Theorem dalempnes 34937

Description: Lemma for dath 35022. 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 ) ) ) ) )
dalemc.l	|- .<_ = ( le ` K )
dalemc.j	|- .\/ = ( join ` K )
dalemc.a	|- A = ( Atoms ` K )
dalempnes.o	|- O = ( LPlanes ` K )
dalempnes.y	|- Y = ( ( P .\/ Q ) .\/ R )

Assertion

Ref	Expression
dalempnes	|- ( ph -> P =/= S )

Proof of Theorem dalempnes

Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 |- ( 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	dalemkelat 34910	. . 3 |- ( ph -> K e. Lat )
3		dalemc.a	. . . 4 |- A = ( Atoms ` K )
4	1, 3	dalemceb 34924	. . 3 |- ( ph -> C e. ( Base ` K ) )
5	1, 3	dalemseb 34928	. . 3 |- ( ph -> S e. ( Base ` K ) )
6	1, 3	dalemteb 34929	. . 3 |- ( ph -> T e. ( Base ` K ) )
7		simp321 1211	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( S .\/ T ) )
8	1, 7	sylbi 207	. . 3 |- ( ph -> -. C .<_ ( S .\/ T ) )
9		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
10		dalemc.l	. . . 4 |- .<_ = ( le ` K )
11		dalemc.j	. . . 4 |- .\/ = ( join ` K )
12	9, 10, 11	latnlej2l 17072	. . 3 |- ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ S e. ( Base ` K ) /\ T e. ( Base ` K ) ) /\ -. C .<_ ( S .\/ T ) ) -> -. C .<_ S )
13	2, 4, 5, 6, 8, 12	syl131anc 1339	. 2 |- ( ph -> -. C .<_ S )
14	1	dalemclpjs 34920	. . . . 5 |- ( ph -> C .<_ ( P .\/ S ) )
15		oveq1 6657	. . . . . 6 |- ( P = S -> ( P .\/ S ) = ( S .\/ S ) )
16	15	breq2d 4665	. . . . 5 |- ( P = S -> ( C .<_ ( P .\/ S ) <-> C .<_ ( S .\/ S ) ) )
17	14, 16	syl5ibcom 235	. . . 4 |- ( ph -> ( P = S -> C .<_ ( S .\/ S ) ) )
18	1	dalemkehl 34909	. . . . . 6 |- ( ph -> K e. HL )
19	1	dalemsea 34915	. . . . . 6 |- ( ph -> S e. A )
20	11, 3	hlatjidm 34655	. . . . . 6 |- ( ( K e. HL /\ S e. A ) -> ( S .\/ S ) = S )
21	18, 19, 20	syl2anc 693	. . . . 5 |- ( ph -> ( S .\/ S ) = S )
22	21	breq2d 4665	. . . 4 |- ( ph -> ( C .<_ ( S .\/ S ) <-> C .<_ S ) )
23	17, 22	sylibd 229	. . 3 |- ( ph -> ( P = S -> C .<_ S ) )
24	23	necon3bd 2808	. 2 |- ( ph -> ( -. C .<_ S -> P =/= S ) )
25	13, 24	mpd 15	1 |- ( ph -> P =/= S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   LPlanesclpl 34778

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  dalempjsen  34939  dalem24  34983