Theorem dalempnes 33603

Description: Lemma for dath 33688. 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 33576	. . 3 |- ( ph -> K e. Lat )
3		dalemc.a	. . . 4 |- A = ( Atoms ` K )
4	1, 3	dalemceb 33590	. . 3 |- ( ph -> C e. ( Base ` K ) )
5	1, 3	dalemseb 33594	. . 3 |- ( ph -> S e. ( Base ` K ) )
6	1, 3	dalemteb 33595	. . 3 |- ( ph -> T e. ( Base ` K ) )
7		simp321 1138	. . . 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 195	. . 3 |- ( ph -> -. C .<_ ( S .\/ T ) )
9		eqid 2451	. . . 4 |- ( Base ` K ) = ( Base ` K )
10		dalemc.l	. . . 4 |- .<_ = ( le ` K )
11		dalemc.j	. . . 4 |- .\/ = ( join ` K )
12	9, 10, 11	latnlej2l 15346	. . 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 1232	. 2 |- ( ph -> -. C .<_ S )
14	1	dalemclpjs 33586	. . . . 5 |- ( ph -> C .<_ ( P .\/ S ) )
15		oveq1 6199	. . . . . 6 |- ( P = S -> ( P .\/ S ) = ( S .\/ S ) )
16	15	breq2d 4404	. . . . 5 |- ( P = S -> ( C .<_ ( P .\/ S ) <-> C .<_ ( S .\/ S ) ) )
17	14, 16	syl5ibcom 220	. . . 4 |- ( ph -> ( P = S -> C .<_ ( S .\/ S ) ) )
18	1	dalemkehl 33575	. . . . . 6 |- ( ph -> K e. HL )
19	1	dalemsea 33581	. . . . . 6 |- ( ph -> S e. A )
20	11, 3	hlatjidm 33321	. . . . . 6 |- ( ( K e. HL /\ S e. A ) -> ( S .\/ S ) = S )
21	18, 19, 20	syl2anc 661	. . . . 5 |- ( ph -> ( S .\/ S ) = S )
22	21	breq2d 4404	. . . 4 |- ( ph -> ( C .<_ ( S .\/ S ) <-> C .<_ S ) )
23	17, 22	sylibd 214	. . 3 |- ( ph -> ( P = S -> C .<_ S ) )
24	23	necon3bd 2660	. 2 |- ( ph -> ( -. C .<_ S -> P =/= S ) )
25	13, 24	mpd 15	1 |- ( ph -> P =/= S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   Latclat 15319   Atomscatm 33216   HLchlt 33303   LPlanesclpl 33444

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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-lat 15320  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304

This theorem is referenced by:  dalempjsen  33605  dalem24  33649