Theorem pexmidN 35255

Description: Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 35239. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 35253. (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 e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( X .+ ( ._|_ ` X ) ) = A )

Proof of Theorem pexmidN

Step	Hyp	Ref	Expression
1		simpll 790	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> K e. HL )
2		simplr 792	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> X C_ A )
3		pexmid.a	. . . . . . 7 |- A = ( Atoms ` K )
4		pexmid.o	. . . . . . 7 |- ._|_ = ( _|_P ` K )
5	3, 4	polssatN 35194	. . . . . 6 |- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) C_ A )
6	5	adantr 481	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` X ) C_ A )
7		pexmid.p	. . . . . 6 |- .+ = ( +P ` K )
8	3, 7, 4	poldmj1N 35214	. . . . 5 |- ( ( K e. HL /\ X C_ A /\ ( ._|_ ` X ) C_ A ) -> ( ._|_ ` ( X .+ ( ._|_ ` X ) ) ) = ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) )
9	1, 2, 6, 8	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( X .+ ( ._|_ ` X ) ) ) = ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) )
10	3, 4	pnonsingN 35219	. . . . 5 |- ( ( K e. HL /\ ( ._|_ ` X ) C_ A ) -> ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) = (/) )
11	1, 6, 10	syl2anc 693	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ( ._|_ ` X ) i^i ( ._|_ ` ( ._|_ ` X ) ) ) = (/) )
12	9, 11	eqtrd 2656	. . 3 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( X .+ ( ._|_ ` X ) ) ) = (/) )
13	12	fveq2d 6195	. 2 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` ( X .+ ( ._|_ ` X ) ) ) ) = ( ._|_ ` (/) ) )
14		simpr 477	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
15		eqid 2622	. . . . . . 7 |- ( PSubCl ` K ) = ( PSubCl ` K )
16	3, 4, 15	ispsubclN 35223	. . . . . 6 |- ( K e. HL -> ( X e. ( PSubCl ` K ) <-> ( X C_ A /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) )
17	16	ad2antrr 762	. . . . 5 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( X e. ( PSubCl ` K ) <-> ( X C_ A /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) )
18	2, 14, 17	mpbir2and 957	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> X e. ( PSubCl ` K ) )
19	3, 4, 15	polsubclN 35238	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. ( PSubCl ` K ) )
20	19	adantr 481	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` X ) e. ( PSubCl ` K ) )
21	3, 4	2polssN 35201	. . . . 5 |- ( ( K e. HL /\ X C_ A ) -> X C_ ( ._|_ ` ( ._|_ ` X ) ) )
22	21	adantr 481	. . . 4 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> X C_ ( ._|_ ` ( ._|_ ` X ) ) )
23	7, 4, 15	osumclN 35253	. . . 4 |- ( ( ( K e. HL /\ X e. ( PSubCl ` K ) /\ ( ._|_ ` X ) e. ( PSubCl ` K ) ) /\ X C_ ( ._|_ ` ( ._|_ ` X ) ) ) -> ( X .+ ( ._|_ ` X ) ) e. ( PSubCl ` K ) )
24	1, 18, 20, 22, 23	syl31anc 1329	. . 3 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( X .+ ( ._|_ ` X ) ) e. ( PSubCl ` K ) )
25	4, 15	psubcli2N 35225	. . 3 |- ( ( K e. HL /\ ( X .+ ( ._|_ ` X ) ) e. ( PSubCl ` K ) ) -> ( ._|_ ` ( ._|_ ` ( X .+ ( ._|_ ` X ) ) ) ) = ( X .+ ( ._|_ ` X ) ) )
26	1, 24, 25	syl2anc 693	. 2 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` ( X .+ ( ._|_ ` X ) ) ) ) = ( X .+ ( ._|_ ` X ) ) )
27	3, 4	pol0N 35195	. . 3 |- ( K e. HL -> ( ._|_ ` (/) ) = A )
28	27	ad2antrr 762	. 2 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` (/) ) = A )
29	13, 26, 28	3eqtr3d 2664	1 |- ( ( ( K e. HL /\ X C_ A ) /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( X .+ ( ._|_ ` X ) ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Atomscatm 34550   HLchlt 34637   +Pcpadd 35081   _|_PcpolN 35188   PSubClcpscN 35220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by: (None)