Theorem 4atexlemtlw 35353

Description: Lemma for 4atexlem7 35361. (Contributed by NM, 24-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	|- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
4thatlem0.l	|- .<_ = ( le ` K )
4thatlem0.j	|- .\/ = ( join ` K )
4thatlem0.m	|- ./\ = ( meet ` K )
4thatlem0.a	|- A = ( Atoms ` K )
4thatlem0.h	|- H = ( LHyp ` K )
4thatlem0.u	|- U = ( ( P .\/ Q ) ./\ W )
4thatlem0.v	|- V = ( ( P .\/ S ) ./\ W )

Assertion

Ref	Expression
4atexlemtlw	|- ( ph -> T .<_ W )

Proof of Theorem 4atexlemtlw

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` K ) = ( Base ` K )
2		4thatlem0.l	. 2 |- .<_ = ( le ` K )
3		4thatlem.ph	. . 3 |- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
4	3	4atexlemkl 35343	. 2 |- ( ph -> K e. Lat )
5	3	4atexlemt 35339	. . 3 |- ( ph -> T e. A )
6		4thatlem0.a	. . . 4 |- A = ( Atoms ` K )
7	1, 6	atbase 34576	. . 3 |- ( T e. A -> T e. ( Base ` K ) )
8	5, 7	syl 17	. 2 |- ( ph -> T e. ( Base ` K ) )
9	3	4atexlemk 35333	. . 3 |- ( ph -> K e. HL )
10		4thatlem0.j	. . . 4 |- .\/ = ( join ` K )
11		4thatlem0.m	. . . 4 |- ./\ = ( meet ` K )
12		4thatlem0.h	. . . 4 |- H = ( LHyp ` K )
13		4thatlem0.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
14	3, 2, 10, 11, 6, 12, 13	4atexlemu 35350	. . 3 |- ( ph -> U e. A )
15		4thatlem0.v	. . . 4 |- V = ( ( P .\/ S ) ./\ W )
16	3, 2, 10, 11, 6, 12, 13, 15	4atexlemv 35351	. . 3 |- ( ph -> V e. A )
17	1, 10, 6	hlatjcl 34653	. . 3 |- ( ( K e. HL /\ U e. A /\ V e. A ) -> ( U .\/ V ) e. ( Base ` K ) )
18	9, 14, 16, 17	syl3anc 1326	. 2 |- ( ph -> ( U .\/ V ) e. ( Base ` K ) )
19	3, 12	4atexlemwb 35345	. 2 |- ( ph -> W e. ( Base ` K ) )
20	3	4atexlemkc 35344	. . 3 |- ( ph -> K e. CvLat )
21	3, 2, 10, 11, 6, 12, 13, 15	4atexlemunv 35352	. . 3 |- ( ph -> U =/= V )
22	3	4atexlemutvt 35340	. . 3 |- ( ph -> ( U .\/ T ) = ( V .\/ T ) )
23	6, 2, 10	cvlsupr4 34632	. . 3 |- ( ( K e. CvLat /\ ( U e. A /\ V e. A /\ T e. A ) /\ ( U =/= V /\ ( U .\/ T ) = ( V .\/ T ) ) ) -> T .<_ ( U .\/ V ) )
24	20, 14, 16, 5, 21, 22, 23	syl132anc 1344	. 2 |- ( ph -> T .<_ ( U .\/ V ) )
25	3	4atexlemp 35336	. . . . . 6 |- ( ph -> P e. A )
26	3	4atexlemq 35337	. . . . . 6 |- ( ph -> Q e. A )
27	1, 10, 6	hlatjcl 34653	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
28	9, 25, 26, 27	syl3anc 1326	. . . . 5 |- ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
29	1, 2, 11	latmle2 17077	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
30	4, 28, 19, 29	syl3anc 1326	. . . 4 |- ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ W )
31	13, 30	syl5eqbr 4688	. . 3 |- ( ph -> U .<_ W )
32	3, 10, 6	4atexlempsb 35346	. . . . 5 |- ( ph -> ( P .\/ S ) e. ( Base ` K ) )
33	1, 2, 11	latmle2 17077	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
34	4, 32, 19, 33	syl3anc 1326	. . . 4 |- ( ph -> ( ( P .\/ S ) ./\ W ) .<_ W )
35	15, 34	syl5eqbr 4688	. . 3 |- ( ph -> V .<_ W )
36	1, 6	atbase 34576	. . . . 5 |- ( U e. A -> U e. ( Base ` K ) )
37	14, 36	syl 17	. . . 4 |- ( ph -> U e. ( Base ` K ) )
38	1, 6	atbase 34576	. . . . 5 |- ( V e. A -> V e. ( Base ` K ) )
39	16, 38	syl 17	. . . 4 |- ( ph -> V e. ( Base ` K ) )
40	1, 2, 10	latjle12 17062	. . . 4 |- ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ V e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( U .<_ W /\ V .<_ W ) <-> ( U .\/ V ) .<_ W ) )
41	4, 37, 39, 19, 40	syl13anc 1328	. . 3 |- ( ph -> ( ( U .<_ W /\ V .<_ W ) <-> ( U .\/ V ) .<_ W ) )
42	31, 35, 41	mpbi2and 956	. 2 |- ( ph -> ( U .\/ V ) .<_ W )
43	1, 2, 4, 8, 18, 19, 24, 42	lattrd 17058	1 |- ( ph -> T .<_ W )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   CvLatclc 34552   HLchlt 34637   LHypclh 35270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274

This theorem is referenced by:  4atexlemntlpq  35354  4atexlemnclw  35356  4atexlemcnd  35358