Theorem cdleme0ex2N 35020

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 996	. . 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 1022	. . 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 1065	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4		simp3 998	. . 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 35019	. . 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 1233	. 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 1107	. . . . . . . 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 34156	. . . . . . . 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 1063	. . . . . . . 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 1017	. . . . . . 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 1017	. . . . . . 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 997	. . . . . . 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 1028	. . . . . . 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 34140	. . . . . . 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 1241	. . . . . 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 998	. . . . . . . . . 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 1064	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
25	24	3ad2ant1 1017	. . . . . . . . . 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 4465	. . . . . . . . . 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 1066	. . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
29	28	3ad2ant1 1017	. . . . . . . . . 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 4465	. . . . . . . . . 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 975	. . . . . . 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 1198	. . . 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 2970	. 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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14558   joincjn 15427   meetcmee 15428   Atomscatm 34060   CvLatclc 34062   HLchlt 34147   LHypclh 34780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784

This theorem is referenced by: (None)