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Theorem pexmidN 34072
Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 34056. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 34070. (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 785 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝐾 ∈ HL)
2 simplr 787 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋𝐴)
3 pexmid.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
4 pexmid.o . . . . . . 7 = (⊥𝑃𝐾)
53, 4polssatN 34011 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ⊆ 𝐴)
65adantr 479 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ⊆ 𝐴)
7 pexmid.p . . . . . 6 + = (+𝑃𝐾)
83, 7, 4poldmj1N 34031 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ ( 𝑋) ⊆ 𝐴) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
91, 2, 6, 8syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = (( 𝑋) ∩ ( ‘( 𝑋))))
103, 4pnonsingN 34036 . . . . 5 ((𝐾 ∈ HL ∧ ( 𝑋) ⊆ 𝐴) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
111, 6, 10syl2anc 690 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (( 𝑋) ∩ ( ‘( 𝑋))) = ∅)
129, 11eqtrd 2639 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘(𝑋 + ( 𝑋))) = ∅)
1312fveq2d 6088 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = ( ‘∅))
14 simpr 475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( 𝑋)) = 𝑋)
15 eqid 2605 . . . . . . 7 (PSubCl‘𝐾) = (PSubCl‘𝐾)
163, 4, 15ispsubclN 34040 . . . . . 6 (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
1716ad2antrr 757 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
182, 14, 17mpbir2and 958 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
193, 4, 15polsubclN 34055 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ (PSubCl‘𝐾))
2019adantr 479 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( 𝑋) ∈ (PSubCl‘𝐾))
213, 42polssN 34018 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → 𝑋 ⊆ ( ‘( 𝑋)))
2221adantr 479 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ⊆ ( ‘( 𝑋)))
237, 4, 15osumclN 34070 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( 𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ‘( 𝑋))) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
241, 18, 20, 22, 23syl31anc 1320 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾))
254, 15psubcli2N 34042 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + ( 𝑋)) ∈ (PSubCl‘𝐾)) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
261, 24, 25syl2anc 690 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘( ‘(𝑋 + ( 𝑋)))) = (𝑋 + ( 𝑋)))
273, 4pol0N 34012 . . 3 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
2827ad2antrr 757 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → ( ‘∅) = 𝐴)
2913, 26, 283eqtr3d 2647 1 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  cin 3534  wss 3535  c0 3869  cfv 5786  (class class class)co 6523  Atomscatm 33367  HLchlt 33454  +𝑃cpadd 33898  𝑃cpolN 34005  PSubClcpscN 34037
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 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-riotaBAD 33056
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 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-undef 7259  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-p1 16805  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-psubsp 33606  df-pmap 33607  df-padd 33899  df-polarityN 34006  df-psubclN 34038
This theorem is referenced by: (None)
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