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Theorem polcon3N 34017
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 1055 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑋𝑌)
2 iinss1 4463 . . 3 (𝑋𝑌 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) ⊆ 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))
3 sslin 3800 . . 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 2609 . . . 4 (oc‘𝐾) = (oc‘𝐾)
6 2polss.a . . . 4 𝐴 = (Atoms‘𝐾)
7 eqid 2609 . . . 4 (pmap‘𝐾) = (pmap‘𝐾)
8 2polss.p . . . 4 = (⊥𝑃𝐾)
95, 6, 7, 8polvalN 34005 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴) → ( 𝑌) = (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
1093adant3 1073 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) = (𝐴 𝑝𝑌 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
11 simp1 1053 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝐾 ∈ HL)
12 simp2 1054 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑌𝐴)
131, 12sstrd 3577 . . 3 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → 𝑋𝐴)
145, 6, 7, 8polvalN 34005 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) = (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
1511, 13, 14syl2anc 690 . 2 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑋) = (𝐴 𝑝𝑋 ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))))
164, 10, 153sstr4d 3610 1 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  cin 3538  wss 3539   ciin 4450  cfv 5790  occoc 15722  Atomscatm 33364  HLchlt 33451  pmapcpmap 33597  𝑃cpolN 34002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-polarityN 34003
This theorem is referenced by:  2polcon4bN  34018  polcon2N  34019  paddunN  34027
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