Theorem cdleme0ex2N 33760

Description: Part of proof of Lemma E in [Crawley] p. 113. Note that ( P .\/ u ) = ( Q .\/ u ) is a shorter way to express u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme0.l	|- .<_ = ( le ` K )
cdleme0.j	|- .\/ = ( join ` K )
cdleme0.m	|- ./\ = ( meet ` K )
cdleme0.a	|- A = ( Atoms ` K )
cdleme0.h	|- H = ( LHyp ` K )
cdleme0.u	|- U = ( ( P .\/ Q ) ./\ W )

Assertion

Ref	Expression
cdleme0ex2N	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. u e. A ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) )

Distinct variable groups:   u, A   u, .\/   u, .<_   u, P   u, Q   u, U   u, W   u, H   u, K

Allowed substitution hint:   ./\ ( u)

Proof of Theorem cdleme0ex2N

Step	Hyp	Ref	Expression
1		simp1 1005	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
2		simp2l 1031	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
3		simp2rl 1074	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp3 1007	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
5		cdleme0.l	. . . 4 |- .<_ = ( le ` K )
6		cdleme0.j	. . . 4 |- .\/ = ( join ` K )
7		cdleme0.m	. . . 4 |- ./\ = ( meet ` K )
8		cdleme0.a	. . . 4 |- A = ( Atoms ` K )
9		cdleme0.h	. . . 4 |- H = ( LHyp ` K )
10		cdleme0.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
11	5, 6, 7, 8, 9, 10	cdleme0ex1N 33759	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )
12	1, 2, 3, 4, 11	syl121anc 1269	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )
13		simp11l 1116	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> K e. HL )
14		hlcvl 32895	. . . . . . . 8 |- ( K e. HL -> K e. CvLat )
15	13, 14	syl 17	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> K e. CvLat )
16		simp2ll 1072	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
17	16	3ad2ant1 1026	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> P e. A )
18	3	3ad2ant1 1026	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> Q e. A )
19		simp2 1006	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u e. A )
20		simp13 1037	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> P =/= Q )
21	8, 5, 6	cvlsupr2 32879	. . . . . . 7 |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ u e. A ) /\ P =/= Q ) -> ( ( P .\/ u ) = ( Q .\/ u ) <-> ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) ) )
22	15, 17, 18, 19, 20, 21	syl131anc 1277	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( ( P .\/ u ) = ( Q .\/ u ) <-> ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) ) )
23		simp3 1007	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u .<_ W )
24		simp2lr 1073	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
25	24	3ad2ant1 1026	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> -. P .<_ W )
26		nbrne2 4442	. . . . . . . . . 10 |- ( ( u .<_ W /\ -. P .<_ W ) -> u =/= P )
27	23, 25, 26	syl2anc 665	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u =/= P )
28		simp2rr 1075	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
29	28	3ad2ant1 1026	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> -. Q .<_ W )
30		nbrne2 4442	. . . . . . . . . 10 |- ( ( u .<_ W /\ -. Q .<_ W ) -> u =/= Q )
31	23, 29, 30	syl2anc 665	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u =/= Q )
32	27, 31	jca 534	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( u =/= P /\ u =/= Q ) )
33	32	biantrurd 510	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( u .<_ ( P .\/ Q ) <-> ( ( u =/= P /\ u =/= Q ) /\ u .<_ ( P .\/ Q ) ) ) )
34		df-3an 984	. . . . . . 7 |- ( ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) <-> ( ( u =/= P /\ u =/= Q ) /\ u .<_ ( P .\/ Q ) ) )
35	33, 34	syl6rbbr 267	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) <-> u .<_ ( P .\/ Q ) ) )
36	22, 35	bitrd 256	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( ( P .\/ u ) = ( Q .\/ u ) <-> u .<_ ( P .\/ Q ) ) )
37	36	3expia 1207	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A ) -> ( u .<_ W -> ( ( P .\/ u ) = ( Q .\/ u ) <-> u .<_ ( P .\/ Q ) ) ) )
38	37	pm5.32rd 644	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A ) -> ( ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) <-> ( u .<_ ( P .\/ Q ) /\ u .<_ W ) ) )
39	38	rexbidva 2933	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( E. u e. A ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) <-> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) ) )
40	12, 39	mpbird 235	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. u e. A ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2614   E.wrex 2772   class class class wbr 4423   ` cfv 5601  (class class class)co 6306   lecple 15197   joincjn 16189   meetcmee 16190   Atomscatm 32799   CvLatclc 32801   HLchlt 32886   LHypclh 33519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32712  df-ol 32714  df-oml 32715  df-covers 32802  df-ats 32803  df-atl 32834  df-cvlat 32858  df-hlat 32887  df-lhyp 33523

This theorem is referenced by: (None)