Theorem pexmidALTN 40424

Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 40399. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem pexmidALTN

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑋 = ∅ → 𝑋 = ∅)
2		fveq2 6840	. . . 4 ⊢ (𝑋 = ∅ → ( ⊥ ‘𝑋) = ( ⊥ ‘∅))
3	1, 2	oveq12d 7385	. . 3 ⊢ (𝑋 = ∅ → (𝑋 + ( ⊥ ‘𝑋)) = (∅ + ( ⊥ ‘∅)))
4		pexmidALT.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5		pexmidALT.o	. . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾)
6	4, 5	pol0N 40355	. . . . . . 7 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
7		eqimss 3980	. . . . . . 7 ⊢ (( ⊥ ‘∅) = 𝐴 → ( ⊥ ‘∅) ⊆ 𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) ⊆ 𝐴)
9		pexmidALT.p	. . . . . . 7 ⊢ + = (+𝑃‘𝐾)
10	4, 9	padd02 40258	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘∅) ⊆ 𝐴) → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅))
11	8, 10	mpdan 688	. . . . 5 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅))
12	11, 6	eqtrd 2771	. . . 4 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = 𝐴)
13	12	ad2antrr 727	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (∅ + ( ⊥ ‘∅)) = 𝐴)
14	3, 13	sylan9eqr 2793	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
15	4, 9, 5	pexmidlem8N 40423	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅)) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
16	15	anassrs 467	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
17	14, 16	pm2.61dane 3019	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   ⊆ wss 3889  ∅c0 4273  ‘cfv 6498  (class class class)co 7367  Atomscatm 39709  HLchlt 39796  +_𝑃cpadd 40241  ⊥_𝑃cpolN 40348

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)