Theorem cdleme0ex2N 34191

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 988	. . 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 1014	. . 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 1057	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp3 990	. . 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 34190	. . 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 1224	. 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 1099	. . . . . . . 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 33327	. . . . . . . 8 |- ( K e. HL -> K e. CvLat )
15	13, 14	syl 16	. . . . . . 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 1055	. . . . . . . 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 1009	. . . . . . 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 1009	. . . . . . 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 989	. . . . . . 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 1020	. . . . . . 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 33311	. . . . . . 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 1232	. . . . . 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 990	. . . . . . . . . 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 1056	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
25	24	3ad2ant1 1009	. . . . . . . . . 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 4417	. . . . . . . . . 10 |- ( ( u .<_ W /\ -. P .<_ W ) -> u =/= P )
27	23, 25, 26	syl2anc 661	. . . . . . . . 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 1058	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
29	28	3ad2ant1 1009	. . . . . . . . . 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 4417	. . . . . . . . . 10 |- ( ( u .<_ W /\ -. Q .<_ W ) -> u =/= Q )
31	23, 29, 30	syl2anc 661	. . . . . . . . 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 532	. . . . . . . 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 508	. . . . . . 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 967	. . . . . . 7 |- ( ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) <-> ( ( u =/= P /\ u =/= Q ) /\ u .<_ ( P .\/ Q ) ) )
35	33, 34	syl6rbbr 264	. . . . . 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 253	. . . . 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 1190	. . . 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 640	. . 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 2861	. 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 232	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 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2647   E.wrex 2799   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   lecple 14363   joincjn 15232   meetcmee 15233   Atomscatm 33231   CvLatclc 33233   HLchlt 33318   LHypclh 33951

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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481

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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-lhyp 33955

This theorem is referenced by: (None)