Theorem 4atlem0ae 32960

Description: Lemma for 4at 32979. (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 1012	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ Q ) )
2		simp1 983	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
3		simp22 1017	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A )
4		simp23 1018	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A )
5		simp21 1016	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. A )
6		simp3l 1011	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= Q )
7	6	necomd 2693	. . 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 32761	. . 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 1226	. 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 960   = wceq 1364   e. wcel 1761   =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   lecple 14241   joincjn 15110   Atomscatm 32630   HLchlt 32717

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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718

This theorem is referenced by:  4atlem11  32975  2llnma2  33155  4atexlemc  33435