Theorem dalempnes 33260

Description: Lemma for dath 33345. 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 33233	. . 3 |- ( ph -> K e. Lat )
3		dalemc.a	. . . 4 |- A = ( Atoms ` K )
4	1, 3	dalemceb 33247	. . 3 |- ( ph -> C e. ( Base ` K ) )
5	1, 3	dalemseb 33251	. . 3 |- ( ph -> S e. ( Base ` K ) )
6	1, 3	dalemteb 33252	. . 3 |- ( ph -> T e. ( Base ` K ) )
7		simp321 1164	. . . 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 200	. . 3 |- ( ph -> -. C .<_ ( S .\/ T ) )
9		eqid 2461	. . . 4 |- ( Base ` K ) = ( Base ` K )
10		dalemc.l	. . . 4 |- .<_ = ( le ` K )
11		dalemc.j	. . . 4 |- .\/ = ( join ` K )
12	9, 10, 11	latnlej2l 16366	. . 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 1289	. 2 |- ( ph -> -. C .<_ S )
14	1	dalemclpjs 33243	. . . . 5 |- ( ph -> C .<_ ( P .\/ S ) )
15		oveq1 6321	. . . . . 6 |- ( P = S -> ( P .\/ S ) = ( S .\/ S ) )
16	15	breq2d 4427	. . . . 5 |- ( P = S -> ( C .<_ ( P .\/ S ) <-> C .<_ ( S .\/ S ) ) )
17	14, 16	syl5ibcom 228	. . . 4 |- ( ph -> ( P = S -> C .<_ ( S .\/ S ) ) )
18	1	dalemkehl 33232	. . . . . 6 |- ( ph -> K e. HL )
19	1	dalemsea 33238	. . . . . 6 |- ( ph -> S e. A )
20	11, 3	hlatjidm 32978	. . . . . 6 |- ( ( K e. HL /\ S e. A ) -> ( S .\/ S ) = S )
21	18, 19, 20	syl2anc 671	. . . . 5 |- ( ph -> ( S .\/ S ) = S )
22	21	breq2d 4427	. . . 4 |- ( ph -> ( C .<_ ( S .\/ S ) <-> C .<_ S ) )
23	17, 22	sylibd 222	. . 3 |- ( ph -> ( P = S -> C .<_ S ) )
24	23	necon3bd 2649	. 2 |- ( ph -> ( -. C .<_ S -> P =/= S ) )
25	13, 24	mpd 15	1 |- ( ph -> P =/= S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   Latclat 16339   Atomscatm 32873   HLchlt 32960   LPlanesclpl 33101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-preset 16221  df-poset 16239  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-lat 16340  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961

This theorem is referenced by:  dalempjsen  33262  dalem24  33306