Theorem hlsupr 34672

Description: A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)

Hypotheses

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

Assertion

Ref	Expression
hlsupr	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) )

Distinct variable groups:   A, r   K, r   P, r   Q, r

Allowed substitution hints:   .\/ ( r)   .<_ ( r)

Proof of Theorem hlsupr

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
2		hlsupr.l	. . . 4 |- .<_ = ( le ` K )
3		hlsupr.j	. . . 4 |- .\/ = ( join ` K )
4		hlsupr.a	. . . 4 |- A = ( Atoms ` K )
5	1, 2, 3, 4	hlsuprexch 34667	. . 3 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P =/= Q -> E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) /\ A. r e. ( Base ` K ) ( ( -. P .<_ r /\ P .<_ ( r .\/ Q ) ) -> Q .<_ ( r .\/ P ) ) ) )
6	5	simpld 475	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q -> E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) ) )
7	6	imp 445	1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> E. r e. A ( r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Atomscatm 34550   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-ne 2795  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:  hlsupr2  34673  atbtwnexOLDN  34733  atbtwnex  34734  cdlemb  35080  lhpexle2lem  35295  lhpexle3lem  35297  cdlemf1  35849  cdlemg35  36001