Theorem 4atexlemunv 35352

Description: Lemma for 4atexlem7 35361. (Contributed by NM, 21-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
4atexlemunv	|- ( ph -> U =/= V )

Proof of Theorem 4atexlemunv

Step	Hyp	Ref	Expression
1		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 ) ) ) )
2	1	4atexlemnslpq 35342	. 2 |- ( ph -> -. S .<_ ( P .\/ Q ) )
3	1	4atexlemk 35333	. . . . . . 7 |- ( ph -> K e. HL )
4	1	4atexlemp 35336	. . . . . . 7 |- ( ph -> P e. A )
5	1	4atexlems 35338	. . . . . . 7 |- ( ph -> S e. A )
6		4thatlem0.l	. . . . . . . 8 |- .<_ = ( le ` K )
7		4thatlem0.j	. . . . . . . 8 |- .\/ = ( join ` K )
8		4thatlem0.a	. . . . . . . 8 |- A = ( Atoms ` K )
9	6, 7, 8	hlatlej2 34662	. . . . . . 7 |- ( ( K e. HL /\ P e. A /\ S e. A ) -> S .<_ ( P .\/ S ) )
10	3, 4, 5, 9	syl3anc 1326	. . . . . 6 |- ( ph -> S .<_ ( P .\/ S ) )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ U = V ) -> S .<_ ( P .\/ S ) )
12		4thatlem0.v	. . . . . . . . 9 |- V = ( ( P .\/ S ) ./\ W )
13	1	4atexlemkl 35343	. . . . . . . . . 10 |- ( ph -> K e. Lat )
14	1, 7, 8	4atexlempsb 35346	. . . . . . . . . 10 |- ( ph -> ( P .\/ S ) e. ( Base ` K ) )
15		4thatlem0.h	. . . . . . . . . . 11 |- H = ( LHyp ` K )
16	1, 15	4atexlemwb 35345	. . . . . . . . . 10 |- ( ph -> W e. ( Base ` K ) )
17		eqid 2622	. . . . . . . . . . 11 |- ( Base ` K ) = ( Base ` K )
18		4thatlem0.m	. . . . . . . . . . 11 |- ./\ = ( meet ` K )
19	17, 6, 18	latmle1 17076	. . . . . . . . . 10 |- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
20	13, 14, 16, 19	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
21	12, 20	syl5eqbr 4688	. . . . . . . 8 |- ( ph -> V .<_ ( P .\/ S ) )
22	1	4atexlemkc 35344	. . . . . . . . 9 |- ( ph -> K e. CvLat )
23		4thatlem0.u	. . . . . . . . . 10 |- U = ( ( P .\/ Q ) ./\ W )
24	1, 6, 7, 18, 8, 15, 23, 12	4atexlemv 35351	. . . . . . . . 9 |- ( ph -> V e. A )
25	17, 6, 18	latmle2 17077	. . . . . . . . . . . 12 |- ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
26	13, 14, 16, 25	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( ( P .\/ S ) ./\ W ) .<_ W )
27	12, 26	syl5eqbr 4688	. . . . . . . . . 10 |- ( ph -> V .<_ W )
28	1	4atexlempw 35335	. . . . . . . . . . 11 |- ( ph -> ( P e. A /\ -. P .<_ W ) )
29	28	simprd 479	. . . . . . . . . 10 |- ( ph -> -. P .<_ W )
30		nbrne2 4673	. . . . . . . . . 10 |- ( ( V .<_ W /\ -. P .<_ W ) -> V =/= P )
31	27, 29, 30	syl2anc 693	. . . . . . . . 9 |- ( ph -> V =/= P )
32	6, 7, 8	cvlatexchb1 34621	. . . . . . . . 9 |- ( ( K e. CvLat /\ ( V e. A /\ S e. A /\ P e. A ) /\ V =/= P ) -> ( V .<_ ( P .\/ S ) <-> ( P .\/ V ) = ( P .\/ S ) ) )
33	22, 24, 5, 4, 31, 32	syl131anc 1339	. . . . . . . 8 |- ( ph -> ( V .<_ ( P .\/ S ) <-> ( P .\/ V ) = ( P .\/ S ) ) )
34	21, 33	mpbid 222	. . . . . . 7 |- ( ph -> ( P .\/ V ) = ( P .\/ S ) )
35	34	adantr 481	. . . . . 6 |- ( ( ph /\ U = V ) -> ( P .\/ V ) = ( P .\/ S ) )
36		oveq2 6658	. . . . . . . 8 |- ( U = V -> ( P .\/ U ) = ( P .\/ V ) )
37	36	eqcomd 2628	. . . . . . 7 |- ( U = V -> ( P .\/ V ) = ( P .\/ U ) )
38	1	4atexlemq 35337	. . . . . . . . . . 11 |- ( ph -> Q e. A )
39	17, 7, 8	hlatjcl 34653	. . . . . . . . . . 11 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
40	3, 4, 38, 39	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
41	17, 6, 18	latmle1 17076	. . . . . . . . . 10 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
42	13, 40, 16, 41	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
43	23, 42	syl5eqbr 4688	. . . . . . . 8 |- ( ph -> U .<_ ( P .\/ Q ) )
44	1, 6, 7, 18, 8, 15, 23	4atexlemu 35350	. . . . . . . . 9 |- ( ph -> U e. A )
45	17, 6, 18	latmle2 17077	. . . . . . . . . . . 12 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
46	13, 40, 16, 45	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ W )
47	23, 46	syl5eqbr 4688	. . . . . . . . . 10 |- ( ph -> U .<_ W )
48		nbrne2 4673	. . . . . . . . . 10 |- ( ( U .<_ W /\ -. P .<_ W ) -> U =/= P )
49	47, 29, 48	syl2anc 693	. . . . . . . . 9 |- ( ph -> U =/= P )
50	6, 7, 8	cvlatexchb1 34621	. . . . . . . . 9 |- ( ( K e. CvLat /\ ( U e. A /\ Q e. A /\ P e. A ) /\ U =/= P ) -> ( U .<_ ( P .\/ Q ) <-> ( P .\/ U ) = ( P .\/ Q ) ) )
51	22, 44, 38, 4, 49, 50	syl131anc 1339	. . . . . . . 8 |- ( ph -> ( U .<_ ( P .\/ Q ) <-> ( P .\/ U ) = ( P .\/ Q ) ) )
52	43, 51	mpbid 222	. . . . . . 7 |- ( ph -> ( P .\/ U ) = ( P .\/ Q ) )
53	37, 52	sylan9eqr 2678	. . . . . 6 |- ( ( ph /\ U = V ) -> ( P .\/ V ) = ( P .\/ Q ) )
54	35, 53	eqtr3d 2658	. . . . 5 |- ( ( ph /\ U = V ) -> ( P .\/ S ) = ( P .\/ Q ) )
55	11, 54	breqtrd 4679	. . . 4 |- ( ( ph /\ U = V ) -> S .<_ ( P .\/ Q ) )
56	55	ex 450	. . 3 |- ( ph -> ( U = V -> S .<_ ( P .\/ Q ) ) )
57	56	necon3bd 2808	. 2 |- ( ph -> ( -. S .<_ ( P .\/ Q ) -> U =/= V ) )
58	2, 57	mpd 15	1 |- ( ph -> U =/= V )

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:  4atexlemtlw  35353  4atexlemntlpq  35354  4atexlemc  35355  4atexlemnclw  35356