Theorem pexmidALTN 37274

Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 37249. 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 6645	. . . 4 ⊢ (𝑋 = ∅ → ( ⊥ ‘𝑋) = ( ⊥ ‘∅))
3	1, 2	oveq12d 7153	. . 3 ⊢ (𝑋 = ∅ → (𝑋 + ( ⊥ ‘𝑋)) = (∅ + ( ⊥ ‘∅)))
4		pexmidALT.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5		pexmidALT.o	. . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾)
6	4, 5	pol0N 37205	. . . . . . 7 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
7		eqimss 3971	. . . . . . 7 ⊢ (( ⊥ ‘∅) = 𝐴 → ( ⊥ ‘∅) ⊆ 𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) ⊆ 𝐴)
9		pexmidALT.p	. . . . . . 7 ⊢ + = (+𝑃‘𝐾)
10	4, 9	padd02 37108	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘∅) ⊆ 𝐴) → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅))
11	8, 10	mpdan 686	. . . . 5 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅))
12	11, 6	eqtrd 2833	. . . 4 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = 𝐴)
13	12	ad2antrr 725	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (∅ + ( ⊥ ‘∅)) = 𝐴)
14	3, 13	sylan9eqr 2855	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
15	4, 9, 5	pexmidlem8N 37273	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅)) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
16	15	anassrs 471	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)
17	14, 16	pm2.61dane 3074	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ⊆ wss 3881  ∅c0 4243  ‘cfv 6324  (class class class)co 7135  Atomscatm 36559  HLchlt 36646  +_𝑃cpadd 37091  ⊥_𝑃cpolN 37198

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-riotaBAD 36249

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-undef 7922  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-psubsp 36799  df-pmap 36800  df-padd 37092  df-polarityN 37199  df-psubclN 37231

This theorem is referenced by: (None)