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Theorem pexmidALTN 34080
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 34055. 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 6083 . . . 4 (𝑋 = ∅ → ( 𝑋) = ( ‘∅))
31, 2oveq12d 6540 . . 3 (𝑋 = ∅ → (𝑋 + ( 𝑋)) = (∅ + ( ‘∅)))
4 pexmidALT.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
5 pexmidALT.o . . . . . . . 8 = (⊥𝑃𝐾)
64, 5pol0N 34011 . . . . . . 7 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
7 eqimss 3614 . . . . . . 7 (( ‘∅) = 𝐴 → ( ‘∅) ⊆ 𝐴)
86, 7syl 17 . . . . . 6 (𝐾 ∈ HL → ( ‘∅) ⊆ 𝐴)
9 pexmidALT.p . . . . . . 7 + = (+𝑃𝐾)
104, 9padd02 33914 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘∅) ⊆ 𝐴) → (∅ + ( ‘∅)) = ( ‘∅))
118, 10mpdan 698 . . . . 5 (𝐾 ∈ HL → (∅ + ( ‘∅)) = ( ‘∅))
1211, 6eqtrd 2638 . . . 4 (𝐾 ∈ HL → (∅ + ( ‘∅)) = 𝐴)
1312ad2antrr 757 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (∅ + ( ‘∅)) = 𝐴)
143, 13sylan9eqr 2660 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( 𝑋)) = 𝐴)
154, 9, 5pexmidlem8N 34079 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅)) → (𝑋 + ( 𝑋)) = 𝐴)
1615anassrs 677 . 2 ((((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( 𝑋)) = 𝐴)
1714, 16pm2.61dane 2863 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wne 2774  wss 3534  c0 3868  cfv 5785  (class class class)co 6522  Atomscatm 33366  HLchlt 33453  +𝑃cpadd 33897  𝑃cpolN 34004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-riotaBAD 33055
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-iin 4447  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-undef 7258  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-p1 16804  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-psubsp 33605  df-pmap 33606  df-padd 33898  df-polarityN 34005  df-psubclN 34037
This theorem is referenced by: (None)
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