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Theorem pexmidALTN 35999
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 35974. 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 6411 . . . 4 (𝑋 = ∅ → ( 𝑋) = ( ‘∅))
31, 2oveq12d 6896 . . 3 (𝑋 = ∅ → (𝑋 + ( 𝑋)) = (∅ + ( ‘∅)))
4 pexmidALT.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
5 pexmidALT.o . . . . . . . 8 = (⊥𝑃𝐾)
64, 5pol0N 35930 . . . . . . 7 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
7 eqimss 3853 . . . . . . 7 (( ‘∅) = 𝐴 → ( ‘∅) ⊆ 𝐴)
86, 7syl 17 . . . . . 6 (𝐾 ∈ HL → ( ‘∅) ⊆ 𝐴)
9 pexmidALT.p . . . . . . 7 + = (+𝑃𝐾)
104, 9padd02 35833 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘∅) ⊆ 𝐴) → (∅ + ( ‘∅)) = ( ‘∅))
118, 10mpdan 679 . . . . 5 (𝐾 ∈ HL → (∅ + ( ‘∅)) = ( ‘∅))
1211, 6eqtrd 2833 . . . 4 (𝐾 ∈ HL → (∅ + ( ‘∅)) = 𝐴)
1312ad2antrr 718 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (∅ + ( ‘∅)) = 𝐴)
143, 13sylan9eqr 2855 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( 𝑋)) = 𝐴)
154, 9, 5pexmidlem8N 35998 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅)) → (𝑋 + ( 𝑋)) = 𝐴)
1615anassrs 460 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( 𝑋)) = 𝐴)
1714, 16pm2.61dane 3058 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2971  wss 3769  c0 4115  cfv 6101  (class class class)co 6878  Atomscatm 35284  HLchlt 35371  +𝑃cpadd 35816  𝑃cpolN 35923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-riotaBAD 34974
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-undef 7637  df-proset 17243  df-poset 17261  df-plt 17273  df-lub 17289  df-glb 17290  df-join 17291  df-meet 17292  df-p0 17354  df-p1 17355  df-lat 17361  df-clat 17423  df-oposet 35197  df-ol 35199  df-oml 35200  df-covers 35287  df-ats 35288  df-atl 35319  df-cvlat 35343  df-hlat 35372  df-psubsp 35524  df-pmap 35525  df-padd 35817  df-polarityN 35924  df-psubclN 35956
This theorem is referenced by: (None)
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