Theorem pexmidN 40415

Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 40399. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 40413. (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 40354	. . . . . 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 40374	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))
9	1, 2, 6, 8	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))
10	3, 4	pnonsingN 40379	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ∅)
11	1, 6, 10	syl2anc 585	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ∅)
12	9, 11	eqtrd 2771	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = ∅)
13	12	fveq2d 6844	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = ( ⊥ ‘∅))
14		simpr 484	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
15		eqid 2736	. . . . . . 7 ⊢ (PSubCl‘𝐾) = (PSubCl‘𝐾)
16	3, 4, 15	ispsubclN 40383	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
17	16	ad2antrr 727	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
18	2, 14, 17	mpbir2and 714	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
19	3, 4, 15	polsubclN 40398	. . . . 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 40361	. . . . 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 40413	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) → (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾))
24	1, 18, 20, 22, 23	syl31anc 1376	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾))
25	4, 15	psubcli2N 40385	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾)) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = (𝑋 + ( ⊥ ‘𝑋)))
26	1, 24, 25	syl2anc 585	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = (𝑋 + ( ⊥ ‘𝑋)))
27	3, 4	pol0N 40355	. . 3 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
28	27	ad2antrr 727	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘∅) = 𝐴)
29	13, 26, 28	3eqtr3d 2779	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ‘cfv 6498  (class class class)co 7367  Atomscatm 39709  HLchlt 39796  +_𝑃cpadd 40241  ⊥_𝑃cpolN 40348  PSubClcpscN 40380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-p1 18390  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-psubsp 39949  df-pmap 39950  df-padd 40242  df-polarityN 40349  df-psubclN 40381

This theorem is referenced by: (None)