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Theorem pexmidALTN 37000
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 36975. 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 6669 . . . 4 (𝑋 = ∅ → ( 𝑋) = ( ‘∅))
31, 2oveq12d 7168 . . 3 (𝑋 = ∅ → (𝑋 + ( 𝑋)) = (∅ + ( ‘∅)))
4 pexmidALT.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
5 pexmidALT.o . . . . . . . 8 = (⊥𝑃𝐾)
64, 5pol0N 36931 . . . . . . 7 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
7 eqimss 4027 . . . . . . 7 (( ‘∅) = 𝐴 → ( ‘∅) ⊆ 𝐴)
86, 7syl 17 . . . . . 6 (𝐾 ∈ HL → ( ‘∅) ⊆ 𝐴)
9 pexmidALT.p . . . . . . 7 + = (+𝑃𝐾)
104, 9padd02 36834 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘∅) ⊆ 𝐴) → (∅ + ( ‘∅)) = ( ‘∅))
118, 10mpdan 683 . . . . 5 (𝐾 ∈ HL → (∅ + ( ‘∅)) = ( ‘∅))
1211, 6eqtrd 2861 . . . 4 (𝐾 ∈ HL → (∅ + ( ‘∅)) = 𝐴)
1312ad2antrr 722 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (∅ + ( ‘∅)) = 𝐴)
143, 13sylan9eqr 2883 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( 𝑋)) = 𝐴)
154, 9, 5pexmidlem8N 36999 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅)) → (𝑋 + ( 𝑋)) = 𝐴)
1615anassrs 468 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( 𝑋)) = 𝐴)
1714, 16pm2.61dane 3109 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3021  wss 3940  c0 4295  cfv 6354  (class class class)co 7150  Atomscatm 36285  HLchlt 36372  +𝑃cpadd 36817  𝑃cpolN 36924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-riotaBAD 35975
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-undef 7935  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-psubsp 36525  df-pmap 36526  df-padd 36818  df-polarityN 36925  df-psubclN 36957
This theorem is referenced by: (None)
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