Theorem hlexch1 34668

Description: A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)

Hypotheses

Ref	Expression
hlsuprexch.b	|- B = ( Base ` K )
hlsuprexch.l	|- .<_ = ( le ` K )
hlsuprexch.j	|- .\/ = ( join ` K )
hlsuprexch.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
hlexch1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) )

Proof of Theorem hlexch1

Step	Hyp	Ref	Expression
1		hlcvl 34646	. 2 |- ( K e. HL -> K e. CvLat )
2		hlsuprexch.b	. . 3 |- B = ( Base ` K )
3		hlsuprexch.l	. . 3 |- .<_ = ( le ` K )
4		hlsuprexch.j	. . 3 |- .\/ = ( join ` K )
5		hlsuprexch.a	. . 3 |- A = ( Atoms ` K )
6	2, 3, 4, 5	cvlexch1 34615	. 2 |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) )
7	1, 6	syl3an1 1359	1 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ -. P .<_ X ) -> ( P .<_ ( X .\/ Q ) -> Q .<_ ( X .\/ P ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Atomscatm 34550   CvLatclc 34552   HLchlt 34637

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  cvratlem  34707  4noncolr3  34739  3dimlem4a  34749  3dimlem4OLDN  34751  ps-2  34764  4atlem0a  34879