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Theorem pexmidALTN 34282
 Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 34257. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pexmidALT.a 𝐴 = (Atoms‘𝐾)
pexmidALT.p + = (+𝑃𝐾)
pexmidALT.o = (⊥𝑃𝐾)
Assertion
Ref Expression
pexmidALTN (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)

Proof of Theorem pexmidALTN
StepHypRef Expression
1 id 22 . . . 4 (𝑋 = ∅ → 𝑋 = ∅)
2 fveq2 6103 . . . 4 (𝑋 = ∅ → ( 𝑋) = ( ‘∅))
31, 2oveq12d 6567 . . 3 (𝑋 = ∅ → (𝑋 + ( 𝑋)) = (∅ + ( ‘∅)))
4 pexmidALT.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
5 pexmidALT.o . . . . . . . 8 = (⊥𝑃𝐾)
64, 5pol0N 34213 . . . . . . 7 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
7 eqimss 3620 . . . . . . 7 (( ‘∅) = 𝐴 → ( ‘∅) ⊆ 𝐴)
86, 7syl 17 . . . . . 6 (𝐾 ∈ HL → ( ‘∅) ⊆ 𝐴)
9 pexmidALT.p . . . . . . 7 + = (+𝑃𝐾)
104, 9padd02 34116 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘∅) ⊆ 𝐴) → (∅ + ( ‘∅)) = ( ‘∅))
118, 10mpdan 699 . . . . 5 (𝐾 ∈ HL → (∅ + ( ‘∅)) = ( ‘∅))
1211, 6eqtrd 2644 . . . 4 (𝐾 ∈ HL → (∅ + ( ‘∅)) = 𝐴)
1312ad2antrr 758 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (∅ + ( ‘∅)) = 𝐴)
143, 13sylan9eqr 2666 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( 𝑋)) = 𝐴)
154, 9, 5pexmidlem8N 34281 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅)) → (𝑋 + ( 𝑋)) = 𝐴)
1615anassrs 678 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( 𝑋)) = 𝐴)
1714, 16pm2.61dane 2869 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Atomscatm 33568  HLchlt 33655  +𝑃cpadd 34099  ⊥𝑃cpolN 34206 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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