Theorem cdleme0ex1N 35510

Description: Part of proof of Lemma E in [Crawley] p. 113. (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
cdleme0ex1N	|- ( ( ( 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 ) )

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

Allowed substitution hints:   H( u)   K( u)   ./\ ( u)

Proof of Theorem cdleme0ex1N

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
2		simp2l 1087	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
3		simp2r 1088	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> Q e. A )
4		simp3 1063	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ 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	lhpat2 35331	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
12	1, 2, 3, 4, 11	syl112anc 1330	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U e. A )
13		simp2ll 1128	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> P e. A )
14	5, 6, 7, 8, 9, 10	cdlemeulpq 35507	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )
15	1, 13, 3, 14	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U .<_ ( P .\/ Q ) )
16		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> K e. HL )
17		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
18	16, 17	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> K e. Lat )
19		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
20	19, 6, 8	hlatjcl 34653	. . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
21	16, 13, 3, 20	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) )
22		simp1r 1086	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> W e. H )
23	19, 9	lhpbase 35284	. . . . 5 |- ( W e. H -> W e. ( Base ` K ) )
24	22, 23	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> W e. ( Base ` K ) )
25	19, 5, 7	latmle2 17077	. . . 4 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
26	18, 21, 24, 25	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
27	10, 26	syl5eqbr 4688	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U .<_ W )
28		breq1 4656	. . . 4 |- ( u = U -> ( u .<_ ( P .\/ Q ) <-> U .<_ ( P .\/ Q ) ) )
29		breq1 4656	. . . 4 |- ( u = U -> ( u .<_ W <-> U .<_ W ) )
30	28, 29	anbi12d 747	. . 3 |- ( u = U -> ( ( u .<_ ( P .\/ Q ) /\ u .<_ W ) <-> ( U .<_ ( P .\/ Q ) /\ U .<_ W ) ) )
31	30	rspcev 3309	. 2 |- ( ( U e. A /\ ( U .<_ ( P .\/ Q ) /\ U .<_ W ) ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )
32	12, 15, 27, 31	syl12anc 1324	1 |- ( ( ( 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 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   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:  cdleme0ex2N  35511