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Theorem pexmidN 34273
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 34257. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 34271. (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 786 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝐾 ∈ HL)
2 simplr 788 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋𝐴)
3 pexmid.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
4 pexmid.o . . . . . . 7 = (⊥𝑃𝐾)
53, 4polssatN 34212 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ⊆ 𝐴)
65adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ⊆ 𝐴)
7 pexmid.p . . . . . 6 + = (+𝑃𝐾)
83, 7, 4poldmj1N 34232 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ ( 𝑋) ⊆ 𝐴) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
91, 2, 6, 8syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
103, 4pnonsingN 34237 . . . . 5 ((𝐾 ∈ HL ∧ ( 𝑋) ⊆ 𝐴) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
111, 6, 10syl2anc 691 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
129, 11eqtrd 2644 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = ∅)
1312fveq2d 6107 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = ( ‘∅))
14 simpr 476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( 𝑋)) = 𝑋)
15 eqid 2610 . . . . . . 7 (PSubCl‘𝐾) = (PSubCl‘𝐾)
163, 4, 15ispsubclN 34241 . . . . . 6 (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
1716ad2antrr 758 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
182, 14, 17mpbir2and 959 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
193, 4, 15polsubclN 34256 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ (PSubCl‘𝐾))
2019adantr 480 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ∈ (PSubCl‘𝐾))
213, 42polssN 34219 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → 𝑋 ⊆ ( ‘( 𝑋)))
2221adantr 480 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ⊆ ( ‘( 𝑋)))
237, 4, 15osumclN 34271 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( 𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ‘( 𝑋))) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
241, 18, 20, 22, 23syl31anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
254, 15psubcli2N 34243 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾)) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
261, 24, 25syl2anc 691 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
273, 4pol0N 34213 . . 3 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
2827ad2antrr 758 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘∅) = 𝐴)
2913, 26, 283eqtr3d 2652 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cin 3539  wss 3540  c0 3874  cfv 5804  (class class class)co 6549  Atomscatm 33568  HLchlt 33655  +𝑃cpadd 34099  𝑃cpolN 34206  PSubClcpscN 34238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-polarityN 34207  df-psubclN 34239
This theorem is referenced by: (None)
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