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Theorem pexmidN 37213
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 37197. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 37211. (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 766 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝐾 ∈ HL)
2 simplr 768 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋𝐴)
3 pexmid.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
4 pexmid.o . . . . . . 7 = (⊥𝑃𝐾)
53, 4polssatN 37152 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ⊆ 𝐴)
65adantr 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ⊆ 𝐴)
7 pexmid.p . . . . . 6 + = (+𝑃𝐾)
83, 7, 4poldmj1N 37172 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ ( 𝑋) ⊆ 𝐴) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
91, 2, 6, 8syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
103, 4pnonsingN 37177 . . . . 5 ((𝐾 ∈ HL ∧ ( 𝑋) ⊆ 𝐴) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
111, 6, 10syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
129, 11eqtrd 2859 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = ∅)
1312fveq2d 6665 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = ( ‘∅))
14 simpr 488 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( 𝑋)) = 𝑋)
15 eqid 2824 . . . . . . 7 (PSubCl‘𝐾) = (PSubCl‘𝐾)
163, 4, 15ispsubclN 37181 . . . . . 6 (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
1716ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
182, 14, 17mpbir2and 712 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
193, 4, 15polsubclN 37196 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ (PSubCl‘𝐾))
2019adantr 484 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ∈ (PSubCl‘𝐾))
213, 42polssN 37159 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → 𝑋 ⊆ ( ‘( 𝑋)))
2221adantr 484 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ⊆ ( ‘( 𝑋)))
237, 4, 15osumclN 37211 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( 𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ‘( 𝑋))) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
241, 18, 20, 22, 23syl31anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
254, 15psubcli2N 37183 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾)) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
261, 24, 25syl2anc 587 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
273, 4pol0N 37153 . . 3 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
2827ad2antrr 725 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘∅) = 𝐴)
2913, 26, 283eqtr3d 2867 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3918  wss 3919  c0 4276  cfv 6343  (class class class)co 7149  Atomscatm 36507  HLchlt 36594  +𝑃cpadd 37039  𝑃cpolN 37146  PSubClcpscN 37178
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-riotaBAD 36197
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-undef 7935  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-oposet 36420  df-ol 36422  df-oml 36423  df-covers 36510  df-ats 36511  df-atl 36542  df-cvlat 36566  df-hlat 36595  df-psubsp 36747  df-pmap 36748  df-padd 37040  df-polarityN 37147  df-psubclN 37179
This theorem is referenced by: (None)
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