Theorem pexmidN 39952

Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 39936. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 39950. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pexmid.a	⊢ 𝐴 = (Atoms‘𝐾)
pexmid.p	⊢ + = (+𝑃‘𝐾)
pexmid.o	⊢ ⊥ = (⊥𝑃‘𝐾)

Assertion

Ref	Expression
pexmidN	⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Proof of Theorem pexmidN

Step	Hyp	Ref	Expression
1		simpll 767	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝐾 ∈ HL)
2		simplr 769	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ⊆ 𝐴)
3		pexmid.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4		pexmid.o	. . . . . . 7 ⊢ ⊥ = (⊥𝑃‘𝐾)
5	3, 4	polssatN 39891	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴)
6	5	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ⊆ 𝐴)
7		pexmid.p	. . . . . 6 ⊢ + = (+𝑃‘𝐾)
8	3, 7, 4	poldmj1N 39911	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))
9	1, 2, 6, 8	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))
10	3, 4	pnonsingN 39916	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ∅)
11	1, 6, 10	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ∅)
12	9, 11	eqtrd 2775	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = ∅)
13	12	fveq2d 6911	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = ( ⊥ ‘∅))
14		simpr 484	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
15		eqid 2735	. . . . . . 7 ⊢ (PSubCl‘𝐾) = (PSubCl‘𝐾)
16	3, 4, 15	ispsubclN 39920	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
17	16	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
18	2, 14, 17	mpbir2and 713	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
19	3, 4, 15	polsubclN 39935	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾))
20	19	adantr 480	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾))
21	3, 4	2polssN 39898	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
22	21	adantr 480	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
23	7, 4, 15	osumclN 39950	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) → (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾))
24	1, 18, 20, 22, 23	syl31anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾))
25	4, 15	psubcli2N 39922	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾)) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = (𝑋 + ( ⊥ ‘𝑋)))
26	1, 24, 25	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = (𝑋 + ( ⊥ ‘𝑋)))
27	3, 4	pol0N 39892	. . 3 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
28	27	ad2antrr 726	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘∅) = 𝐴)
29	13, 26, 28	3eqtr3d 2783	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ‘cfv 6563  (class class class)co 7431  Atomscatm 39245  HLchlt 39332  +_𝑃cpadd 39778  ⊥_𝑃cpolN 39885  PSubClcpscN 39917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-polarityN 39886  df-psubclN 39918

This theorem is referenced by: (None)