Theorem dalempjqeb 33608

Description: Lemma for dath 33699. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Hypotheses

Ref	Expression
dalema.ph	|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalemb.j	|- .\/ = ( join ` K )
dalemb.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
dalempjqeb	|- ( ph -> ( P .\/ Q ) e. ( Base ` K ) )

Proof of Theorem dalempjqeb

Step	Hyp	Ref	Expression
1		dalema.ph	. . 3 |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
2	1	dalemkehl 33586	. 2 |- ( ph -> K e. HL )
3	1	dalempea 33589	. 2 |- ( ph -> P e. A )
4	1	dalemqea 33590	. 2 |- ( ph -> Q e. A )
5		eqid 2452	. . 3 |- ( Base ` K ) = ( Base ` K )
6		dalemb.j	. . 3 |- .\/ = ( join ` K )
7		dalemb.a	. . 3 |- A = ( Atoms ` K )
8	5, 6, 7	hlatjcl 33330	. 2 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
9	2, 3, 4, 8	syl3anc 1219	1 |- ( ph -> ( P .\/ Q ) e. ( Base ` K ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   joincjn 15228   Atomscatm 33227   HLchlt 33314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315

This theorem is referenced by:  dalemcea  33623  dalem3  33627  dalem4  33628  dalem5  33630  dalem-cly  33634  dalem10  33636  dalem17  33643  dalem38  33673  dalem44  33679  dalem48  33683  dalem54  33689  dalem55  33690  dalem57  33692