Theorem cdleme19c 35593

Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. D, F represent s₂, f(s). We prove f(s) =/= s₂. (Contributed by NM, 13-Nov-2012.)

Hypotheses

Ref	Expression
cdleme19.l	|- .<_ = ( le ` K )
cdleme19.j	|- .\/ = ( join ` K )
cdleme19.m	|- ./\ = ( meet ` K )
cdleme19.a	|- A = ( Atoms ` K )
cdleme19.h	|- H = ( LHyp ` K )
cdleme19.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g	|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d	|- D = ( ( R .\/ S ) ./\ W )
cdleme19.y	|- Y = ( ( R .\/ T ) ./\ W )

Assertion

Ref	Expression
cdleme19c	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= D )

Proof of Theorem cdleme19c

Step	Hyp	Ref	Expression
1		cdleme19.d	. . . 4 |- D = ( ( R .\/ S ) ./\ W )
2		simp1l 1085	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
3		hllat 34650	. . . . . 6 |- ( K e. HL -> K e. Lat )
4	2, 3	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
5		simp31 1097	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
6		simp23l 1182	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
7		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
8		cdleme19.j	. . . . . . 7 |- .\/ = ( join ` K )
9		cdleme19.a	. . . . . . 7 |- A = ( Atoms ` K )
10	7, 8, 9	hlatjcl 34653	. . . . . 6 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
11	2, 5, 6, 10	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
12		simp1r 1086	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
13		cdleme19.h	. . . . . . 7 |- H = ( LHyp ` K )
14	7, 13	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. ( Base ` K ) )
15	12, 14	syl 17	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
16		cdleme19.l	. . . . . 6 |- .<_ = ( le ` K )
17		cdleme19.m	. . . . . 6 |- ./\ = ( meet ` K )
18	7, 16, 17	latmle2 17077	. . . . 5 |- ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
19	4, 11, 15, 18	syl3anc 1326	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
20	1, 19	syl5eqbr 4688	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> D .<_ W )
21		simp32 1098	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
22		simp33 1099	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
23	21, 22	jca 554	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) )
24		cdleme19.u	. . . . 5 |- U = ( ( P .\/ Q ) ./\ W )
25		cdleme19.f	. . . . 5 |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
26	16, 8, 17, 9, 13, 24, 25	cdleme3 35524	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W )
27	23, 26	syld3an3 1371	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W )
28		nbrne2 4673	. . 3 |- ( ( D .<_ W /\ -. F .<_ W ) -> D =/= F )
29	20, 27, 28	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> D =/= F )
30	29	necomd 2849	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= D )

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   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-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-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274

This theorem is referenced by:  cdleme19d  35594  cdleme20l1  35608