Theorem hlomcmat 34651

Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)

Assertion

Ref	Expression
hlomcmat	|- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )

Proof of Theorem hlomcmat

Step	Hyp	Ref	Expression
1		hloml 34644	. 2 |- ( K e. HL -> K e. OML )
2		hlclat 34645	. 2 |- ( K e. HL -> K e. CLat )
3		hlatl 34647	. 2 |- ( K e. HL -> K e. AtLat )
4	1, 2, 3	3jca 1242	1 |- ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   CLatccla 17107   OMLcoml 34462   AtLatcal 34551   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:  hlatmstcOLDN  34683  hlatle  34684  hlrelat1  34686  pmaple  35047  pol1N  35196  polpmapN  35198  pmaplubN  35210