Theorem dalemkelat 34910

Description: Lemma for dath 35022. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Hypothesis

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 ) ) ) ) )

Assertion

Ref	Expression
dalemkelat	|- ( ph -> K e. Lat )

Proof of Theorem dalemkelat

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 34909	. 2 |- ( ph -> K e. HL )
3		hllat 34650	. 2 |- ( K e. HL -> K e. Lat )
4	2, 3	syl 17	1 |- ( ph -> K e. Lat )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Latclat 17045   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-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  dalemcnes  34936  dalempnes  34937  dalemqnet  34938  dalemply  34940  dalemsly  34941  dalem1  34945  dalemcea  34946  dalem3  34950  dalem4  34951  dalem5  34953  dalem8  34956  dalem-cly  34957  dalem10  34959  dalem13  34962  dalem16  34965  dalem17  34966  dalem21  34980  dalem25  34984  dalem27  34985  dalem38  34996  dalem39  34997  dalem43  35001  dalem44  35002  dalem45  35003  dalem48  35006  dalem54  35012  dalem55  35013  dalem56  35014  dalem57  35015  dalem60  35018