Theorem 4atlem0ae 33544

Description: Lemma for 4at 33563. (Contributed by NM, 10-Jul-2012.)

Hypotheses

Ref	Expression
4at.l	|- .<_ = ( le ` K )
4at.j	|- .\/ = ( join ` K )
4at.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
4atlem0ae	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. Q .<_ ( P .\/ R ) )

Proof of Theorem 4atlem0ae

Step	Hyp	Ref	Expression
1		simp3r 1017	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ Q ) )
2		simp1 988	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
3		simp22 1022	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A )
4		simp23 1023	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A )
5		simp21 1021	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. A )
6		simp3l 1016	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= Q )
7	6	necomd 2719	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q =/= P )
8		4at.l	. . . 4 |- .<_ = ( le ` K )
9		4at.j	. . . 4 |- .\/ = ( join ` K )
10		4at.a	. . . 4 |- A = ( Atoms ` K )
11	8, 9, 10	hlatexch1 33345	. . 3 |- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) )
12	2, 3, 4, 5, 7, 11	syl131anc 1232	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( Q .<_ ( P .\/ R ) -> R .<_ ( P .\/ Q ) ) )
13	1, 12	mtod 177	1 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. Q .<_ ( P .\/ R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   lecple 14347   joincjn 15216   Atomscatm 33214   HLchlt 33301

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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

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 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302

This theorem is referenced by:  4atlem11  33559  2llnma2  33739  4atexlemc  34019