Theorem pexmidN 40461

Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 40445. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 40459. (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 772	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝐾 ∈ HL)
2		simplr 774	. . . . 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 40400	. . . . . 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 40420	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))
9	1, 2, 6, 8	syl3anc 1379	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))))
10	3, 4	pnonsingN 40425	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘𝑋) ⊆ 𝐴) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ∅)
11	1, 6, 10	syl2anc 590	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (( ⊥ ‘𝑋) ∩ ( ⊥ ‘( ⊥ ‘𝑋))) = ∅)
12	9, 11	eqtrd 2774	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋))) = ∅)
13	12	fveq2d 6831	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = ( ⊥ ‘∅))
14		simpr 485	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
15		eqid 2739	. . . . . . 7 ⊢ (PSubCl‘𝐾) = (PSubCl‘𝐾)
16	3, 4, 15	ispsubclN 40429	. . . . . 6 ⊢ (𝐾 ∈ HL → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
17	16	ad2antrr 732	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 ∈ (PSubCl‘𝐾) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
18	2, 14, 17	mpbir2and 719	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ (PSubCl‘𝐾))
19	3, 4, 15	polsubclN 40444	. . . . 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 40407	. . . . 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 40459	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ (PSubCl‘𝐾) ∧ ( ⊥ ‘𝑋) ∈ (PSubCl‘𝐾)) ∧ 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) → (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾))
24	1, 18, 20, 22, 23	syl31anc 1381	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾))
25	4, 15	psubcli2N 40431	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + ( ⊥ ‘𝑋)) ∈ (PSubCl‘𝐾)) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = (𝑋 + ( ⊥ ‘𝑋)))
26	1, 24, 25	syl2anc 590	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘( ⊥ ‘(𝑋 + ( ⊥ ‘𝑋)))) = (𝑋 + ( ⊥ ‘𝑋)))
27	3, 4	pol0N 40401	. . 3 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
28	27	ad2antrr 732	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → ( ⊥ ‘∅) = 𝐴)
29	13, 26, 28	3eqtr3d 2782	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ‘cfv 6485  (class class class)co 7356  Atomscatm 39755  HLchlt 39842  +_𝑃cpadd 40287  ⊥_𝑃cpolN 40394  PSubClcpscN 40426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-psubsp 39995  df-pmap 39996  df-padd 40288  df-polarityN 40395  df-psubclN 40427

This theorem is referenced by: (None)