Theorem pexmidN 37983

Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 37967. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 37981. (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 764	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝐾 ∈ HL)
2		simplr 766	. . . . 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 37922	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴)
6	5	adantr 481	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ⊆ 𝐴)
7		pexmid.p	. . . . . 6 ⊢ + = (+𝑃‘𝐾)
8	3, 7, 4	poldmj1N 37942	. . . . 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 37947	. . . . 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 2778	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = ∅)
13	12	fveq2d 6778	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = ( ⊥ ‘∅))
14		simpr 485	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
15		eqid 2738	. . . . . . 7 ⊢ (PSubCl‘𝐾) = (PSubCl‘𝐾)
16	3, 4, 15	ispsubclN 37951	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
17	16	ad2antrr 723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
18	2, 14, 17	mpbir2and 710	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
19	3, 4, 15	polsubclN 37966	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾))
20	19	adantr 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾))
21	3, 4	2polssN 37929	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
22	21	adantr 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
23	7, 4, 15	osumclN 37981	. . . 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 37953	. . 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 37923	. . 3 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
28	27	ad2antrr 723	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘∅) = 𝐴)
29	13, 26, 28	3eqtr3d 2786	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ‘cfv 6433  (class class class)co 7275  Atomscatm 37277  HLchlt 37364  +_𝑃cpadd 37809  ⊥_𝑃cpolN 37916  PSubClcpscN 37948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-riotaBAD 36967

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-undef 8089  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-polarityN 37917  df-psubclN 37949

This theorem is referenced by: (None)