Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  polcon3N Structured version   Visualization version   GIF version

Theorem polcon3N 39920
Description: Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polss.a 𝐴 = (Atoms‘𝐾)
2polss.p = (⊥𝑃𝐾)
Assertion
Ref Expression
polcon3N ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))

Proof of Theorem polcon3N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simp3 1138 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑋𝑌)
2 iinss1 5006 . . 3 (𝑋𝑌 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) ⊆ 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))
3 sslin 4242 . . 3 ( 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) ⊆ 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) → (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) ⊆ (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
41, 2, 33syl 18 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) ⊆ (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
5 eqid 2736 . . . 4 (oc‘𝐾) = (oc‘𝐾)
6 2polss.a . . . 4 𝐴 = (Atoms‘𝐾)
7 eqid 2736 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
8 2polss.p . . . 4 = (⊥𝑃𝐾)
95, 6, 7, 8polvalN 39908 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴) → ( 𝑌) = (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
1093adant3 1132 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) = (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
11 simp1 1136 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝐾 ∈ HL)
12 simp2 1137 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑌𝐴)
131, 12sstrd 3993 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑋𝐴)
145, 6, 7, 8polvalN 39908 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) = (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
1511, 13, 14syl2anc 584 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑋) = (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
164, 10, 153sstr4d 4038 1 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2107  cin 3949  wss 3950   ciin 4991  cfv 6560  occoc 17306  Atomscatm 39265  HLchlt 39352  pmapcpmap 39500  𝑃cpolN 39905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-polarityN 39906
This theorem is referenced by:  2polcon4bN  39921  polcon2N  39922  paddunN  39930
  Copyright terms: Public domain W3C validator