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Theorem pexmidN 35777
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 35761. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 35775. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmid.a 𝐴 = (Atoms‘𝐾)
pexmid.p + = (+𝑃𝐾)
pexmid.o = (⊥𝑃𝐾)
Assertion
Ref Expression
pexmidN (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)

Proof of Theorem pexmidN
StepHypRef Expression
1 simpll 750 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝐾 ∈ HL)
2 simplr 752 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋𝐴)
3 pexmid.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
4 pexmid.o . . . . . . 7 = (⊥𝑃𝐾)
53, 4polssatN 35716 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ⊆ 𝐴)
65adantr 466 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ⊆ 𝐴)
7 pexmid.p . . . . . 6 + = (+𝑃𝐾)
83, 7, 4poldmj1N 35736 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ ( 𝑋) ⊆ 𝐴) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
91, 2, 6, 8syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
103, 4pnonsingN 35741 . . . . 5 ((𝐾 ∈ HL ∧ ( 𝑋) ⊆ 𝐴) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
111, 6, 10syl2anc 573 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
129, 11eqtrd 2805 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = ∅)
1312fveq2d 6336 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = ( ‘∅))
14 simpr 471 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( 𝑋)) = 𝑋)
15 eqid 2771 . . . . . . 7 (PSubCl‘𝐾) = (PSubCl‘𝐾)
163, 4, 15ispsubclN 35745 . . . . . 6 (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
1716ad2antrr 705 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
182, 14, 17mpbir2and 692 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
193, 4, 15polsubclN 35760 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ (PSubCl‘𝐾))
2019adantr 466 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ∈ (PSubCl‘𝐾))
213, 42polssN 35723 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → 𝑋 ⊆ ( ‘( 𝑋)))
2221adantr 466 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ⊆ ( ‘( 𝑋)))
237, 4, 15osumclN 35775 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( 𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ‘( 𝑋))) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
241, 18, 20, 22, 23syl31anc 1479 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
254, 15psubcli2N 35747 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾)) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
261, 24, 25syl2anc 573 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
273, 4pol0N 35717 . . 3 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
2827ad2antrr 705 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘∅) = 𝐴)
2913, 26, 283eqtr3d 2813 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cin 3722  wss 3723  c0 4063  cfv 6031  (class class class)co 6793  Atomscatm 35072  HLchlt 35159  +𝑃cpadd 35603  𝑃cpolN 35710  PSubClcpscN 35742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-riotaBAD 34761
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-undef 7551  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-polarityN 35711  df-psubclN 35743
This theorem is referenced by: (None)
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