Theorem cdleme42i 35771

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)

Hypotheses

Ref	Expression
cdleme41.b	|- B = ( Base ` K )
cdleme41.l	|- .<_ = ( le ` K )
cdleme41.j	|- .\/ = ( join ` K )
cdleme41.m	|- ./\ = ( meet ` K )
cdleme41.a	|- A = ( Atoms ` K )
cdleme41.h	|- H = ( LHyp ` K )
cdleme41.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme41.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme41.e	|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme41.g	|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme41.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme41.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme41.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme41.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdleme34e.v	|- V = ( ( R .\/ S ) ./\ W )

Assertion

Ref	Expression
cdleme42i	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( ( F ` R ) .\/ ( F ` S ) ) .<_ ( ( F ` R ) .\/ V ) )

Distinct variable groups:   A, s   .\/ , s   .<_ , s   ./\ , s   P, s   Q, s   R, s   S, s   U, s   W, s   y, t, A, s   B, s, t, y   y, D   y, G   E, s, y   H, s, t, y   t, .\/ , y   K, s, t, y   t, .<_ , y   t, ./\ , y   t, P, y   t, Q, y   t, R, y   t, S, y   t, U, y   t, W, y   x, z, A   x, B, z   z, E, s   z, H   x, .\/ , z   z, K   x, .<_ , z   x, ./\ , z   x, N, z   x, P, z   x, Q, z   x, R, z   x, S, z   x, U, z   x, W, z, s, t, y   V, s, t, x, z

Allowed substitution hints:   D( x, z, t, s)   E( x, t)   F( x, y, z, t, s)   G( x, z, t, s)   H( x)   I( x, y, z, t, s)   K( x)   N( y, t, s)   O( x, y, z, t, s)   V( y)

Proof of Theorem cdleme42i

Step	Hyp	Ref	Expression
1		simp11l 1172	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> K e. HL )
2		hllat 34650	. . . 4 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 17	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> K e. Lat )
4		simp1 1061	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
5		simp2ll 1128	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> R e. A )
6		cdleme41.b	. . . . . 6 |- B = ( Base ` K )
7		cdleme41.a	. . . . . 6 |- A = ( Atoms ` K )
8	6, 7	atbase 34576	. . . . 5 |- ( R e. A -> R e. B )
9	5, 8	syl 17	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> R e. B )
10		cdleme41.l	. . . . 5 |- .<_ = ( le ` K )
11		cdleme41.j	. . . . 5 |- .\/ = ( join ` K )
12		cdleme41.m	. . . . 5 |- ./\ = ( meet ` K )
13		cdleme41.h	. . . . 5 |- H = ( LHyp ` K )
14		cdleme41.u	. . . . 5 |- U = ( ( P .\/ Q ) ./\ W )
15		cdleme41.d	. . . . 5 |- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
16		cdleme41.e	. . . . 5 |- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
17		cdleme41.g	. . . . 5 |- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
18		cdleme41.i	. . . . 5 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
19		cdleme41.n	. . . . 5 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
20		cdleme41.o	. . . . 5 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
21		cdleme41.f	. . . . 5 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
22	6, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21	cdleme32fvcl 35728	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ R e. B ) -> ( F ` R ) e. B )
23	4, 9, 22	syl2anc 693	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` R ) e. B )
24		cdleme34e.v	. . . 4 |- V = ( ( R .\/ S ) ./\ W )
25		simp2rl 1130	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> S e. A )
26	6, 11, 7	hlatjcl 34653	. . . . . 6 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. B )
27	1, 5, 25, 26	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( R .\/ S ) e. B )
28		simp11r 1173	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> W e. H )
29	6, 13	lhpbase 35284	. . . . . 6 |- ( W e. H -> W e. B )
30	28, 29	syl 17	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> W e. B )
31	6, 12	latmcl 17052	. . . . 5 |- ( ( K e. Lat /\ ( R .\/ S ) e. B /\ W e. B ) -> ( ( R .\/ S ) ./\ W ) e. B )
32	3, 27, 30, 31	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( ( R .\/ S ) ./\ W ) e. B )
33	24, 32	syl5eqel 2705	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> V e. B )
34	6, 10, 11	latlej1 17060	. . 3 |- ( ( K e. Lat /\ ( F ` R ) e. B /\ V e. B ) -> ( F ` R ) .<_ ( ( F ` R ) .\/ V ) )
35	3, 23, 33, 34	syl3anc 1326	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` R ) .<_ ( ( F ` R ) .\/ V ) )
36	6, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24	cdleme42h 35770	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` S ) .<_ ( ( F ` R ) .\/ V ) )
37	6, 7	atbase 34576	. . . . 5 |- ( S e. A -> S e. B )
38	25, 37	syl 17	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> S e. B )
39	6, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21	cdleme32fvcl 35728	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ S e. B ) -> ( F ` S ) e. B )
40	4, 38, 39	syl2anc 693	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` S ) e. B )
41	6, 11	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ ( F ` R ) e. B /\ V e. B ) -> ( ( F ` R ) .\/ V ) e. B )
42	3, 23, 33, 41	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( ( F ` R ) .\/ V ) e. B )
43	6, 10, 11	latjle12 17062	. . 3 |- ( ( K e. Lat /\ ( ( F ` R ) e. B /\ ( F ` S ) e. B /\ ( ( F ` R ) .\/ V ) e. B ) ) -> ( ( ( F ` R ) .<_ ( ( F ` R ) .\/ V ) /\ ( F ` S ) .<_ ( ( F ` R ) .\/ V ) ) <-> ( ( F ` R ) .\/ ( F ` S ) ) .<_ ( ( F ` R ) .\/ V ) ) )
44	3, 23, 40, 42, 43	syl13anc 1328	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( ( ( F ` R ) .<_ ( ( F ` R ) .\/ V ) /\ ( F ` S ) .<_ ( ( F ` R ) .\/ V ) ) <-> ( ( F ` R ) .\/ ( F ` S ) ) .<_ ( ( F ` R ) .\/ V ) ) )
45	35, 36, 44	mpbi2and 956	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( ( F ` R ) .\/ ( F ` S ) ) .<_ ( ( F ` R ) .\/ V ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-undef 7399  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274

This theorem is referenced by:  cdleme42k  35772