ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lspss GIF version
Theorem lspss 13490
Description: Span preserves subset ordering. (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 1002 . . . . 5 (((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) ∧ 𝑑 ∈ (LSubSpβ€˜π‘Š)) β†’ 𝑇 βŠ† π‘ˆ)
2 sstr2 3164 . . . . 5 (𝑇 βŠ† π‘ˆ β†’ (π‘ˆ βŠ† 𝑑 β†’ 𝑇 βŠ† 𝑑))
31, 2syl 14 . . . 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 3238 . . 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 3867 . . 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 14 . 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 997 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ π‘Š ∈ LMod)
8 simp3 999 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ 𝑇 βŠ† π‘ˆ)
9 simp2 998 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ π‘ˆ βŠ† 𝑉)
108, 9sstrd 3167 . . 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 2177 . . . 4 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
13 lspss.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
1411, 12, 13lspval 13482 . . 3 ((π‘Š ∈ LMod ∧ 𝑇 βŠ† 𝑉) β†’ (π‘β€˜π‘‡) = ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ 𝑇 βŠ† 𝑑})
157, 10, 14syl2anc 411 . 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 13482 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ (LSubSpβ€˜π‘Š) ∣ π‘ˆ βŠ† 𝑑})
17163adant3 1017 . 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 3202 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {crab 2459   βŠ† wss 3131  βˆ© cint 3846  β€˜cfv 5218  Basecbs 12464  LModclmod 13382  LSubSpclss 13447  LSpanclspn 13478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-mulr 12552  df-sca 12554  df-vsca 12555  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-lmod 13384  df-lssm 13448  df-lsp 13479
This theorem is referenced by:  lspun  13493  lspssp  13494  lspprid1  13502
  Copyright terms: Public domain W3C validator