulrss1  32887  idlsrgmulrss2  32888  lindsunlem  32985  dimkerim  32988  lindsadd  36784  lssats  38185  lpssat  38186  lssatle  38188  lssat  38189  dvhdimlem  40618  dvh3dim3N  40623  mapdindp2  40895  lspindp5  40944
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