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Theorem polcon3N 37364
 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 1135 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑋𝑌)
2 iinss1 4900 . . 3 (𝑋𝑌 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) ⊆ 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))
3 sslin 4164 . . 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 2798 . . . 4 (oc‘𝐾) = (oc‘𝐾)
6 2polss.a . . . 4 𝐴 = (Atoms‘𝐾)
7 eqid 2798 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
8 2polss.p . . . 4 = (⊥𝑃𝐾)
95, 6, 7, 8polvalN 37352 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴) → ( 𝑌) = (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
1093adant3 1129 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) = (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
11 simp1 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝐾 ∈ HL)
12 simp2 1134 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑌𝐴)
131, 12sstrd 3927 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑋𝐴)
145, 6, 7, 8polvalN 37352 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) = (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
1511, 13, 14syl2anc 587 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑋) = (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
164, 10, 153sstr4d 3964 1 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3882   ⊆ wss 3883  ∩ ciin 4886  ‘cfv 6332  occoc 16585  Atomscatm 36710  HLchlt 36797  pmapcpmap 36944  ⊥𝑃cpolN 37349 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-polarityN 37350 This theorem is referenced by:  2polcon4bN  37365  polcon2N  37366  paddunN  37374
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