Theorem dalempnes 34322

Description: Lemma for dath 34407. 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 34295	. . 3 |- ( ph -> K e. Lat )
3		dalemc.a	. . . 4 |- A = ( Atoms ` K )
4	1, 3	dalemceb 34309	. . 3 |- ( ph -> C e. ( Base ` K ) )
5	1, 3	dalemseb 34313	. . 3 |- ( ph -> S e. ( Base ` K ) )
6	1, 3	dalemteb 34314	. . 3 |- ( ph -> T e. ( Base ` K ) )
7		simp321 1141	. . . 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 2460	. . . 4 |- ( Base ` K ) = ( Base ` K )
10		dalemc.l	. . . 4 |- .<_ = ( le ` K )
11		dalemc.j	. . . 4 |- .\/ = ( join ` K )
12	9, 10, 11	latnlej2l 15548	. . 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 1236	. 2 |- ( ph -> -. C .<_ S )
14	1	dalemclpjs 34305	. . . . 5 |- ( ph -> C .<_ ( P .\/ S ) )
15		oveq1 6282	. . . . . 6 |- ( P = S -> ( P .\/ S ) = ( S .\/ S ) )
16	15	breq2d 4452	. . . . 5 |- ( P = S -> ( C .<_ ( P .\/ S ) <-> C .<_ ( S .\/ S ) ) )
17	14, 16	syl5ibcom 220	. . . 4 |- ( ph -> ( P = S -> C .<_ ( S .\/ S ) ) )
18	1	dalemkehl 34294	. . . . . 6 |- ( ph -> K e. HL )
19	1	dalemsea 34300	. . . . . 6 |- ( ph -> S e. A )
20	11, 3	hlatjidm 34040	. . . . . 6 |- ( ( K e. HL /\ S e. A ) -> ( S .\/ S ) = S )
21	18, 19, 20	syl2anc 661	. . . . 5 |- ( ph -> ( S .\/ S ) = S )
22	21	breq2d 4452	. . . 4 |- ( ph -> ( C .<_ ( S .\/ S ) <-> C .<_ S ) )
23	17, 22	sylibd 214	. . 3 |- ( ph -> ( P = S -> C .<_ S ) )
24	23	necon3bd 2672	. 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 968   = wceq 1374   e. wcel 1762   =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521   Atomscatm 33935   HLchlt 34022   LPlanesclpl 34163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-lat 15522  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023

This theorem is referenced by:  dalempjsen  34324  dalem24  34368