Theorem cdlemednuN 35587

Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. D represents s₂. (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
cdlemednuN	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D =/= U )

Proof of Theorem cdlemednuN

Step	Hyp	Ref	Expression
1		cdlemeda.l	. . 3 |- .<_ = ( le ` K )
2		cdlemeda.j	. . 3 |- .\/ = ( join ` K )
3		cdlemeda.m	. . 3 |- ./\ = ( meet ` K )
4		cdlemeda.a	. . 3 |- A = ( Atoms ` K )
5		cdlemeda.h	. . 3 |- H = ( LHyp ` K )
6		cdlemeda.d	. . 3 |- D = ( ( R .\/ S ) ./\ W )
7	1, 2, 3, 4, 5, 6	cdlemednpq 35586	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. D .<_ ( P .\/ Q ) )
8		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
9		simp1r 1086	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
10		simp21 1094	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
11		simp22 1095	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
12		cdlemednu.u	. . . . . 6 |- U = ( ( P .\/ Q ) ./\ W )
13	1, 2, 3, 4, 5, 12	cdlemeulpq 35507	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )
14	8, 9, 10, 11, 13	syl22anc 1327	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U .<_ ( P .\/ Q ) )
15		breq1 4656	. . . 4 |- ( D = U -> ( D .<_ ( P .\/ Q ) <-> U .<_ ( P .\/ Q ) ) )
16	14, 15	syl5ibrcom 237	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( D = U -> D .<_ ( P .\/ Q ) ) )
17	16	necon3bd 2808	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( -. D .<_ ( P .\/ Q ) -> D =/= U ) )
18	7, 17	mpd 15	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D =/= U )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   meetcmee 16945   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-iin 4523  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-mpt2 6655  df-1st 7168  df-2nd 7169  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-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274

This theorem is referenced by: (None)