Theorem dalempnes 35115

Description: Lemma for dath 35200. 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 35088	. . 3 |- ( ph -> K e. Lat )
3		dalemc.a	. . . 4 |- A = ( Atoms ` K )
4	1, 3	dalemceb 35102	. . 3 |- ( ph -> C e. ( Base ` K ) )
5	1, 3	dalemseb 35106	. . 3 |- ( ph -> S e. ( Base ` K ) )
6	1, 3	dalemteb 35107	. . 3 |- ( ph -> T e. ( Base ` K ) )
7		simp321 1147	. . . 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 2443	. . . 4 |- ( Base ` K ) = ( Base ` K )
10		dalemc.l	. . . 4 |- .<_ = ( le ` K )
11		dalemc.j	. . . 4 |- .\/ = ( join ` K )
12	9, 10, 11	latnlej2l 15576	. . 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 1242	. 2 |- ( ph -> -. C .<_ S )
14	1	dalemclpjs 35098	. . . . 5 |- ( ph -> C .<_ ( P .\/ S ) )
15		oveq1 6288	. . . . . 6 |- ( P = S -> ( P .\/ S ) = ( S .\/ S ) )
16	15	breq2d 4449	. . . . 5 |- ( P = S -> ( C .<_ ( P .\/ S ) <-> C .<_ ( S .\/ S ) ) )
17	14, 16	syl5ibcom 220	. . . 4 |- ( ph -> ( P = S -> C .<_ ( S .\/ S ) ) )
18	1	dalemkehl 35087	. . . . . 6 |- ( ph -> K e. HL )
19	1	dalemsea 35093	. . . . . 6 |- ( ph -> S e. A )
20	11, 3	hlatjidm 34833	. . . . . 6 |- ( ( K e. HL /\ S e. A ) -> ( S .\/ S ) = S )
21	18, 19, 20	syl2anc 661	. . . . 5 |- ( ph -> ( S .\/ S ) = S )
22	21	breq2d 4449	. . . 4 |- ( ph -> ( C .<_ ( S .\/ S ) <-> C .<_ S ) )
23	17, 22	sylibd 214	. . 3 |- ( ph -> ( P = S -> C .<_ S ) )
24	23	necon3bd 2655	. 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 974   = wceq 1383   e. wcel 1804   =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   Latclat 15549   Atomscatm 34728   HLchlt 34815   LPlanesclpl 34956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15431  df-poset 15449  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-lat 15550  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816

This theorem is referenced by:  dalempjsen  35117  dalem24  35161